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LIE ALCEBRASNathan Jacobson

Lie group theory, developed by M. Sophus Lie in the lgth centur.|, ranksamong the more important developments in modern mathematics. Liealgebras comprise a significant part of Lie group theory and are beingactively studied today. This book, by Professor Nathan Jacobson of Yale,is the definitive treatment of the subject and can be used as a textbookfor graduate courses.

Chapter I introduces basic concepts that are necessary for an understandingof structure theory, while the following three chapters present the theoryitself: solvable and nilpotent Lie algebras, Cartan's criterion and its con-sequences, and split semi-simple Lie algebras. Chapter 5, on universalenveloping algebras, provides the abstract concepts underlying represerrta-tion theory. Then the basic results on representation theory are given inthree succeeding chapters: the theorem of Ado-Iwasalva, classification ofirreducible modules, and characters of the irreducible modules. In Chapter9 the automorphisms of semi-simple Lie algebras over an algebraically closedfield of characteristic zero are determined. These results are applied inChapter l0 to the problems of sorting out the simple Lie algebras over anarbitrary field. The reader, to fully benefit from this tenth chapter, shouldhave some knowledge about the notions of Galois theory and some of theresults of the Wedderburn structure theory of associative algebras.

Nathan Jacobson, presently Henry Ford II Professor of Mathematicsat Yale University, is a well-known authority in the field of abstractalgebra. His book, Lie Algebras, is a classic handbook both for researchersand students. Though it presupposes a knowledge of linear algebra, it isnot overly theoretical and can be readily used for self-study.

Unabridged, corrected (1979) republication of the original (1962)Bibliography. Index. ix a 331pp. \s/B x 8/4. Paperbound.

A DOVER EDITION DESIGNED FOR YEARS OF USE!We have made every effort to make this the best book possible. Our paperis opaque, rvith minimal shorv-through; it rvil l not discolor or beconre britt lewith age. Pages are servn in signatures, in the method traditionally used forthe best books, and will not drop out, as often happens u'ith paperbacks heldtogether with glue. Books open flat for easy reference. The binding will notcrack or split. This is a permanent book.

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LIEALGEERASby

I.TATHAN JACOBSONHenry Ford II Professor of MathematicsYale University, New Haven, Connecticut

Dover Publications, Inc.New'\brk

Copyright O 1962 by NathanJacobson.All rights reserved under Pan American and Inter-

national Copyright Conventions.

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Published in the United Kingdom by Constable aildCompany, Ltd., l0 Orange Street, London WCZH 7EG.

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PREFACE

The present book is based on lectures which the author has givenat Yale during the past ten years, especially those given duringthe academic year 1959-1960. It is primarily a textbook to bestudied by students on their own or to be used for a course onLie algebras. Besides the usual general knowledge of algebraicconcepts, a good acquaintance with linear algebra (linear trans-formations, bilinear forms, tensor products) is presupposed. More-over, this is about all the equipment needed for an understandingof the first nine chapters. For the tenth chapter, we require alsoa knowledge of the notions of Galois theory and some of theresults of the Wedderburn structure theory of associative algebras.

The subject of Lie algebras has much to recommend it as a.subject for study immediately following courses on general abstractalgebra and linear algebra, both because of the beauty of itsresults and its structure, and because of its many contacts withother branches of mathematics (group theory, differential geometry,differential equations, topology). In this exposition we'have triedto avoid rnaking the treatment too abstract and have consistentlyfollowed the point of view of treating the theory as a branch oflinear algebra. The general abstract notions occur in two groups:the first, adequate for the structure theory, in Chapter I; and thesecond, adequate for representation theory, in Chapter V. ChaptersI through IV give the structure theory, which culminates in theclassification of the so-called "split simple Lie algebras." Thebasic results on representation theory are given in Chapters VIthrough VIII. In Chapter IX the automorphisms of semi-simpleLie algebras over an algebraically closed field of characteristiczero are determined. These results are applied in Chapter X tothe problem of sorting out the simple Lie algebras over an arbitraryfield.

No attempt has been made to indicate the historical develop-ment of the subject or to give credit for individual contributionsto it. In this respect we have confined ourselves to brief indica-tions here and there of the names of those responsible for themain ideas. It is well to record here the author's own indbbted-ness to one of the great creators of the theory, Professor HermannWeyl, whose lectures at the Institute for Advanced Study in 1933-

l v l

v l PREFACE

1934 were truly inspiring and led to the author's research in thisfield. It should be noted also that in these lectureB ProfessorWeyl, although primarily concerned with the Lie theory of con-tinuous groups, set the subject of Lie algebras on its own indepen-dent course by introducing for the first time the term "Lie algebra"as a substitute for "infinitesimal group," which had been used ex-clusively until then.

A fairly extensive bibliography is included; howevef, this is byno means complete. The primary aim in compiling the bibli-ography has been to indicate the avenues for further jstudy of thetopics of the book and those which are immediately rrelated to it.

I am very much indebted to my colleague George $eligman forcarefully reading the various versions of the maguscript andoffering many suggestions for improving the expo$ition. Drs.Paul Cohn and Ancel Mewborn have also made valuable comments,and all three have assisted with the proofreading. I take thisopportunity to offer all three my sincere thanks.

N.lrs.$.r JlconsouMay 28, 1961New Hauen, Connecticut

CONTENTS

Cnlrrpn I

Basic Concepts

1. Definition and construction of Lie and associativealgebras 2

2. Algebras of linear transformations. Derivations 5

3. Inner derivations of associative and Lie algebras . 9

4. Determination of the Lie algebras of low dimensionalities 11

5. Representations and modules . 14

6. Some basic module oPerations t9

7. Ideals, solvability, nilpotency . 23

8. Extension of the base field . 26

CulprnR II

Solvable and Nilpotent Lie Algebras

1. Weakly closed subsets of an associative algebra2. Nil weakly closed sets3. Engel's theorem4. Primary components. Weight spaces5. Lie algebras with serni-simple enveloping associative

algebras6. Lie's theorems7. Applications to abstract Lie algebras. Some counter

examples .

31333637

4348

51

Cn.l.rtsn III

Cartan's Criterion and Its Consequences

1. Cartan subalgebras . 572. Products of weight spaces 613. An example 644. Cartan's criteria 66

5. Structure of semi-simple algebras 706. Derivations. 73

7. Complete reducibility of the representations ofsemi-sinrple algebra 75

l v i i l

Viii CONTENTS

8. Representations of the split three-dimensional simpleLie algebra.

9. The theorems of Levi and Malcev-Harish-Chandra .10. Cohomology groups of a Lie algebra .11. More on complete reducibility . . .

Cn^l,rtpn IV

Split Semi-simple Lie Algebras

1. Properties of roots and root spaces. .i . . 10g2. A basic theorem on representations and its

consequences for the structure theory . .; . . Ilz3. Simple systems of roots . . . 1194. The isomorphism theorem. Simplicity . . . .IZT5. The determination of the Cartan matrices . LZg6. Construction of the algebras . 13S7. Compact forms . . 146

83869396

Cn.c.prpn V

Universal Envetoping Algebras

Definition and basic properties . .The Poincar6-Birkhoff-Witt theoremFiltration and graded algebraFree Lie algebrasThe Campbell-Hausdorff formula .Cohomology of Lie algebras. The standard complexRestricted Lie algebras of characteristic pAbelian restricted Lie algebras .

151156163t67170174185r92

CHlrrnn VI

The Theorem of Ado-Iwasawa

1. Preliminary results .2. The characteristic zero case . Z}L3 . T h e c h a r a c t e r i s t i c p c a s e . . . . Z } J

CHAPTER. VII

Classification of Irreducible Modules

1. Definition of certain Lie algebras . ,f .207

CONTENTS ix

2. On certain cyctic modules for E .2123. Finite-dimensional irreducible modules .zLs4. Existence theorem and isomorphism theorem

for semi-simple Lie algebras . .2205. Existence of E, and Ea . .2236. Basic irreducible modules .225

Cn.lprpn VIII

Characterg of the lrreducible Modules

1. Some properties of the Weyl group2. Freudenthal's formula3. Weyl's character formula4. Some examples .5. Applications and further results

240243249257259

CHlrrnn IX

Automorphisms

1. Lemmas from algebraic geometry .zffi2. Conjugacy of Cartan subalgebras . .2713. Non-isomorphism of the split simple Lie algebras .2744. Automorphisms of semi-simple Lie algebras over

an algebraically closed field .2755. Explicit determirtation of the automorphisms

for the simple Lie algebras .281

Cru,prpn X

Simple Lie Algebras oyer an Arbitrary Field

1. Multiplication algebra and centroid ofa non-associative algebra . . 290

2. Isomorphism of extension algebras . .2953. Simple Lie algebras of types A-D . 2984. Conditions for isomorphism .3035. Completeness theorems . . 3086. A closer look at the isomorphism conditions . 3117. Central simple real Lie algebras . 313

B i b l i o g r a p h y . . . 3 1 9

Index .329

LIEALGEBRAS

CHAPTER I

Basic Concepts

The theory of Lie algebras is an outgrowth of the Lie theory of

continuous groups. The main result of the latter is the reduction of

"local" problems concerning Lie groups to corresponding problems

on Lie algebras, thus to problems in linear algebra. One associates

with every Lie group a Lie algebra over the reals or complexes

and one establishes a correspondence between the analytic subgroups

of the Lie group and the subalgebras of its Lie algebra, in which

invariant subgroups correspond to ideals, abelian subgroups to abelian

subalgebras, etc. Isomorphism of the Lie algebras is equivalent to

local isomorphism of the corresponding Lie groups. We shall not

discuss these matters in detail since excellent modern accounts of

the Lie theory are available. The reader may consult one of the

following books: Chevalley's Theorl of Li,e Groufs, Cohn's Lie

Groups, Pontrjagin's Topological Groups.More recently, two other types crf group theory have been aided

by the introduction of appropriate Lie algebras in their study. The

first of these is the theory of free groups which can be studied by

means of free Lie algebras using a method which was originated

by Magnus. Although the connection here is not so close as in

the Lie theory, significant results on free groups and other types

of discrete groups have been obtained using Lie algebras. Particu'

larly noteworthy are the results on the so-called restricted Burnsideproblem: Is there a bound for the orders of the finite groups with

a fixed number r of generators and satisfying the relation tr* : I,

m a fixed, positive integer? It is worth mentioning that Lie algebrasof prime characteristic play an important role in these applicationsto discrete group theory. Again we shall not enter into the details

but refer the interested reader to two articles which give a good

account of this method in group theory. These are: Lazard tzland Higman [1].

The type of correspondence between subgroups of a Lie group

and subalgebras of its Lie algebra which obtains in the Lie theory

t l l

2 YIE ALGEBRAS

has a counterpart in chevalley's theory of linear algebraic groups.Roughly speaking, a linear algebraic group is a subdroup of thegroup of non-singular n x n matrices which is specified,by a set ofpolynomial equations in the entries of the matrices. An exampleis the orthogonal group which is defined by the set of equationsX i r l ? i : l , \ i d i i d i t t : A , i + h , j , k - 1 , . . . t l t t o n t h e e h t r i e s e ; ; o fthe matrix (a;1). with each linear algebraic group chevalley hasdefined a corresponding Lie algebra (see Chevalley t2l) hrhictr givesuseful information on the group and is decisive in the theory oflinear algebraic groups of characteristic zero. I

In view of all this group theoretic background it is inot surpris-ing that the basic concepts in the theory of Lie algdbras have agroup-theoretic flavor. This should be kept in mindr throughoutthe. study of Lie algebras and particularly in this chdpter, whichgives the foundations that are adequate for the mafin structuretheory to be developed in Chapters II to IV. euestioris on founda-tions are taken up again in Chapter V. These concern some con-cepts that are necessary for the representation theoryl which willbe treated in Chapters VI and VII.

1. Defrnition and construction of Lieand aasociatiae algebras

We recall the definition of a non-associative algebra (:not neces-sarily associative algebra) ?I over a field, a. This is jirst a vectorspace ll over o in which a bilinear composition is defined. Thusfor every pair (r, !), r,y in ?I, we can associate a pfoduct rJ elland this satisfies the bilinearity conditions

( x r + x i l : h ! * r z ! x(y, + !r): rlt * *!z

x(ay) , a€O .a(xy): (ax)y -

A similar definition can be given for a non-associative algebra overa commutative ring t0 having an identity element (unit) 1. This isa left O-module with a product ry € ?I satisfying (1) iand (2). Weshall be interested mainty in the case of algebras over fields and,in fact, in such algebras which are finite-dimensiorlal as vectorspaces. For such an algebra we have a basis (er,er,.)..,en) and wecan write €r€i:}I=ffure* where the 7's are in A. The zt faiy arte,called the constants of multiplication of the algebra (relative to the

( 1 )( 2 )

chosen basis). They give the values of every prduet e;e1,

I. BASIC CONCEPTS

1,2, . . , , n. Moreover, these products determine every product in ?I.Thus let r and y be any two elements of tr and write r: ZEi€;,! :>Tiei, Errt i€O. Then, bV (1) and (2),

xy - ( T,{'erX l,n ie i) : l,(E ie)Qt e i). t r , t

- \E;@{nP)): \E;nr(e&t) ,

and this is determined by the e&t.This reasoning indicates a universal construction for finite-dimen-

sional non-associative algebras. We begin with any vector spacelI and a basis (er) in t. For every pair (l,l) we define in any waywe please *gt ds an element of lt. Then if x : LTEee; y - Zlv$twe define

try: f Erro@oun)i , i : L

( 3 )

( 5 )

One checks immediately that this is bilinear in the sense that (1)

and (2) are valid. The choice of e1e1 is equivalent to the choiceof the elements Tux in @ such that e&t:}Taixan.

The notion of a non-associative algebra is too general to lead tointeresting structurai results. In order to obtain such results onemust impose some further conditions on the multiplication. Themost important ones-and the ones which, will concern us here, arethe associative laws and the Lie conditions.

Dprrurrrorv 1. A non-associative algebra lI is said to be associatiueif its multiplication satisfies the associative law

( 4 ) x(vz) .

A non-associative algebra lI is said to be a Li,e algebra if itsmu'ltiplication satisfies the Lie conditions

x , : 0 , ( x y ) z + ( y z ) x + ( z r ) y - 0 .

The second of these is called the Jacobi identity.Since these types of non-associative'algebras are defined by

identities, it is clear that subalgebras and homomorphic imagesare of the same type, i.e., associative or Lie. If lI is a Lie algebraandx,ye l l , then 0 - . ( r * y ) ' : x '+ xy + l t c * ! ' : x l +yr so tha t

( 6 ) - y x

4 LIE ALGEBRAS

holds in any Lie algebra. Conversely, if this condition holds then2tr' :0, so that, if the characteristic is not two, then rz't: 0. Hencefor algebras of characteristic * 2 the condition (6) canj be used forthe first of (5) in the definition of a Lie algebra.

Pnoroslrlox 1. A non-associatiue algebra W, with basis (er,er,. ..,eo)ouer O is associ,atiae if and only if (ep)er-er(e$) for i,j,k:1,2, . -.,n. If eaei:\,Tu,e, these conditions are equiaalent to

(7 ) \TurT,*r : lTurT1,, , i , j , k , s : 1, 2, " . , ? .

The algebra 2l is Lie if and only if e',t :0, e&i: - ei01,,

(ete)er * (eie)er * @p)e1 : g

. . ., n. These conditions are equiaalentt to

T u r : 0 , T u x : - T i t : r ,

Z(TotTrr, * T i*,Tru * T*rrTr.rr) : 0 .

f o r i , j , k : 1 , 2 ,

( 8 )

Proof: If ?I is associative, then (ep)er,: e;(epx). Conversely,assume these conditions hold for the et. lf x: ),E;ev, ! :Zqpt,z : 2($t, then (xy)z : 2E iv{n(e&)er and x(yz) : 2iniCpr(ep*).Hence (ry)z - x(yz) and lI is associative. If €;€1 : ET;t&,, then(ep)e*: Xr, eT;irT*cqc and et(ep*) : X", tTirtTirr€t. Hence the linearindependence of the ei implies that the conditions (e;b)er,: e;(eq*)are equivalent to (7). The proof in the Lie case is slmilar to theforegoing and will be omitted.

In actual practice the general procedure we have indicated is notoften used in constructing examples of associative and of Lie algebrasexcept for algebras of low dimensionalities. We shall employ this indetermining the Lie algebras of one, two, and three dimpnsions in g4.There are a couple of simplifying remarks that can be made in theLie case. First, we note that if e? :0 and e;ei : -' ,ru, in an al-gebra, then the validity of (e;e)en * (ep*)er + @p)et: 0 for a parti-cular triple i, j, k implies (eie;)erc * (e;e)et * (eret)e;: 0, Since cyclicpermutations of i, j, k are clearly allowed it follows that the Jacobiidentity for (e;,e;')4, is valid for it, j',k', a permutation of. i, j,h.Next let i : j . Then e?e* + (e;e*)e; * (ene;)e;: 0 * (e;et)er - (e;er)e; - O.Hence e?: 0, ei01 : - ri€i or, what is the same thing, x' : 0 in ?Iimplies that the Jacobi identities are satisfied for e;, e;,, €i. In parti-cular, the Jacobi identities are consequences of x' :0 if dim A < 2

I. BASIC CONCEPTS 5

and if dim t[ :3, then the only identity we have to check is(ep)es * (eze)er * (eser)ez = 0.

2. Algebrat of linear tranaformations. Deriaations

Actually, it is unnecessary to sit down and construct examples

of associative and Lie algebras by the method of bases and multi-plication tables since these algebras occur "in nature." The prime

examples of associative algebras are obtained as follows. Let lll

be a vector space over a field O and let @ denote the set of linear trans'

formations of Dt into itself. We recall that if. A, BeE and ae0,

then A + B, aA and AB are defined by x(A + B) : xA * xB,

x(aA): s.(rA), x(AB): (rA)B for r in lll. Then it is well known

that € is a vector space relative to * and the scalar multiplication

and that multiptication is associative and satisfies (1) and (2). Hence

6 is an associative algebra. It is well known also that if tJt is

rz-dimensional, n, 1 @, then O is m'-dimensional over O. lf. (er,

€2,...,2^) is a basis for fi over o, then the linear transformations

.E; l such tha t e tE t i :e i ,0 rE; i :0 i f . r+ i , i , i -1 , " ' , / r t , fo rm a

basis for O over O. If AeE, then we can write 0;A - }ia;Pti -- l,- . . t fri, and (a) : (ar) is the matrix of. A relative to the ba'

sis (ec). The correspondence A-+ (a) is an isomorphism of 0 onto

the algebt? O^ of. m x m matrices with entries ait in O.

The atgebra @ is called the (associatiae) algebra of linear transform'

ations in llt over O. Any subalgebra l[ of 0, that is, a subspace

of O which is closed under multiplication, is called an algebra of

linear tr ansformations.If lt is an arbitrary non-associative algebra and a e lI, then the

mapping ar which sends any x into ra is a linear transformation.

It is well known and easy to check tl:pt (a * D)" : aB * bn, (aa)p:

aan "nd'

if U is associative, (ab)": anba. Hence if U is an as'

sociative algebra, the mapping a -> an is a homomorphism of ?I into

the algebra @ of linear transformations in the vector space ?I. If

lI has an identity (or unit) 1, then a -+ an is an isomorphism of lI

into @. Hence 2t is isomorphic to an algebra of linear transform'ations. If lI does not have an identity, w€ can adjoin one in a

simple way to get an algebra, ?I* with an identity such that dimlI* : dim ?I + 1 (cf. Jacobson l2l, vol. I, p. 84). Since lI* is

isomorphic to an algebra of linear transformations, the same is

true for lt. If U is finite-dimensional, the argument shows that

LIE ALGEBRAS

l[ is isomorphic to an algebra of lineardimensional vector space.

transformation$ in a finite-

Lie algebras arise from associative algebras inway. Let lI be an associative algebra. lf x,yell,the Li,e product or (additive) commutator of. x and y

( 9 ) txy l : xy - l x .

One checks immediately that

ln * rcz, lf : [x'yj * Ixryl,lr, !, * yrl : Ixyrl * [x!r] ,alxyl: lax, yf : lr, dyl .

Moreover,

l x r l : t c t - x ' : 0 ,

llxylzl + [yzlxl + llzxl yl: (xy - yr)z - z(xy - yr) + (yz - zy)r

- r(yz - zt) * kx - rz)y - tkx -',t"z) - 0 .

fhus the produ ct Ixyl satisfies all the conditions on tire product ina Lie algebra. The Lie algebra obtained in this way I is called, theLie algebra of the associative algebra lI. We shall defrote this Liealgebra as 2Is. In particular, we have the Lie algebrd @z obtainedfrom G. Any subalgebra I of CIz is called a Li,e algebra of lineartransformations. we shall see later that every Lie algebra isisomorphic to a subalgebra of a Lie algebra ?Ir, g[ associative. Inview of the result just proved on associative algebras this is equi-valent to showing that every Lie algebra is isomordric to a Liealgebra of linear transformations.

we shall consider now some important instances od subalgebrasof Lie algebras @r, @ the associative algebra of linehr transform-ations in a vector space llt over a field O.

orthogonal Lie algebra. Let !)t be equipped with a nbn-degeneratesymmetric bilinear form (x, y) and assume !)t finite-dimensional.Then any linear transformation A in !)l has an adjoint ..4* relativeto (x,y); that is, Ax is linear and satisfies: (rd, y) : (r, tA*). Themapping A -. A* is an anti-automorphism in the algebra G:(A+ B)*: A* + B*, (aA1*: aA*, (AB)*: B*A*. l l -et € denotethe set of Ae @ which are skew in the sense that A* * - A. Then

is a subspace of @ and if. A* - - A, B* : - B, fhen lABl* :

a ltrery simplethqn we defineAS

I. BASIC CONCEPTS 7

(AB - BA)* -- B*A* - A*B* : BA - AB: lBAl: -LABI. Hence

tABle 6 and 6 is a subalgebra of 0r.If @ is the field of real numbers, then the Lie algebta 6 is the

Lie algebra of the orthogonal group of llt relative to (r' y). This

is the group of linear transformations O in 9]t which are orthogonal

in the sense that (xO,yO):(x,!), r,! in !It. For this reason we

shall call the orthogonal Li.e algebra relative to (r, y).

Syruplectic Lie algebra. Here we suppose (r, y) is a non-degeneratealternate form: (r,x):0 and again dim l}t < oo. We recall thatthese conditions imply that dim tn - 2l is even. Again let ,4* be

the adjoint of ,4( e O) relative to (r, y). Then the set 6 of skew(A* : - A) linear transformations is a subalgebra of @2. This is

related to the symplectic group and so we shall call it the symPlectic

Lie algebra 6 of the alternate form (x, y).

Triangular linear transfarmations. Let 0c1]tr cl}tzc''' c!]l-*Dt

be a chain of subspaces of 9Jt such that dim IJtt: I and let ! bethe set of linear transformations T such that lltrT G TJtt. It is clearthat E is a subalgebra of the associative algebra @: hence E; is asubalgebra of @2.

'We can choose a basis (xr, rr, "', x*) for lll so

that (trr, trr, "', x;) is a basis for fii. Then if TeT, Tft;T S !]himplies that the matrix of. T relative to (xr, Nr, "', x*) is of theform

, : [ : :

: ' : ]Such a matrix is called triangular and correspondingly we shallcall any TeT a triangular linear transformation.

Deriaation algebras. Let ?I be an arbitrary non-associative algebra.A deriaation D in ?I is a linear mapping of lI into ?I satisfying

(11) (xy)D: (xD)y * x(yD) .

Let D(?I) denote the set of derivations in !I. lf Dr,DzeS([),then

(xy)(D, * D,) - (ry)D1 + @y)Dz - (rD,)t

* r(!D) * (xD)y * x(yD,): (r(Dr * Dr))y * x(y(h + Dr))

(10)

LIE ALGEBRAS

Hence Dt * Dz e D(U;. Similarly, one checks that a,DrQS(U) if

a,e0. We have

(xy)D,Dz: (@Db * x(yD,))Dz: (xD,Dz)y * (xD,1lyD') + (xDr)(tD,) + x(vD,Dr) '

Interchange of l, 2 and subtraction gives i

(ry)lD,Dzl: @lDD'l)Y + r(YlD'D,l) .

Hence lDrDzle E(lI) and so S(U) is a subalgebra of Gz, where € is

the algebra of linear transformations in the vector space lI. We

shalt call this the Lie algebra of deriaations or deriadtion algebra

of a.The Lie algebra lD(U) is the Lie algebra of the group of automor-

phisms of lI if lt is a finite-dimensional algebra over the field of

real numbers. We shall not prove any of our asseftions on the

relation between Lie groups and Lie algebras but refef the reader

to the literature on Lie groups for this. However, in rthe present

instance we shall indicate the link between the group;of automor'

phisms and the Lie algebra of derivations.Let D be a derivation. Then induction on n gives the Leibniz

rule:

(12)

If the

(12')

(13)

,"O#:E(0 we can divide bY n! and obtain

+.D)(hyD".).

(ry) D" : V^("r)rx

D i)( v D'- t)

characteristic of O is

If ll is finite-dimensional over the field of reals, then it is easy to

prove (cf. Jacobson [2], vol. II, p. 197) that the seriesl

r+ D*+++ + "'converges for every linear mapping D in l[, and the linear mapping

exp D defined by (13) is 1: 1. Also it is easy to see, using (12',),

that if D is a derivation, then G : exP D satisfies (ry)G : (rGXyG).

Hence G is an automorphism of ?I.

A connection between automorphisms and derivations can be

established in a purely algebraic setting which has lmportant ap'

plications. Here we suppose the base field of U iS arbitrary of

G:expD:L+D++*.. .+d+

W e w r i t e t h i s a s G : L * 2 , Z : D + ( D ' z p D + . . . + ( D n - ' l ( N - 1 ) ! )a n d n o t e t h a t Z N : 0 . H e n c e G : 1 * Z h a s t h e i n v e r s e L - Z +

Z' + ... -f Z"-' anld so G is 1: 1 of lI onto lI. We have

(rG)(yG) : (E+X,!,#)zry_z/ r_/

"D'11 yo"-' 11: ,FoEr

i t 11n-tyt))zJy_2

, D": E@y ,_n=o

' n l

.lv-l D"

n=o n !

-- (xv)G -Hence G is an automorphism of lI.

I. BASIC CONCEPTS

characteristic 0. Let D be a nilpotent derivation, sY, DN :0.

Consider the mapping

(14)

(by r?t)

(15)

3. fnner deriiations of aesoeiatiue and Lie algebras

If a is any element of a non-associative algebra lI, then a deter'mines two mappings az: x -+ ax and ani x -, xQ of lI into itself.These are called the left multiplication and right multiplication deter'mined by a. The defining conditions (1) and (2) for an algebrashow that at and aa are linear mappings and the mappings a + (zL,

a ---> an are linear of !t into the space 0 of linear transformationsin !t. Now let ?t be associative and set D" : (zn - av Hence Dois the linear mapping x--+ tut - &r. We have

rja - axy -- (ra - ax)t * x(ya - ay) ;

hence D" is a derivation in the associative algebra ?I. We shallcall this the i,nner deriaation determined by a.

Next let I be a Lie algebra. Because of the way Lie algebrasarise from associative ones it is customary to denote the product

in 8 by [ry] and we shall do this from now on. Also, it is usualto denote the right multiptication an (: - az since lxal - - lax)) byad a and to call this the adioint mapping determind by a. Wehave

IO LIE ALGEBRAS

tlxyial + llyalx) + llaxJ yl : 0,

llxy)a) - - llyalrl - lLar) yl - lrlyall + llxa)y1 ;

hence ad 6 v -,fxaf is a derivation. We call this also the i'nnerderiaation determined by ae8,.

A subset E of a non-associative algebra ?I is called i an ideal if(1) 8 is a subspace of the vector space ?I, (2) ab, ba€ E for any ain ll, 6 in E. Consider the set of elements of the forfn l,a;bt ai,

Dt in ?I. We denote this set as lI2 and we can check ,that this is

an ideal in lI. If ?I : 8 is a Lie algebra, then it is customary to

write 8' for 8' and to call this the deriued algebra (or:ideal) of 8.

If 8 is a Lie algebra, then the skew symmetry of the multiplica-

tion implies that a subspace E of I is an ideal if and only if labl(or [Da]) is in E for every aeSJ, DeE. It follows that the subsetG of elements c such that [ac] - 0 for all a e I is an lideal. Thisis called the center of 8. 8 is called abelian if I == 6, which is

equivalent to 8' : 0.

Pnorosrrrox 2. If lI is associatiue or Lie, then the innQr deriuations

forrn an ideal 3(ll) iz the deriaation algebra D(?l). l

Proof: In any non'associative algebra we have (a + b)r,: ar, * bt,(aa)": dazt @ * b)n : aR * bn, (aa)*-- dan. Hence if ' D"-- an - at;

then D,+ a: D, * Dt, Dro - aDo and the inner deriv]ations of anassociative or of a Lie algebra form a subspace of S(U). Let D bea derivation in lI. Then (ax)D - (aD)x * a(xD), ot (ax)D - a(xD):(aD)x. In operator form this reads (xar)D - (xD)az',: x(aD)r' or

lapl: atD - Da, - (aD)r Similarly, [a*Dl-@D)n and tconsequentlyalso [D"D] : D,n. These formulas show that if ll is dssociative or

Lie and .I is an inner derivation and D any derivatiori, then UDI is

an inner derivation. Hence S0I) is an ideal in O(U).

Erample. Let I be the algebra with basis (e, f) such that [efl -

e - - ffel and all other products of base elements are 0. Then

laa):O in Il and since dim I :2, 8 is a Lie algebra. The derived

algebra g' : Ae. If D is a derivation in any algebra ?I, then ?IzD g ?Iz.

Hence if D is a derivation in I then eD: 6e. Also a[(D/) has theproperty e(ad D/) : le,6fl -- 6e. Hence if E: D - adp/, then E is

a derivation and eE: A. Then e - lefl gives g - [e,fpl. It follows

that fE -- Te. Now ad(- re) satisfies e ad(- re) :0' /ad(- Te) :

lf, - yel-ylefl: re. Hence E-ad(- re) is inner and D: E*adDf

is inner. Thus every derivation of [l : Ae + Af is infier.

I. BASIC CONCEPTS 11

In group theory one defines a group to be complete if all of

its automorphisms are inner and its center is the identity. If H

is complete and invariant in G then ^Ff is a direct factor of G.

By analogy we shall call a Lie algebta complete if its derivations

are all inner and its center is 0.

Pnorostuon 3. If R is complete and an ideal in t., then 8': S @ E

where E is an ideal.Proof: We note first that if .f is an ideal in IJ, then the centralizer

E of s, that is, the set of elements D such that [&D] : 0 for all

&eS is an ideal. E is evidently a subspace and if DeE and

a € g, then lhlbal) - - [alkb]l - lbtakll : 0 - [b, k'7, ft' : fak) c Ri

hence tklball:0 for all k e 0 and [ba]e$. Hence E is an ideal.

Now let ,R be complete. If c e S n E, then c is in the center of fr

and so c : 0. Hence S n E - 0. Next let a € 8. Since S is an

ideal in 8, ad a maps F into itself and hence it induces a deriva'

tion D in S. This is inner and so we have a & e R such that

r D - - l r a l - - l x k l f . o r e v e r y r € S . T h e n b : a - k e E a n d a : b * k ,

beE, f t e n . Thus I - S *E - n@E as requ i red .

Eram\le. The algebra Ae + Af of the last example is complete.

4, Determination of the Lie algebrasof low dimcnsionalities

We shall now determine all the Lie algebras IJ such that dim

8 S 3 . l f (e r ,€2 , . . . ,e , ) i s a bas is fo raL iea lgebraS, then le ;e r ] :0

and le;ei\:. - leie;\. Hence in giving the multiplication table for

the basis, it suffices to give the products fe;ei f-or i < i. We shall

use these abbreviated multiplication tables in our discussion.I. dim I : 1. Then 8 :Oe, leel - g.

I I . d i m S - 2 .(a) 8' : 0, I is abelian.(b) 8' + 0. Since I -- Oe + Af,8' - a[ef] is one-dimensional. We

may choose e so that 8/ : Oe. Then lefl - ae * 0 and replacement

of. f by a-'f permits us to take lefl - e. Then I is the algebra of

the example of $ 3. This can now be characterized as the non'

abelian two-dimensional Lie algebra.III. dim 13 - 3.(a) 8' :0, I abelian.(b) dim 8/ : 1, 8' S S the center. If 8' -- oe we write 8,: oe *

Af + Ag. Then 8,' :0[fg7. Hence we may suppose lfgl - e. Thus

L2 LIE ALGEBRAS

8 has basis (e,f,g), with multiplication table

(16) l f s f - e , le f l -0 , Ieg l -0 .

We have only one Lie algebra satisfying our conditions. (If wehave (16), then the Jacobi condition is satisfied.)

(c) dim 13' : 1, !' E 0 the center. If 8' - O€, then, there is an/ such that lef) + 0. Then [ef] - Fe + 0 and we may supposelefj - e. Hence Oe t Af is the non-abelian two'dimensional algebran. SinceR=8' ,Sis an ideat and s inceSiscomplete, 8-SOE,E:Og. Hence IJ has basis (e,f,g) with multiplicatiorl table

(17) lefl : e, lesl - o, [fsl -- o .

(d) dim 8' :2. 8' cannot be the non-abelian two-dimensionalL iea lgebraS. For then 8 :S@E and 8 ' : S ' - A . Bu t S / c S .Hence we have 8' abelian. Let 8/ : Oe + Af and I - Ae + af + Og.Then 8t:alegl+o17g1 and so adg induces a 1:1 linear mappingin 8'. Hence we have basis (e,f, g) with

(18) lefl - 0, [esl : ae * Ff , lfsl - re * 8f

where "

: (; il t. a non-singular matrix. Convergely, in any

space I with basis (e,f, g) we can define a product [aD] so that

[aa] - 0 and (18) holds. Then tteflsl + [tfg]el + llselfl = 0 and henceI is a Lie algebra. What changes can be made in the multiplica'tion table (18)? Our choice of basis amounts to thiS: We havechosen a basis (e,f) for It and supplemented this witt'. a 9 to get

a basis for 8. A change of basis in 8' will change A to a similarmatrix M-'AM. The type of change allowable f.or g [s to replacei t b y p g + x , p + O i n O , r i n S / . T h e n l e , p g * x l : p l e g l , ' l f , p g * r l :plfgl so this changes A to pA. Hence the different matrices .4which can be used in (18) are the non-zero multiples of ithe matricessimilar to A. This means that we have a 1 : 1 correspondence be'tween the algebras I satisfying dim 8 : 3, dim 8' :2 and the con'jugacy classes in the two dimensional collineation grotrp.

If the field is algebraically closed we can choose A in one of thefollowing forms:

( l ) a*0, ( ; f ) o+o :

These give the multiplication tables

I. BASIC CONCEPTS T3

l e f l : 0 , I e g ) : e , l f g l : a f

le f l -0 , leg l : e + B f , l fg l - f '

Difierent choices of a give different algebras unless ad' : \. Hence

we get an infinite number of non-isomorphic algebras.

Gj dim 8' : 3. Let (eb €2, €s) be a basis and set fere'l: f',

Vrrrl - fr, ferezl : fr. Then (fr, fr,/t) is a basis' Write f i - [i='eiiet'-4 -- (a;i) non-singular. The only Jacobi condition which has to be

imposed is that lf 'erl * lfrerl * [/gea] : 0. This gives

0 : arzlezer) * argleserf * azrLe4zl * azslegezf * asrlererl * aazlezesl

- - anfg * arrf, * dnfs - azsft - asrfz * atrft '

Hence aii = a4; "nd,

so A is a symmetric matrix. Let (7yEz,ds)_be

a second basis where 4 : Lpitrt, M : (piil non'singular. Set /1 :

l,rdrl, fr: [d&rl, ir: [er,i- We have f'or (i, i, k) any cyclic per'

mutat ion of . (L,2,3)

7, - ldprl - [l,rr &,,I,pr,eJ - Epi,ttn,f0,0,l

: (pizltrs ' tttsltrr)f ,' * (pisttt , - piillnsJfz * (pnp"' - ttizpt)fg

:}v; ' f ' '

The matrix N: (vi.i) - adj M' : (M')-' det Mt . The matrix relat'

ing the /'s to the a's is A and that relating the e's to the 7's is

M-1. Hence if A is the matrix (au) such that f t, -- \d.;id1, then

(19) { - (det M'Y.M')-'AM-' .

Two matrices A, Bare called multi.pticatiuely cogredient if' B:pN' AN

where N is non-singular and p + 0 in O. In this case we may

write B - po'(o-tN)t A@-tN), o : p det N and if the matrices are

of three rows and columns, then we take llrf - oN-' and fi -

p(M-')'AM-t, 7c : poz : det M. _Thus we have the relation (19).

Thus the conditions on A and A are that these symmetric matrices

are multiplicatively cogredient. Hence with each I satisfying our

conditions we can associate a unique class of non'singular multi'plicativety cogredient symmetric matrices. We have as many alge'

bras as there are classes of such matrices. For the remainder of

this section we assume the characteristic is not two. Then each co-gredience class contains a diagonal matrix of the form diag {4, LL},aF + 0. This implies that the basis can be chosen so that

(20) fere2l: es , legel) -- Pes .

T4 LIE ALGEBRAS

If the base field is the field of reals, then we have two differentalgebras obtained by taking a, : 9: 1 and a -- - l, p::1. If thefield is algebraically closed we can take a: I : l.

We shall now single out a particular algebra in thb family ofalgebras satisfying dim I : 3 : dim 8'. We impose the conditionthat I contains an element h such that ad lz has a characteristicroot a #0 belonging b A. Then we have a vector e*0 suchthat

feh l -eadh:d€ ;e0 and s ince lhh l -U, e and h are l inear ly in -dependent and are part of a basis (?,, €2,0a)=(€,h,f). lf (f ' ,fr,ft)are defined as before, the symmetric matrix (461) is now

(21)/dt dp du\

( o * d z z o l .\ a 0 0 /

Then we have tehl - ae, lhhl:0, tfhl- -af - dt€ - dph, whichimplies that the characteristic roots of ad h are 0, a and - &. Wemay replace f by a characteristic vector belonging to the root - a

of. ad h. This is linearly independent of (e, h) and may be usedf.or f. Hence we may suppose ttla;t Lehl: &€s lfhl: + df. If wereplace h by Za-'h we obtain [eh] - 2e, lfhl - - 2f. The form of(21) now gives [ef] - Ph + 0. If we replace f by Fl'f, then weobtain the basis (e,f, h) such that

(22) lehl:2n , l fhl : - 2f , lef l -- 11 . i

Thus we see that there is a unique algebra satisfying our condition.We shall see soon that any I such that dim I * dimS' = 3 is simple,that is, 8 has no ideals other than itself and 0 and B' + 0. Theparticular algebra we have singled out by the condition that itcontains ft with ad ft having a non-zero characteristic foot in O is

called thp sblit three-dimensional simple Lie algebra. It will play

an important role in the sequel.

5. Reprewntations and modules

If lI is an associative algebra over a field O, then a representation

of lI is a homomorphism of tr into an algebra 0 of linear trans-

formations of a vector space llt over O. lf. a + A, b'+ B in the

representation, then, by definition, a *b-+ A * B, qn-+aA, ae O,

and, ab -+ AB. A right \-module for the associative al$ebra lI is a

vector space l}t over @ together with a binary product of llt x U

I. BASTC CONCEPTS 15

into IJt mapping (r,a), r CTft, a e \, into an element ra eIIt such

that

1. (rr * xz)d: rta * xza , x(a, * az): xar * xoz ,

2. a(xa) -- (ax)a: x(aa) , d Q O ,

3. x(ab) - (xa)b .

lf. a -- A is a representation of U, then the representation space llt

can be made into a right ?I-moduie by defining tca: rA. Thus we

will have

(x, + xz)a: (xt + xr)A: xtA * xzA: xfl * *za

x(ar * az): r(At * Ar): xA, * tr42: tch * frat

a(ra): a(xA): (ar)A - (&, f la

a(xA) - x(aA): x(aa)

x(ab) - x(AB) - (xA)B - (ra)b .

Conversely, if llt is any right ll-module, then for any 4 e lI we let

A denote the mapping x -) tca. Then the first part of 1 and the

first part o! 2 show that A is a linear transformation in IJt over

o. The rest of the conditions in 1, 2, and 3 imply that a --+ A is a

representation.In the theory of representations and in other parts of the theory

of associative algebras, algebras with an identity play a pre'

ponderant role. In fact, for most considerations it is convenient

to confine attention to these algebras and to consider only those

homomorphisms which map the identity into the identity of the image

algebra. In the sequel we shall find it useful at times to deal with

associative algebras which need not have identities. We shall

therefore adopt the following conventions on terminology: " Algebra"

without any modifier witt fnean "associatiae algebra with an i'dentity."For these "subalgebra" will mean subalgebra in the usual sense

containing the identity, and "homomorphism" will mean homo-

morphism in the usual sense mapping 1 into 1. In particular, this

will be understood for representations. The corresponding notion

of a module is defined by 1 through 3 above, together with the

condition

4 , t ( I : x , r ; e D t .

If we wish to allow the possibility that t does not have an identity

then we shalt speak of the "associative algebra" lI and if we wish

(23)

We nowlinearity

16 LIE ALGEBRAS

to drop 4, then we shall speak of a "module for associative alge'bra" rather than a module for "the algebra ?[."

The algebra lI can be considered as a right ll-module by takingxa to be the product as defined in lI. Then 1, 2, and 3 hold asa consequence of the axioms for an algebra and 4. holds since 1 isthe identity. The representation a -).4 where A is the linear trans-formation x -+ xa is called the regular Tepresentation; We haveseen ($ 2) tl:ort the regular representation is faithful, that is, an iso-morphism.

Now let 8 bei a Lie algebra. Then we define a reprvsentation of.I3 to be a homomorphism I -- L of I into a Lie algebro Gz, @ the

algebra of linear transforr-nations of a vector space Dt over O. Theconditions here are that i| t, -. Lr, lr-+ lz, then

l, * l, --' Lt * Lr, al1-+ a2,

llrlrJ - [L'Lz) : LrLz - LrL' .

define xl f.or .r e !Dt, I e I by rl : xL. Then (23) and theof. L gives the following conditions:

1. (x, * xr)l : xrl * rzl ,

2. a(xl) - (ax)l : x(al) ,

3. xllJl - (rl)lz - @l)tr

x(1, * lr): xl, * rlz ,

We shatl now use these conditions to define the concept of an

8-modute, [J a Lie algebra. This is a vector space I]t over O with

a mapping of Ift x 8 to llt such that the result r/ satisfies 1, 2,

and 3 above.As in the associative case, the concepts of module and represent-

ation are equivalent. Thus we have indicated that if I -+ L is a

representation, then the representation space IJt can be considered

as a module. On the other hand, if tlt is any module, then for any

/ e 8 we let L denote the mapping x -'fl. Then Z is linear in

I)t over O and I --' L is a representation of I by lineaf transform-

ations in TJI over O.We note next that 53 itself can be considered as an F-module by

taking rl to be the product [r/] in 8. Then 1 and 2 are conse-

quences of the axioms for an algebra and 3. follows frotn the Jacobi

identity and skew symmetry. We have denoted the representing

transformation of. l:1s'-lxll by adl. The representation l-+ad'l

determined by this module is called the adjoint reprdsentation of.

I. BASIC CONCEPTS

8. 'We

recall that the mappings ad I are derivations in 8.

If tJt is a module for the Lie algebra I then we can consider TJt

as an abelian Lie algebra with product [xy]:0. Then the mappings

r --+ xl are linear transformations in 9)t and because of the triviality

of the multiplication in llt, these are derivations. More generally,

we suppose now that fi is an !-module which is at the same time

a Lie algebra, and we assume that the module mappings x -t xl are

derivations in St. Thus, in addition to the axioms for a Lie alge-

bra in I and in l}t and 1, 2, and 3 above, we have also

4. lxrxzll - fxrl, rrl * lxr, xzll .

Now let S be the direct sum of the two vector spaces I and !Ut'

We introduce in R a multiplication luul by means of the formula

[r, + lr, x, * lrl: lrrxzl * rJz - xzlt + Utlr] .

clear that this composition is bilinear, so it makes the vector

S into a non'associative algebra. Moreover,

lx * l , x * I ) : 0 * x l - r la 0 : 0 .

[[rr * lt, xz * lzlxs * 1'1 - l[xaix'] * l[xJr]x'l

+ l?,xzlx'1 1 [[/'/'lra] * [[rrrz]/al+ llrrlzllsl + [U'r,]ral + [U'rrUJ

- llx*t)xrl + [xrlr, rs] - lxzlr, xsl - xtllJrl

+ lrrxzlls * (x,l)h - (xzl)h + t[r'rr]r'l: llxfizlxsf * lxJz, xsf - lrzlb xal - (r"lr)lz

* (xJil, a lrJa, rrl I fxr, xzhJ

* (xrlr)lg - (xzl)I, + IUJrU'l .

If we permute L, 2, and 3 cyclically and add, then the terms in-

volving three ,'s or three /'s add up to 0 by the Jacobi identity

in I and in 'lt. The terms involving two r's and one / are

lxtlz, xsl - lx2lu rrl * txrlg, xz]l * lrt, xzlsl

* lxzlg, xl - fxrlz, xr) * fxzlv xt) * lx2, xslt)

* fxslb xrl - f,xtlu xzl * lxslz, rtl * lxa, xJzl - 0 .

The terms involving two /'s and one .r are

- (xrlr)lz + (xslz)h t UJ)ls - @zlr)ls- (xrlr)I" * (xJilz * @zl")l' - (rslrV,

- (rzl)1, * (xzl)h * (xalr)lz - (rrl')lz - $ .

t7

(24)

It isspace

18 LIE ALGEBRAS

This shows that ft - I @ l}t is a Lie algebra. It is imrhediate from(24) and, the fact that [r/] € !ft for r in IJt, / in 8, that I is a sub-algebra of R and !)t is an ideal in A. We shall call S' the splitextension of I by fi.

An important special case of this is obtained by taking llt : I

to be any Lie algebra and I - S, the derivation algebra. Since 0

is, by definition, a Lie algebra of linear transforrhations, theidentity mapping is a representation of O in 8. I bepomes a 0'

module by defining the module product lD : lD, / b 8, D e S.

The split extension of E by 8, 6 : D @ 8 is called the holomorph of

8. This is the analogue of the holomorph of a group lwhich is an

extension of the group of automorphisms of a group by the group.

We can make the same construction S - Sr O 8 where 8 is any

Lie algebra and S, is a subalgebra of the derivation algebra. In

particular, itisoften useful to do this for Or: oD, the subalgebra

of multiples of one derivation D of 8.Another important special case of a split extensiort is obtained

by taking IJ and Dt to be arbitrary Lie algebras and considering l)t

as a trivial module for I by defining ml :0, ffi e tJl, / e 8. The

Lie algebra R - I @ llt is the direct sum of Il and lll. ,More gener-

ally, if lJ,, ,8r, ' '', I' are Lie algebras then the direct sum 8 -

tl, @ 8, @ . . . @ 8" is the vector space direct sum of the 8r with

the Lie product [Xi/;, ZTm,i : :,Tllimil. As in group theory, if I]

is a Lie algebra and I contains ideals ,8c such that S:8'@[email protected]", as vector space, thenLl t l t l € 8a f l 81 :0 i f / t € 8t , / r € 8r

and i + j. Then I is isomorphic to the direct sum of the Lie

algebras 8c and we shall say that '8 is a direct sum bf the ideals

IJr of 8.The kernel S' of a homomorphism 11 of a Lie algebr4 I into a Lie

algebra !ft is an ideat in I and the image 8Z is a subalsebra of IIt'

The fundamental theorem of homomorphisms states that 8n " 8/S

under the correspondence / + n - hl. We recall that 8/S is the

vector space 8/ft' considered as an algebra relative to the multiplic-

ation U, + S, /, + Sl : ltJrl + A. This is a Lie algebra. The kernel

of the adjoint representation is the set of elements c suchlthat [rc] : $

for all, x. This is just the center o of 8. The imag€, ! : ad 8 is

the algebra of inner derivations and we have 8/G = E, If 6 : 0,

ad I is a Lie algebra of linear transformations isomdrphic to 8'

Thus in this case we obtain in an easy way a faithfulr representa-

tion of 8. We shall see later that every I has a faithfUl represent-


ation and that every finite'dimensional I has a faithful finite-dimensional representation, that is, a faithful representation actingin a finite-dimensional space.

Eramples. We shall now determine the matrices of the adjointrepresentations of two of our examples.

(a ) S theL iea lgebrawi th bas is (e , f ) , le f l :e . We have eade:0 ,

f ade - - ei eadf : e, f adf - 0. Hence relative to the basis (e,f)

the matrices are

'-(l 3)(b) 8 the Lie algebra with basis (er, er, ee) such that Iepzf : es,

leresl - er, les€tl -- €2. Here 0r ?d €t -- 0, 0r 3d e, -- - qst e, ad €t : €z',4 " d , 0 2 : € s ; q z a d c z : 0 , e g a d € z : - € i € r a d e r : * € z t € z v d € s : € r t

€sades:0. Hence the representation by matrices is

Note that the matrices are skew symmetric and form a basis forthe space of skew symmetric matrices. Hence we see that I isisomorphic to the Lie algebra of skew symmetric matrices in thematrix algebra Os.

6. Some basic module operationa

The notion of a submodule It of a module 9Jl for a Lie or as-sociative algebra is clear: It is a subspace of llt closed under thecomposition by elements of the algebra. If It is a submodule, thenwe obtain the factor module lll/ft which is the coset space tn/ttwith the module compositions (r * Tt)a : xa + tt, a in the algebra.If YJtr and !)tz are two modules for an associative or a Lie algebra,then the spac€ Ut' O fi, is a module relative to the composition(xt * rz)a : xfi * rzar !c; € IJlr. This module is called the directsum Tft, @ Ilt, of the two given modules. A similar constructioncan be given for any number of modules.

The module concepts we have just indicated are applicable to

,-'(-l 3),

( rsils)

/ 00 0 rn , - - ( oo - l ) , €z -

\0 1 0/

10 -1e r - l I 0

\ 00

20 LIE ALGEBRAS

both associative and Lie algebras. We shall now cdnsider some

notions which apply only in the Lie case. These are hnalogous to

well-known concepts of the representation theory of $roups. The

principal ones we shall consider are the tensor product of modules

and the contragredient module.We Suppose first that I -, Lr and / -, Lz are two re$resentations

of a Lie algebra I by linear transformations acting in the same

vector Space Tft over O. We assume' moreover, that if Lt is any

representing transformation from the first representatign and M" is

any representing transformation from the second refresentation,

then [LrMzJ: LrM, - MrL, -- 0. We shall now define a new map-

ping of 8 into the algebra @, of linear transformationF in TJt by

l - + L t * L r .(25)

since this is the sum of the linear mappings l - L, and I - L, it

is linear. Now let m -- Mt, n't. - Mr, so that in the new mapping

we have m - M, * Mz. Then

(26) lL, * Lr, M, + Mzl - [L,M,J * lL,Mz\ + ILzM] "v"[L,M')

: [L1M'l I [LrMzl

and since ltml--,lLrMrl a lLrMr\, the new mapping is ja represent-

ation. (Note: Nothing like this holds in the associativB case')

We suppose next that fir and !]tz are any two modtlles for a Lie

algebra 8. Let Ift - utl 8llt, (= IJlr IrTJt') the tensor (0r Kronecker

oidirect) product of llt, and 9ltz. We recall that if 4; is a linear

transformation in IJli, then the mapping )ri I li l l,xtA'E^ yiAz'

xt € Tltr, yi € [!tz, is a linear transformation At I Az lin ']tt I ffir.

We have the rules

(27)

(Ar + B ' ) 8 Az: e ' 8 A2 + B '8 A '

A ' 8 (A, + Br) : .4 ' I Az * Ar@ B' Ia(Ar6 B,) : aArS B' : Ar@ aBr , a Q O

(,4' 8 Ar\,Br@ Bz) : A'Br@ AzBz '

It is clear from these relations that the mapping Ar+4'8Lr, I,,

the identity in fir, is a homomorphism of the algebfa O(!)t') of

linear transformations in fir into the algebra @(yJt' I tnd). similarly'

Az + lr 8 A, is a homomorphism of €(m') into O(fi' @ IJtr). Now

we have the representation R; determined by the !)ti' The linear

transformation /nd associated with t e I is'r1--+ yol. The resultant

I. BASIC CONCEPTS 2T

of the Lie homomorphism I -, l&i with the associative (hence Lie)homomorphism of @(lltr) into @(!]tr I !lt') is a representation of I actingin fi, I fir. The two representations of I obtained in this way are

( 2 8 ) l - a l B r @ l z a n d l - l t & l * ' .

lf. l, m e 8, then

(29) (/or €) lrXl, I m*r) : l"' I m": (1, 6 mBzylP' I 1r) .

Hence the commutativity condition of the last paragraph holds. Itnow follows that

I - . l * '8 1, + 11 @ lnz

is a representation of I acting in !lt' I TJlz. In this way I)l' I fitis an 8-module with the module composition defined by

(X", &y)l - \xit Syr + I,,n&yl

(30)

(311

(32)

Alsoin 9ltsuch

(33)

The module Dl' S llt' obtained in this way is called the tensorproduct of the two modules fie. The same terminology is appliedto the representation, which we denote as fr, I Rr.

We consider next a Lie module llt and the dual space IJt* oflinear functions on IJt (to the base field). We shall denote thevalue of a linear function y* at the vector r e IJt by (*, t*). Then(x, y*) e o and this product is bilinear:

(n * xr , !*) : (xr ,y*) + (xr , y*)

(r, yl + fi> : (x,yl) + Q, yf)(ox, y*) : a(x,y*) : (x, o!*) .

the product is non-degenerate. Any linear transformation 24,determines an adjoint (or transpose) transformation A* in tJt*that

(xA,J*) : (x, t* A*)

The mapping 4 -, A* is an associative anti-homomorphism of 0(tn)into G(IJI*). Now consider the mapping A -, - A*. This is linearand

[- A*(34)

: A*B* - B*A*(AB)* - IBAI*

[A*, B*l(BA)* -- lABl*

Hence A -. - A* is homomorphism of @(fi)z into O(!]l*)2. If we

22 LIE ALGEBRAS

now take the resultant of the representation I -.lR determined byllt with A -, - A* we obtain a new representation I -+ r--(l8)* of 8.For the corresponding module llt* we have(r'Y*t)

=Y-':iifl.'Hence the characteristic property relating the two modules is

( x , t * l ) * ( x l , y * ) : 0 .(35)

We call the module TJt* defined in this way the contragredient rnodule!lt* of llt and denote the corresponding representation by R*.

We recall that if tJt is a finite-dimensional space, thdn there is anatural isomorphism of !|n S llt* onto the vector spade C(!n). IfXiri I yt e tn S !lt*, then the corresponding linear transformationin Dt is r -, 2;(r, yt)x;. If tlt is a module for 8, then fi* and!m S IJt* are modules. By definition,

(\xa8Yf)t - l ,xal SYi * X"o Qc^Yf I .

If we denote x-'rl by tu then r*-'r*/ is -(r")n. Then in therepresentation in IIt I Dtn the mapping correspondin$ to / sendsErr SyI into }xnlo @yf - 2xi8yi'(/')*. The elemehts of O(!lt)associated with these two transformations are

A: r__ !,(r, yf )x, .B: x -' )(r, y{)xil" - >il, yf (to)*)*n

* \(x,yt)xnl* * Z(*l*,v{)n -

It is clear from these formulas that B : lA,ln). We can interpretthis result as follows: Consider an arbitrary 8-module fi and thealgebra G(9n). If l8 is the representing transformation, of l, thenthe mapping X--+lx,l"l is a representation of I actin$ in G. Theresult we have shown is that this representation is equivalent tothe representation in fi 81]t*; that is, the module O is isomorphicto llt I tJt*.

The result just indicated can be generalized to a pair of vectorspaces [l,,IJlr. We recall that the set of linear tranSformationsO(!ltr, Dtr) of Dtz into lltr is a vector space under usual compositionsof addition and scalar multiplication. If the spaces are finite'dimensional, then there is a natural isomorphisms of !)t, I mf


onto @(Iltr, I)lr) mapping the element LriSyf, .f,; € !Itr, Jf eDti'

into the linear mapping y -+ ><l,lt)x; of Dlz into Dtr. If tnr

and flltz ?ra [-modules, then @(Ilt', Dt') is an 8-module relative

to the composition Xl: XPt - P2X. As for a single space, this

module is isomorphic to m' I gltf under the space isomorphism

we defined.

7. fdeals, solvabilita, nilpotenca

If Er and Ec are subspaces of a Lie algebra 8 then we write

E, n Er, E, * Er, respectively, for the intersection and space

spanned by E, and Ez. The latter is just the collection of elements

of the form b, * br, D6 in Ee. We now define [E'E'] to be the sub'

space spanned by all products \btbrl, Di € E;. It is immediate that

this is the set of sums Zilbrtbsi, &;i € Ei. We assume the reader

is familiar with the (lattice) properties of the set of subspaces

relative to the compositions n and + and we proceed to state

the main properties of the composition [E'Er]. We list these as

follows and leave the veification to the reader.

1. [E'Er] : [Er8rl ,

2. [Er + Er,E] - [E'E'l a [EeEal ,

3. [[E'8,]E'l s [[E'E JE,] * [[E,E']E'J '4. [E, n Er, E'] I [ErEa] n [EzEg] .

A subspace E is an ideal if and only if tESl g E. The inter'section and sum of ideals is an ideal and 3 for Es : I shows thatthe same is true of the (Lie) product of ideals. In particular, it isclear that the terms of the deriued series

8 = 8 ' - [ 8 8 ]= 9 " - [ g ' g ' ] = . . .

= g (& ) : [ ! r&_Dg( r ' - 1 ) ] = . . .

(36)

are ideals. The same is true of the terms of the lawer centralseries

8 = 8 2 - 8 '

3 g s : [ g r g ] = . . .

? 8e : [8 , - 'gJ = . . .(37)

These series are analogous to the derived series and the lower

24 LIE ALGEBRAS

central series of a group. More generally, if E is an ideal in 8,then the derived algebras E'i' and the powers En are ideals. Wenote also that if 7 is a homomorphism of 8 into a second Lie alge'bra then (8'n')z - (82)'i' and (8n)z - (82)t. These are reaJdily proved

by induction on r.A Lie algebra is said to be solaable if 8'n' : 0 for sogle positive

integer &. Any abelian algebra is solvable, and it iq immediatefrom the list of Lie algebras of dimensions S 3 that all these aresolvable with the exception of those such that dim I = 3 : dim [J'.The algebra of triangular matrices is another example df a solvableLie algebra (Exercise 12 below).

Lpuu.c, Euery subalgebra and eaery homomorphic image of asoluable Lie algebra is solaable. If 8, contains a solaable ldeal E suchthat 8,lE is soluable, then 8, is solaable.

Proof: The first two statements are clear. If E is dn ideal suchthat !/E is solvable, then 8'D) g E for some positivd integer h.Thus let a be the canonical homomorphism I -' I * E of 8 onto 8/8.

Then (&'o')n - (82)(") : (8/E)'o' :0 if. h is sufficiently ldrge. Hence

8(a) c E. If E is solvable we have 5Gr - 0 for some &. Hence

!( , ' ' ) c E i rnpl ies gtD+et c E(&) :0 and 8 is solvable.

Pnorosrrrolr 4. The sum of anyid.eal.

two soluable ideals ls a soluable

Proaf: Let Er and Ez be solvableisomorphism theorems Et ft Ez is anE'/(E' n Er). This is solvable sincethe solvable algebra E'. Since E' isprove Er * Ez solvable.

Now suppose 8 is finite-dimensional and let 6 be a solvable ideal

of maximum dimensionality of 8. Then Proposition 4 implies that

if E is any solvable ideal 6 + E is solvable and an ideal. Henceg + E : 9 s i n c e d i m 6 i s m a x i m a l . C o n s e q u e n t l y , 6 = 8 . W e

therefore have the existence of a solvable ideal 6 which contains

every solvable ideal. We call € the radical of 8. Ifl g : 0, that

is, 8 has no non-zero solvable ideal, then I is called t semi"simple.

If t has no ideals + 0 and 8, and if 8' + 0, then I is Aalled simqle.

If ! is simple and 6 is its radical, then we must have I : 6 or

9 : 0 . B u t i f l J : 9 t h e n 6 / c 6 a n d 6 ' i s a n i d e a l s o g / : l J ' : 0

contrary to the definition. Hence 6 :0; so simplicity implies semi-

ideals. By one of the standardideal in E, and (E' * Ez)/Ez =it is a homomorptric image ofsolvable the lemn[a aPPlies to

I. BASIC CONCEPTS

simplicity. If g is the radical, then any'solvable ideal of 8/6 hasthe form El6, E an ideal in 8. But E is solvable by the lemma.Hence E s 6 and E/6 : 0. Thus 8/6 is semi-simple. If A is anon-zero solvable ideal in [J and Strr-tr + 0, E('r] - 0, then 8ta-tt i"an abelian ideal + 0 in 8. Hence 8 is semi-simple if it has nonon-zero abelian ideals.

The three-dimensional Lie algebras 8 with dim 8/ : 3 (or 8' : 8)are simple. For, if E is an ideal in I such that 0 + S + I, thenE and 8/E are both one or two dimensional; hence solvable. Then! is solvable contrary to 8' : 8.

A Lie algebra I is called nilpotent if 8& : 0 for some positiveinteger &.

Pnoposnror.r 5. [8t8'] -- 8d*'.Proof: By definition [,8d8] : gi+'. We assume [8dt1 g 8i*i for

all i. Then

[gigi*', - [gt[g'8]l s tt8d8l8il + [[808']sl s [8i*'9',1 + [8t*'8J c gd+r+t.

This result implies that any product of ft factors I in any as-sociation is contained in 8&. Since gttr is a product of 2& suchfactors it follows that Srrc) cgz&. Hence if I is nilpotent, szy 8'ft:0,then 8(&) :0 and I is solvable. The converse does not hold sincethe two-dimensional non-abelian Lie algebra is solvable but notnilpotent. The set of nil triangular matrices, that is, the triangularmatrices with 0's on the diagonal is a nilpotent subalgebra of O*twhere O^ is the algebra of. n x n matrices.

Pnorosrrrou 6. The sum 0f nilpotent ideals is nilpotent.Proof: We note first that if E is an ideal, then any product

["'[[?t'?tr]U'1"'?l*l in which h of. the ?Ir -E and the remaining?Ii:8 is contained in EDIE0-8;. A simple induction on ft establishesthis result. Now consider two ideals Er and Ez and the ideal Er * Ez.Then (Er + E2)- is contained in a sum of terms [...[[U,?Ir]U'l... l l*lwhere !l; : Er or Ez. Any such term contains lmlzl Er's or lrnlz) Er's,where .lml2) is the integral part of. ml2; hence it is contained inEfnorzt or in E['ttt. Consequently,

(8, + Er)- s E[-r't * p;t*tzt .

It follows that if Er and Ez ?r€ nilpotent, then E, * Es is nilpotent.As for solvability, we can now conclude that if I is finite-

dimensional, then I contains a nilpotent ideal It which contains

26 LIE ALGEBRAS

every nilpotent ideal of 8. We call It the nil radical of. 8. Thisis contained in the radical 9. For the two'dimensional non-abelian

algebra the * Af, lefl: e, g:8 and Tl: Ae. Also 8/It is abelian,

hence nilpotent. Thus we may have g r tt and !/Tt lnay have a

non-zero nil radical.The theory of nilpotent ideals and radicals has a 1 parallel for

associative algebras. If E' and E, are subspaces of aD associative

algebra lI, then ErE, denotes the subspace spanned by all products

brbz, bi € E;. lI is nitpotent if there exists a positive iriteger & such

that lI& : 0 (Ut : ll,lle : ![&-t[). This is equivalent toi saying that

every product of & elements of lI is 0. If tt' and llz tre nilpotent

ideals of lI, then it is easy to see that ltr * ltr is a nilpotent ideal.

Hence if ?I is finite-dimensional, then lI contains a maxi$ral nilpotent

ideal S which contains every nilpotent ideal. The idehl S is called

the radical of. \. The algebra ll/fft is semi,'simple in the sense that

it has no nilpotent ideals + 0. The proofs of these stAtements are

simitar to the corresponding ones for Lie algebras and will be left

as exercises. I

8. Extenaion of the bare field '

We are assuming that the reader is familiar with the basic de'

finitions and results on tensor products and extension of the field

of operators of vector spaces and non-associative algebras. We

now recall without proofs the main properties which \h/ill be need'

ed in the sequel.If lI and E are arbitrary non'associative algebras Qver O, then

the vector space ?I8 S (?I@eE) can be made into a ndl'associative

algebra by definin g (2;at8 D;X)r i e bh - Zas! @ bdbi for ai, a!1 e \1,

b;, bj e E. If lI and E are associative, then ?I I E is assOciative also.

lf P is a field extension of O and ?I is arbitrary, then the O'algebra

P@?I can be considered as a non'associative algebrta over P by

defining p(Ep; g a) : 2pp;8 ar, Q, & e P, a; e \. Wb denote this

"extension" of ?I as llr. Such extensions for Lie 4lgebras willplay an important role from time to time in the sequNl.

We recall some of the main properties of ?Ip and of Utp where

fi is any vector space over O and lllr is PSSJt considefed as vector

space over P relative to p(Ipi @ rr) : I,pp; @ r;, P, Pi E P, tr; € fi,

lf {eili e I} is a basis for !}t over A, then {1 @e;} is a basis for

!]tr over P. The set of A-linear combinations of thiese elements


coincides with the subset {1 I x I x € $t} of $Jti,. This set is a

O-subspace of !ftp which is isomorphic to IJt. We may identify

1 8 r with r and the set {1 I r} with fi. In this way lJt becomes

a @-subspace of D?p which has the following two properties: (1) the

P-space spanned by TJt is fip, (2) any subset of !)t which is linearly

independent over @ is linearly independent over P. These imply

that any basis for 9l? over A is a basis for lJl over P. If ?I is a

non-associative algebra, then the identification just made permits

us to consider lI as a @-subalgebra of llp. The properties (1) and(2) are characteristic. Thus, let fl be any vector space oYer P, O

a subfield of P, so that frl can be considered also as space over O.

Suppose Dt is a subspace of frl ouet @ satisfying (1) and (2). Then

the mapping-Ipr @ x;-Zp;ri,-y; € P, r; e tJl, is an isomorphism

of ulp onto fr|. Similarly, if ?I is a non'associative algebra over P

and ?I is a @-subalgebra satisfying (1) and (2), then we have the isomor'phism indicated of ?Ip onto ?I.

If ?I is associative, then qaa')a" : a(a'a") in lI implies that llr" is

associative. Similarly, if ?I is a Lie algebra, then I a.a] - A, laa'l:- la'a), l laa'ia"l * l la'a"lal + l la" ala'l :0 imply that lI, is a Lie

algebra.If It is a subspace of IJt, then the P-subspace PIt generated by It

may be identified with ltr. If E is a subalgebra (ideal) of lI' then

E"(: PE) is a subalgebra (ideal) of llp. The ideal (21")' of ?I" is the

set of P-linear combinations of the elements aa' , a, a' e 21. Hence(U")' :LjL. If I is a Lie algebra, I ' is the set of l inear combina'

tions of the products 7"' larazl"'a,1, an € 8. It follows that (8')":(8r)'. Similarly, the derived algebra 8" is the set of linear com'

binations of the products of the form [lararJlata)1,9ttt the set of

linear combinations of products

[ [arar][a"a.1], [ [aiaol, [arat l l l , di € 8

etc. , for 8 ' t ' . . . . I t fo l lows that (8") ' ' ' : (8 ' ' ' ) " . These observa-

tions imply that a Lie algebra 8 is commutative, nilpotent, orsolvable if and only if 8, is, respectively, commutative, nilpotent,or solvable.

If J is an extension field of the field P, then we can form lft:and (91?p)'. The first of these is J 8ollt while the second isE &r(P 8o!J?;. It is well known that there is a natural isomorphismof (9ltr).t onto fis mapping o8(p8r)-. '6p&ts, o e 2, p e P, r eTlt.Hence we can identify (!ltr)r with ll?r by means of this isomorphism.

28 LIE ALGEBRAS

lf. A is a linear transformation of llt into a second vector spacerl over o, then .4 has a unique extension, which we sfrall usuallydenote by A again, to a linear transformation of lJlr ifito Ttr. wehave (Xpuxo)A:Zpu(nA), Qr € P, tcr. € tlt. The imagefrrA: (!JtA)pand the kernel of the extension A is Sp, S the kernell of. A in !)t.Hence A is surjective (onto) if and only if its exterlsion is sur-jective and A is 1 : I if and only if its extension is tr: 1. If ?I isa non-associative algebra and A is a homomorphisrn (anti-homo-morphism, derivation), then its extension A in ?Ip is a homomor-phism (anti-homomorphism, derivation).

Now let ll be a Lie algebra, llt an t-module and R [he associat-ed representation. rf a e 8, ao has a unique extensidn as whichis a linear transformation in tlnp. We have (a + b|ra -_ a* * b*,(oa)* - daR, [ab]R : la*,b*j, a € O, for these extenbions. Thisimplies that for Q; € P, a; e 8, the mapping Zpoa,i t 2pnf is ahomomorphism r? of 8p into the Lie algebra O(lltr)" of rlinear trans-formations in TJlp. Thus R has an extension R which is a repre-sentation of [Jp acting in IJtr,. For the module gJto which is deter-mined in this way we have x(\pp): Zpu(rar), Qr)e p, at e g,x e T f t p a n d i f x : Z p l x i , r i e D l , p ' t e p , t h e n x a - - Z b l @ i a ) , a € g .In a similar fashion a representation R of an associ{tive algebralI defines a representation R of the extension ?Ir and a jright moduleDt for lI defines a right module fip for ?Ir.

Exercises

1. Let 2l and E be associative algebras. Show that if a is a lhomomorphismof ?[ into E, then e is a homomorphism of ?Ir into Ez and if e is an anti-homomorphism of l)Id[nto E, then - e is a homomorphism 6f 2r, into Ez.Show that if 0 is an inti-homomorphism of ?l into !)1, then ttte subset 6(2I, e)of e-skew elements (a0 = - o) is a subalgebra of t)tz. show ittrat ir D is aderivation of 2[, then D is a derivation of ?Ir. Give examfles of ?l and Ewhich are neither isomorphic nor anti-isomorphic but IIt.Z $z (= indicatesisomorphism). (Hint: Take 2I, E to be commutative.) Giver an example ofa derivation in 2lz which is not a derivation of lll.

2. If S is a subset of a Lie algebra 8 the centralizer G(SD is the set ofelements c such that [sc] = 0 for all s € S. Show that G(S) is a subalgebra.If E is a subspace of 8, then the nwmalizer of. E is the set of I e g suchthat [bl] € E for every b € A. Show that the normalizer of Eris a subalgebra.

3. Let D be a derivation in a non-associatiative algebrd 2I. Show thatthe set of elements a of ?I satisfyinE zD:0 is a subalgebra. I (Such elementsare called D-constants.) Show that the set of elements z safisfying 2Dt : e

I. BASIC CONCEPTS

for some i is a subalgebra.i. Show that if D is a derivation of a Lie algebra such that D commutes

with every inner derivation, then the image 8D g G, G the center. (If G : 0

this implies that the center of the derivation algebra G(E) : g. Hence also

G(A(t(8)) :0, etc.)5. Show that aly three-dimensional simple Lie algebra of characteristic

not two is complete.6. Prove that any four-dimensional nilpotent Lie algebra has a three'dimen-

sional ideal, Use this to classify the four-dimensional nilpotent Lie algebras.

I. Verify that if I has basis (er, azt . . ., ea) with the multiplication table

teftzi : es fepgf : ea Iep+l = et tepsl -- - ea

fezesl: ee lezetl: ea lezeol = - et Ie*rl = - "

Iesesf : - et [eaei = - aB , all other lercil: g

I o r i < i , l e p d - 0 , f e a e i : - [ e P d ,

8 is a nilpotent Lie algebra.8. A subalgebra E of I is called subinuaria,nt in 8 if there exists a chain

of subalgebras 8: 8r I 8z I ... ? 8r: E such that 8t is an ideal in 8t-r.

Show that if the chain is as indicated, then ['' '[SS]8"'El g E if there are

s E's in the term on the left'hand side.

9. (Schenkman). Show that if !I and E are subspaces of a Lie algebra

then [!lE'?] g t. . .tUSlS. . .El (n, E's). Use this to prove that if E is subinvar-

iant in 8, then E. : O ilr$c is an ideal in 8.

10. Let 8 be a nilpotent Lie algebra. Show that a subset S of 8 generates l!

if and only if the cosets s * !2, s € S, generate 8/82. Hence show that for

finite-dimensional Il the minimum number of generators of Il is dim 8/82-

ll. Show that if I is a nilpotent Lie algebra, then every subalgebra of

I is subinvariant. Show also that if E is a non'zero ideal in 8, then E O

G + 0 for G the center of 8.L2. Let E be the Lie algebra of triangular matrices of fl, rows and columns.

Determine the derived series and the lower central series for t.13. Prove that a finite-dimensional Lie algebra is solvable if and only if

there exists a chain B : gol Sr f 8z f " ' )*^: 0 where dim 8l : n - i

and 8r is an ideal in 8r-r.14. If IJ is a Lie algebra t}ire upper central seri'es 0 S Gr S Gz S "' is

defined as follows: Gr is the center of 8 and Gt is the ideal in I such that

Gclgl-r is the center of 8/Ge-r. Show that the upper central series leads to

8, that is, there is an s such that Gs : 8 if and only if I is nilpotent. Show

that the minimum s satisfying this is the same as the minimum t such that

8r+r = 0 .15. Prove that every nilpotent Lie algebra has an outer (: non'inner)

derivation. (Hint: Write 8: [Jl@@a where [Jl isanideal- I f . z € 8: C(f l I t)

the linear mapping such that Dl -+ 0, e + z is a derivation. If a is chosen in

8 but not in !n+r v[s1e ro satisfies I g 8', I E 8*+t, then the derivation

defined by a is outer.)

29

30 LIE ALGEBRAS

16. I'et A be a linear transformation in an z dimensional space. Suppose24. has ra distinct characteristic roots fr, €2, . .., En. Show that ad A acting inG has t he n2 cha rac te r i s t i c r oo t s h - E l , i , i =L ,2 , . . . , n . ,

17. Let ffi be a finite-dimensional module, fi* the contragredient module.Show that if Il is a submodule of fi, then Il- : tz*l <A,z* ) = 0,2 € !}]is a submodule of !Di*. Hence show that fi is irreducible if adrd only if $?*is irreducible.

18. Let [I] and Si* be as in Exercise 17. Show that !fr18 Sl* contains au * O such that ul : O for all l. Assume Sl irreducible and $uppose Il isanymodulesuchthatlJl@Sl contains a r*0 such that nl:0, I € 8. Showthat !t contains a submodule isomorphic to !]l*.

19. Let ?I be a non-associat ive algebra such that 2l=2[r@!Iz@...@U"where the lll are ideals satisfying W?: Wt. Show that the deritation algebraE : Er @ Dz O .. . O D' where Dr is an ideal in D and is isordorphic to thederivation algebra of 2[1.

20. Show that the derived algebra ALr of, the Lie algebra tDw is the setof matrices of trace 0. Show that the center of.Ot. is the set @[ of multiplesof 1 and that the only ideals in @nz are Qkr and @1 unless n"=2 and thecharacteristic is 2.

21. Give an example of. a Lie algebra over the field C of complex numberswhich is not of the form [Jc where 8 is a Lie algebra over the field .B ofreal numbers. (Hint: Consider the Lie algebras satisfying ditn I = 3, dim8,t = 2.)

22. Let E be an ideal in a non-associative algebra Il, D a derivation in 2[.Show that E + ED is an ideal. Show that if 2l is finite-dimensiodal associativeof characteristic zero with radical S, then nD g fr for every lderivation Dof ?I. (This fails for characteristic p; see p. 75.) Prove thd same resultfor Lie algebras.

28. Show that if G is a commutative, associative algebra (wlth l) and 8 isa Lie algebra, then G @ I is a Lie algebra. Give an example to show thatthe tensor product of an associative algebra and a Lie algebra need not be

a Lie algebra and an example to show that the tensor product of two Lie

algebras need not be a Lie algebra. (Hint: For the first of these, take the

associative algebra to be Oz and, note that Q*8 S = E, wherep is any non-associative algebra and E', is the algebra of n x n matrices with Fntries in E.)

CHAPTER II

Solvable and Nilpotent Lie Algebras

The main theme of the last chapter has been the analogy be-

tween Lie algebras and groups. In this chapter we pursue another

idea, namely, relations between Lie algebras and associative alge'

bras. We consider an associative algebra lt-usually the algebra

of linear transformations in a finite dimensional vector space-and

a subalgebra I of Us. We are interested in studying relations

between the structure of I and of the subalgebra 8* of l[ gener-

ated by 8. We study this particularly for 13 solvable or nilpotent'

The results we obtain in this way include the classical theorems

of Lie and Engel on solvable Lie algebras of linear transformations

and a criterion for complete reducibility of a Lie algebra of linear

transformations. We introduce the notion of weight spaces and

we establish a decomposition into weight spaces for the vector

space of a "split" nilpotent Lie algebra of linear transformations.

These results will play an important role in the structure theory

of the next chapter.

7. Weakta elosed aubsete of an anoeiatioe algebra

Our first results can be established for subsets of an associative

algebra which are more general than Lie algebras. It is not much

-or" difficult to treat these more general systems. Moreover,

occasionally these are useful in the study of Lie algebras them'

selves.

DpnxrnoNs 1. A subset IB of an associative algebra lI over a

field O is called weakly closed if for every ordered pair (a,b),

a, b e 8, there is defined an element r@, b) e @ such that ab *

rl@, b)bo e [8. We assume the mapping (a, b) + T(a, D) is fixed and

write a x b: ab * r(a,b)ba. A subset 1l of S is called a subsystem

i f c x d ,e l l fo r every c ,de l landt l i sa le f t idea l ( idea l ) i f ' ax c eU

(a x c and c x a ell) for every a e [$, c e U.

[ 3 1 ]

32

Exaruples

LIE ALGEBRAS

(1) Any subalgebra I of llz is

r @ , b ) - * l .

weakly closed in ?Ii relative to

(2) If 8(= U) is the Lie algebra with basis (e,f,h) such that

lefl: h, Leh]--2e, [f h]: -2f thenl/{d_: oe u af u oh is a subsystemof 8.

(3) The set of symmetric matrices is weakly closed ]in the alge-bra A, of n x n matrices if we take r(a, b) : I.

(4) Let S : 0 U 6 where 6 is the set of symmetfic matrices

and is the set of skew matrices. Defne r@, b) - 11 if a and b

are symmetric and r@,b) - -1 otherwise. Then [S is weakly

closed and 0 is an ideal in S.(5) If. r@,D) = 0 we have a multiplicative semigroup in S.

DprImrtoN 2. If g is a subset of an associative al$ebra lI (an

algebra - associative algebra with 1) we denote by 6*(6t) the sub-

algebra of lI (subalgebra containing 1) generated by lg. We callg*(gt) the enuelopi.ng associatiae algebra (enuelopi,ng Qlgebra) of @(in Ir).

We shall now note some properties of weakly clqsed systems

which will be needed in the proof of our main thedrem on such

sets.I. If W fs an element of a weakly closed systQm 8, then

3 = {W}* nS is a subsystem of 8 such that 8* : {ry}*.Proof: The enveloping associative algebra {W}* i$ the algebra

of polynomials in W with constant terms 0. If Wr), and Wz are

two such polynomials then Wr x Wz is a polynorfiial. Hence

3 : { W}* n S is a subsystem. Since gs W,,3* = IW}*- Sinceg = 1W1* and the latter is a subalgebra, 3* g {W}*. Hence

3* : {w}*.II. If E is a subsystem of 8 and W is an element of 8 such

that B x W e E* for euery B eE then

E*W=WE*+E* .

Proof: The elements of E* are linear combinations bf monomials

B r B " " ' 8 , , B ; € 8 . I f B e E t h e n B W : - T ( B , W ) W B * B xW e WE* + E*. Induction on r now shows that if Bi € E then

B, . - - B,W e WE* + Ex. This proves (1). I

III. Let E be a subsystem of 8 such that E* is nilpotent and

E*+ s*. Then there exists a WeW such that WeE* but

( 1 )

il. SOLVABLE AND NILPOTENT LIE ALGEBRAS 33

B x W e E* for eaerY B in E.

Proof: The assumption E* + S* implies that there exists a

Wr€&, Wr#E*. I t B x W, € E* for a l l B e E then we take

W : Wr. Otherwise we have a Wz : B, x W, e 8, e E*' We

repeat the argument with W, in place of Wr. Either this can be

taken to be the Wof. the statement or we'obtain Wg: Bzx Wz:

Bz x (8, x I7r) e gB, 6 E*. This procedure leads in a finite number

of steps to the required element W ot else we sbtain an infinite

,"q,r.rra" Wr, Wr, "', W, : B'i.-r X W;*r,where W; e[S but W' 6E*'

We shall show that this last possibility cannot occur and this will

complete the proof. We note that Wr, is a linear combination of

products of. k - 1 B's belonging to E and Wr. Since E* is nilpotent

there exists a positive integer z such that any product of n ele'

ments of A is 0. Now Wro is a linear combination of terms

Cr "' ClWrDr "' Dn where the C' and D, e E and i + k :2n - L'

Either i >: fl or k > rt so that we have Cr "' C5WrDr "' Dt :0'

Hence Wro:0 and Wz*€ Ex contrary to assumption

2. Nil weaklY closed seta

The main result we shall obtain on weakly closed systems is the

following

Tnnonnu 1. Iet 8 be a weakly closed subset of the associ'atiue

algebra @ of tinear transformations of a rtnfie-dimensional aector space

[Il oaer A. Assume euery ]7e S is associatiue nilpotent, that is,

Wk :0 for sorne positi.ue integer k. Then the enaeloping associati,ae

algebra tr\* o/ IB fs nilPotent.Proof: We shall prove the result by induction on dim TJt. The

result is clear if dim fi - 0 or if [S : {0}. Hence we assume

dim Dt > 0, tl$ + {0}. Let Q be the collection of subsystems E of

B such that E* is nilpotent and let E be an element of l] such

that dim E* is maximal (for the elements of J]). We shall showthat Ex : S* and this will imply the theorem. We note first that

E* + 0. Thus let W be a non-zero element of S. Then by I,

3 : Is n { W}* is a subsystem and $* - {w}*. Since {W}* is the

set of polynomials in W with constant term 0, {W}* is nilpotent.

Hence g e 9. Since 3 + 0 it follows that E* + 0. This implies

that the subspace [t spanned by all the vectors tcB*, r e Tlt, B* € E*,

is not 0. Also It + !]t. For otherwise, any x : 2 xoBl , x; e VJt,

The main result we shall obtain on weakly closed systems is the

U LIE ALGEBRAS

aI e Et. If we use similar expressions for the r;i we obtainr:X yfilCf , Bf , Cl in E*. A repetition of this pfocess givesr : x zrBtrtBtr -.. Bt,, Bfi in E*. since E* is nilpoteni tthis im"pliesfi :0, and rlt : 0 contrary to assumption. Hence wg havd llt = ll > 0.Let 6 be the subset of I of elements c such that gtcrs tt. Thenit is clear that 6 is a subsystem containing E. Moreover G inducesweakly closed systems of nilpotent linear transformations in !t andin the factor space rltlyt. Since dim ft, dim !lt/[ < dirn I]t we mayassume that the induced systems have nilpotent envelodinq associa-tive algebras. The nilpotency of the algebra in rJtlrt limpties tnatthere exists a positive integer p such that if r is an5f eLment ofIlt and Cr, Cr, ,.., C, are any C; in 0 then xCtCz. . . Cr e It. Alsothere exists an integer 4 such that if Dr, Dr, . . ., Doe fu and y e ttthen JDr .. . Do : 0. This implies that i f Cr, . . . , Cp+a€ 6 thenCtCr... Cr*o : 0. Thus O* is nilpotent and 6 e g. We can nowconclude that 8* : E*. otherwise, by III., there exists w e w,E E* such that B x W e E* for all B in E. By II., lve have forany r -oU l , B*eE* , xBxW-r (WBl+n l ) ,

-B f eE* . Hence

nWgn so We6, . S ince WdE* , d imG*>d imEf r and s inbeE e ll this contradicts the choice of E. Hence E* : s*iis nilpotent.

If [B is a set of nil triangular matrices, that is, triangularmatrices with 0 diagonal elements, then g* is containda in the as-sociative algebra of nil triangular matrices. The latter is nilpotent;hence 8* is nilpotent. The following result is therefofe essentiallyjust a slightly different formulation of Theorem l.

Tsponnu 1'. Let un be as in Theorem 1. Then there exists abasis for ffi such that the matrices of ail the w e g haae nil tri-angular form relatiae to this basis.

Proof: we may assume rlt + 0. Then the proof ot' Theorem 1shows lhat tn r m8* where tjts* is the space spairned by thevectors fiw*, r € Dl, I,[z* e 8*. In general, if rt is a $ubspacl and6 is a set of linear transformations, then we shall write rt6 forthe subspace spanned by all the vectors ys, y e rt, s e 6. Thenit is immediate that tn(r8*)' : (DruB*)uBx. Also if tnr8* + 0 theargument used before shows that llt8* ) tn(!B*)t. Hdnce we havea chain IJt = UIIB*:Dt(8*; '=f i(S*), =.. .=Dt(IB*)x- ' l ! rJl( IB*)r: 0,if (8t)n : 0 and ([B*)r-t + 0. We now choose a basis (dr, . . ., en) forf i such that (er , . . . tnr ) is a basis for l l t (S*)n-r , (er , . . . ,4n1, . . . ,€n;n2)is a basis for l l t18* ; r -2, . . . , (€r , . . . ,€nr+nz+. . .+np) is la basis for

N. SOLVABLE AND NILPOTENT LIE ALGEBRAS 35

tJt([B*)*-t Then the relations l]l(UB*)x-]US c tn(8*)nr-&+r imply thatthe matrix of any W € S has the form

- [ r "NN

.tut l

h l 0 | n z

: ltz

*

Theorem 1 can be generalized to a theorem about ideals in weaklyclosed sets. For this purpose we shall derive another general proper'ty of enveloping associative algebras of weakly closed sets:

IV. Izt W be a weahb closed subset of an associatiae algebra andlet E be an i,deat in 8, E*, !B* the enueloping associatiae algebrasof E and W, respectiuely. Then

uB*(E*)o g (a*)*a* + (E*)o ,(E*)oS*. 8*(E*)o + (E*)* ,

(uB*E*)* g 8*(E*)t , (E*s*)t E (E*)e$*

Proof: By II. of $1, the condition that Bx W €E for BeE,w es&, implies that E*w s I,l4B* + E*. It follows that ifW r , W r , . . . , W , e S , t h e n E * W r W z . . . W r Q 2 S * E * + E * . H e n c eE*m*.[B*E* + E*. Induction now proves (E*)&IB*gIB*(E*)& + (E*)0.Similarly the condition that S is a left ideal implies thats*(E*)* g (E*)o[B* + (E*)*. Hence (2) holds. Now (3) is clear forh: t and if it holds for some &, then

([B*E*;r*r : ([B*E*).[B*E* g m*(E*)rgr:8*= [s*(g*E*e + E*o)E* g g*(E*)k+' .

Similarly, the second part of (3) holds.We can now prove

Tnnonpn 2" Izt 8 be a weakly closed, set of li,near transforma.tionsacting in a finite-dimensional uector space and let E be an ideal inW such that euery element of E is nitpotent. Then E*(hence E) l'scontained in the radical S o/ S*.

Proof : S*E* + E* is an ideal since S*(!B*E* + E*) E S*E* and

( 2 )

( 3 )

36 LIE ALGEBRA

(m*E* +E*Xsx s [B*([8xE*+E*)+s*E*+E* - IB*E*+E*. Also(EB*E* + E*)o = [B*E* + (E*)0. By Theorem 1, E* is nilpotent.

Hence h can be chosen so that (S*E* + E*)o g [B*$*. On the

other hand (mn5*), g [B*(E*)'and I can be taken so that (S*E*)r : 0.

Thus 8*E* + E* is a nilpotent ideal and so it is conitained in S'

It follows that E* and E are contained in fi,

3. Engel's theorem

Theorem 1 applies in particular to Lie algebras. In this case it

is known as Engel's theorem on Lie algebras of linpar transfor'

mations: If g is a Lie algebra of linear transformatiorls i'n a fi'nite'

dimensional aector space and eaery L e g is nilpotenl, then 8* fs

nilpotent. The conclusion can also be stated that a I basis exists

for the unclerlying space so that all the matrices are nil triangular'

These results can be applied via the adjoint reprdsentation to

arbitrary finite-dimensional Lie algebras. Thus we have

Engel's theorem on abstract Lie algebras. If t, is a,rtnite'dirnen'

sionai Lie algebra, then g is nitbotent if and only if ad E is nilpotent

for euery ae8,.Proof : A Lie algebra 8 is nilpotent if and only if I there exists

a n i n t e g e r N s u c h t h a t I . . . L a & z l a s l - . . 4 ^ n , l : 0 f o t s t e t . T h i s

imp l ies t tu t t . . . l xa la l . . .a l :0 i f the produc t con ta ihs ̂A / - l a 's .

Hence (ad a)tr-t : g. Conversely, let I be finite-dirt'rensional and

assume that ad. a is nilpotent for every a e 8. The set ad I of

linear transformations ad a (acting in 8) is a Lie al[ebra of nil-

potent linear transformations. Hence (ad 8)* is nilpotent. This

means that there isan integer N such that ad a.ad,as" 'adax:0

fo r c r e 8 . Thus l " ' [aar la r1" ' an \ :0 and so 8rn '=+ 0 '

we can apply Theorem 2 of the last section tp obtain two

characterizations of the nil radical of a Lie algebra' '

Tsuonpu 3. Let g be a fi,nite-dimensional Lie algebfa. Then the

nil rad.icat Tt of g can be characterized in the follow\ng two ways:

(I) For eaery a e Tl, ad a (acti.ng in 8) es nilpotent and if E is any

id,eal such that adb (in 8) is nilpotent for eaery beE, then Egtlt.

(2) Tt is the set of elements 6 e I such that ad b e th'e radical ft of

(ad 8)*.Proof: i

( 1 ) I f D e T t a n d a e g , t h e n [ a b ] e I t a n d t " ' L a b l b i l " ' b l : 0 f o r

U. SOLVABLE AND NILPOTENT LIE ALGEBRAS 37

enough ,'s. Hence ad b is nilpotent. Next let E be an ideal such

that ad D is nilpotent for every b e E. Then the restriction ad56

of ad D to E is nilpotent and E is nilpotent by Engel's theorem.

Hence E g tt.(2) The image ad R of It under the adjoint mapping is an ideal

in ad I consisting of ni,lpotent transformations. Hence, by Theo'

rem 2, ad It g S. It is clear that S n ad I is an ideal in ad 8; hence

its inverse image !t, under the adjoint representation is an ideal

in 8. But }tr is the set of elements b such that ad b e S. Every

adylrb, b e Ttr, is nilpotent. Hence Tt, is nilpotent and Ttr g It. We

saw before that ad Tt s ft so that Tt I Tl'. Hence It : Ttr.

4. Primara components. Weight spaces

For a linear transformation Aina finite-dimensional vector space

tjt the well-known Fitting's lemrna asserts that fi:fio.r@fir,

where the lltra are invariant relative to A and the induced transfor-

mation in lJtoe is nilpotent and in IJt,r is an isomorphism. We

shall call fio, and !Il1a, respectively, the Fitting null anq one

component of l}t relative to A. The space lJtrr: flLrl)t,4' and

IJtga : {zl zAr : 0 for some r}. The proof of the decomposition

runs as follows. We have yIt2YItA2tIItAz 2 "'. Hence there

exists an r such that tMA' : TItA'*t : " ' : IJtre. Let 3t :

{z;l a,{ - 0}. Then 3, tr 3, c ' ' ', so we have an s such that $, -

&*r : " ' : Tftoe. Let t _ max (r, s). Then [Jtot: $6 and lJlra:tlftAt. Let r e IIt. Then tcA' : yAzt for some y since tIftAt :TI\A".

Thus x: (x - yA') + yA' and yAt € IJt,e, while (x - yA').4' : 0 so

x - yA'€ lJtor. Hence lJt - TJto,r * tlltta' Let z € TJtoe fl IJlt'' Then

z: wA' and 0 : zA' : roA't. Since u)A" :0, u e Tltsa - $, and

wAt - 0. Hence z :0 and Dto,r o lJtrr - 0; hence sIft - l]to, O Wtr*

Since TJlsa: gr, A' : 0 in [Ilsa. Since [Itla: 'lnA' : TIIA'*' : TItlaA,

4 is surjective in lJtr,r. Hence / is an isomorphism of llt,e.

We recall also another type of decomposition of llt relative to

A, namely, the decomposition into pri,mary cornbonents. Here

9m - Mora$ IJto2,r e ' ' ' @T[to,e where rc i : r ; ( t r ) , { r r ' ( i ) , ' ' ' , t r , (A)}

are the irreducible factors with leading coefficients 1 of the mini-

mum (or of the characteristic) polynomial of A, and if p(,i) is anypolynomial, then

( 4 ) Tltp.t: {zlza(A)' : 0 for some r}

(cf. Jacobson, [2], vol. II, p. 130).

38 LIE ALGEBRAS

lf p(l) : 7q(1) is irreducible with leading coefficient one, then IIt",r :0

unless ft : nri for some i. The space Tftpe is evidently invariant

relative to A. The characteristic polynomial of the restliction of A

to sto1,r has the form n;(f,)'i and dim Wtnaa: zir; where ril : deg zc(t).

If zrr(,i): l, then the characteristic pqlynomial of A ih Dhe is i',

so A' :.0 in llh,r and IJhe € lllor. If zdi) + l, the characteristic

polynomial of the restriction of. A to Wtn4 is not divisible by I' so

this restriction is an isomorphism. Hence [Jtop,:\JtnilA: "', so

Wtx&t E lltrr. Thus Xna+rlltna e E IIts. Since m - !Ito, O $Jtu :

lJh,, O Xoa#r$Jtn6e, it follows that IJtor : IJtr,r, Iltre : Ela+^lltoar. In

particular, we see that the Fitting null component is lthe charac'

teristic space of the characteristic root 0 of ,4 and dirh lltoe is the

multiplicity of the root 0 in the characteristic polynornial of. A,

We shall now extend these results to nilpotent Lie algebras of

linear transformations. It would suffice to consider the primary

decomposition since the result on the Fitting decompdsition could

be deduced from it. However, the Fitting decomposit[on is appli'

cable in other situations (e.g., vector spaces over division rings)'

so it seems to be of interest to treat this separately. We shall

need two important commutation formulas. Let lI be an associative

algebra, a. an elertrent of lI and consider the innef derivation

Dlx->tct= [rc] in lI. We write xtt-(vtk-rt) ', xrc): iv Then one

proves directly by induction the following formulas: t

( 5 )

( 6 )

xah : akr * (l)r-'-' * (t)ao-'r" + "' + xtk' .

'We apply this to linear transformations to prove

Lpuru.r, 1. Let A and. B be linear transformations in a'fi'nite'd'im'en'

sional aector sfage. Sufrbose there exists a bosi.ti'ae intpger N such

that l. . .LlB'AlAl . . . al : O. Then the Fitting componefu.ls fioe, fire

of 'm rebti'ue to A are i.naari.ant under B.Proof : Suppose xAn:0. Then f .or k -N*m-L

rBAh - ,(eon. (f )or'"'+ "' * (r1 ,)ro-"*'r'r'-")

- 0 '

H e n c e x B e l J h r . N e x t l e t r € I l l , r . I f f i s t h e i n t e g e r u s e d i n

II. SOLVABLE AND NILPOTENT LIE ALGEBRAS 39

of Fitting's lemma, then we can write tt = yA'*n-t-the proofThen

+ . . . * ( ;1; t ) r ' ' - "o ' ) er I tA '0: nre

xB -- r4t+nts - t(ae,*Ir-r - ( .

i - r)B' A'**-'

Hence IftroB E[ftrt.

Tsponpu 4. Izt 8, be a nilpotent Lie algebra of linear transfor'

mations in a f,nite'dimensional aector space fll and let Tfto: fl ae8[loe,Wt1 : O L'Ilt(8*)c. Then the tfti are inaariant under 8 (that is, under

eoery B e 8) and Wt - mo CI9]tt. Moreouer; fi1 : XeeB!]tra.Proofz Suppose first that IJt : IJtoe for every A e 8. Then

Tltla:0 for all A and XUlr,r : 0. Also by Engel's theorem (8*)t :0

for some N. Hence Dtr : O L'fi(8*)t : 0. Thus the result holds

in this case. We shall now use induction on dim llt and we may

assume Wtol * Ilt for some A. By Lemma 1, the B e I map lltor

into itself; hence 8 induces a nilpotent Lie algebra of linear transfor-mations in St : l}toe c lll. We can write gt : Ilo O ltr where lto :

11 ,6gTtor, Itr : fl tlt(l!*)d : IaeBTtra "rd

9to and llr are invariantunder 8. Then Tft - tto O tt' O Ultr* It is clear from the definitionsthat llo: Illo: naeglDtoa and Il, * Ilt,e S.Xaegllt,r g nrDt(8*)d.On the other hand, by Engel's theorem the atgebra induced by 8*

in llto is nilpotent so that we have an N such that llt(8*)t :0.

Then

tn(B*)t: Dto(8*)'+ Il,(8*)t + IJl,n(8*)'g !t,(8*)' * rll,r(8*) g Tt' * !)1,,r .

Hence Ilr * Dlrr: Xleegtlt,a: 6rDt(8*)i and the theorem is proved.Our discussion of the primary decomposition will be based on a

criterion for multiple factors of polynomials. Let Ol]1 denote thepolynomial ring in the indeterminate i with coefficients in O. \{edefine a sequence of linear mappings Do - l, Dt, Dz, .'. in O[i] asfollows: D; is the linear mapping in @[,1] whose effect on the ele-ment ^i in the basis (1, l, f,',...) for 0[t] over @ is given by

f,tDr -( 7 ) ( "

where we take (l) :0 if i Thus we have AiDr: i l i - t , so

LIE ALGEBRAS

that D, is the usual formal derivation in All'l. Also ^i Do, -

j ( j - 1) . . . ( j - i + l ) i ' -d : i l ( to) i l -n - i lAiD;. Hence i f thecharac-

teristic is zero, then D.i : $lit)D6t If d(f) e O[i] we' shall write

$e = 6Dr. Then we have the Leibniz rule:k

@0* - E 6;Q*-r .I = O

( 8 )

It suffices to check this for 6 : f,i, Q : f,i' in theThen

basis (1, ,1 , ^ ' , " ' ) .

+ Qr@)r" + "' * dttr" ' ,

transformations in a t fi'nite'dimen'

*, onp*-,: * (t,) (u' - ;)i l- ' ]o' -o*'

: (t (t)G'- o))^'."'-r 'Since >f=r?)G|n) : (t+t'\ the above reduces to (r+l')fi|'-h : (li+t'rr.

Hence (8) is valid.We can now prove

Luurrr.e, 2. If q(l)o*'I p(t), then T()') | p;(i), i : 0,t,2, 1" , k.Proof: This is true f.ot k: 0 since ps(t) : p('i). \iVrite p(^) --

d(f),r(i) where TklO and vl{. Then we mav assurrie that 41p1,

V l f u , j : 0 , ! , 2 , . . . , k - 1 , a n d I l * 0 . T h e n n l @ * ) * = p x b y ( 8 ) .

Let r/(i) - I,;l,--sat,lft and multiply (5) bV d'rc and suml on &. Thisglves

( 9 ) x Q @ ) - < P ( a ) r + Q , @ ) r '

which we shall use to establish

Lpuun 3. Let A, B be li,nPar

sional aector space satisfying B'Nt : t. .-1Bm:iJ E 0 for some

N. Let AD be a fotlnomial and let T[tp,t: {ylyp(A)^',:0 for some

mj. Then lltpe as inuariant under B.

Proof: Let y € ffi*, and suppose ys(A)^ --0. Setl z(i) : P(1)*'

{ (^) :qQ)*: p(^)*N. Then,.by (9),

B* (A) - 4 , (A)B + ,P, (A)B ' + ' ' ' * *n - , (A)Bq- l ) .

By Lemma 2, u1Q)l{{1), i = t't - t-and yBlt(A) = 0. Hence YB e Wno.

Hence yhi@) -P, i = N- 1,

Tnponpu 5. Iat 8, be a nilpotent Lie algebra of

mations i,n a' fi.nite'dimensi.onal aector space tn'Iiloear transfor-

Then we ca.n

il. SOLVABLE AND NILPOTENT LIE ALGEBRAS 4I

decompose [!t: ![t, o!]t, o ... o IIt, where TJ?i is inaariant under B

and. the mini,mum potynomial of the restriction of eaery A e 8' to TIti

i.s a prirne pou)er.

Proof: If the minimum polynomial of every 4 e 8 is a prime

power, then there is nothing to prove. Otherwise' w€ have an

,4 e 8 such that ![t - IJt"r,a O ''' O ffinre, tti,: r+(l) irreducible,

Wtone * 0, s ) 1. By Lemma 3, IftnreB ?Wn# for every B e 8.

This permits us to complete the proof by induction on dim fi.

If the base field is algebraically closed, then the minimum poly-

nomials in the subspaces IItr dr€ of the form (^ - a(A);tir, A € 8.

Setting Z;(A) - A.- a.(A) for the sPac€ Dt; we see that Zi(A) is a

nilpotent transformation in IJI;. Hence every A e 8, is a scalarplus a nilpotent in lJti. We therefore have the following

Conor,urnv (Zassenhaus). If 8, is a nilpotent Lie algebra of linear

transformati,ons in a fi,nite-dimensi.anal aector space oaer afi alge'

braically closed field, then the space lJt can be decomPosed, ts

fi, O tjt. O . . . O T!t, where the Ylt; are inaariant subspa.ces such that

the restriction of eaery A e 8 to IJti is a scalar plus a nilpotent linear

transformation.Consider a decomposition of IJt as in Theorem 5. For each i and

each A we may associate the prime polynomial ne(]) such that

nin(f,)kia is the minimum polynomial of A restricted to lltr. Thenthe mapping tra: A -> nu(f,) is a primary function for 8 in the senseof the following

DonNtnoN 3. Let 8 be a Lie algebra of linear transformationsin IJt. Then a mapping n: A-+rt(A), A in 8, rt()), a prime poly-

nomial with leading coefficient one, is called a primary functi,on of.!)l for 8 if there exists a non-zero vector r such that xn/A)^'"e' - Ofor every ,4 e 8. The set of vectors (zero included) satisfying thiscondition is a subspace called the primary component W" corre'sponding to tt.

Using this terminology we may say that the space llti in Theorem5 is contained in the primary component Dt,,6. By adding togethercertain of the IJtr, if necessary, we may suppose that if i + j, thentt;i A + z;,r(l) and n7:. A --> ntt(A) determined by fi; and fiy aredistinct. We shall now show that in this case IJtr coincides withIJto, and the n are the only primary functions. Thus let r A+rla(l)be a primary function and let x € lll,,. Assume n * r;, so that

42 LIE ALGEBRAS l

there exists A; e Ssuch that reo(]) + tt;,t{}). Write x: lr+ "' { rrt

h e T)ft* Since x e t[Itn there exists a positive integer fe such that

0 : Xr , t t (A i )^ : Xgc t r (A ; )^ + . . . 1y , r1n(A; )^ ' ,

Since the decomposition Yt - Dt' 0 .'' O Un' is direct, we have

ranar(Ai)* : Q and since TrAr(^) * tr;an(]) this implies ' rc;:0. It

f o l l o w s t h a t i f n * r i f o r i : 1 , 2 , , . - , f , t h e n r : 0 a n d t h i s c o n t r a -

dicts the definition of a primary function. Thus thei zt are the

only primary functions. The same argument shows that if tc etlll,,t,

then x € lJtii hence IJti : SJt,,c.The argument just given was based on the following itwo proper-

ties of the decomposition: Ifti tr TJtn, wher€ rct iS a prim{ry function

and ni * rt if. i + j. The existence of such a decomposition is a

consequence of Theorem 5. Hence we have the following

Tnponpu 6. If g is a nil4otent Lie algebra of linear:transforma'

tions in a finite-d.imensi.onal aector space W, then B has Qnly a finite

nurnber of primary functions. The corresponding primary compo'

nents are subrnod.ules and. Tft is a direct sum of these. Moreouer, if

!n - IJt ,o lJtr@.. .@Dt, . is a,n! decofnpoTit ion of 'masadirect sum

of subspac7s Iftt + 0 inuariant under B such that (L) for' each i the

m,inimuru fobnomi.al of the restriction of euery A e 8l to TIt; is a

pri.me power ru())^io and (2) if i + i, there eri'sts an 'F such that

r ;A)) * rc ia()) , then the mappings A'n;o()) , i :L,2, "r , r , are the

primary functions and the TIt; are the corresponding prtimary cofn'

ponents.

It is easy to establish the relation between the primary decom'

position and the Fitting decomposition: IJto is the primary component

ln^ it A--+ t is a primary function and is 0 otherwise; mt is the

sum of the primary components IJt,,, r * ),. We leaVe it to the

reader to verify this. '

We shalt now assume that the nilpotent Lie algebra 8 of linear

transformations has the property that the characteristic roots of

every A e g are in the base field. A Lie algebra of linear transfor-

mations having this property will be called split. Evidently, if the

base field @ is algebraically closed, then any Lie algebra of linear

transformations in a finite-dimensional vector space ovdt A is split.

Now let 8 be nilpotent and split. It is clear that the characteristic

polynomial of the restriction of A to the primary component corre-

sponding to A-rn(l) is of the form ttn())'. Since thi$ is a factor

il. SOLVABLE AND NILPOTENT LIE ALGEBRAS 43

of the characteristic polynomial of. A and zrr(t) is irreducible, our

hypothesis implies that no}) - ] - d.(A), a(A) e A. Under these

ciriumstances it is natural to replace the mapping A -+ z,r(,1) by

the mapping A -+ a(A) of B into O and to formulate the following

Dpr.rNruoN 4. Let 8 be a Lie algebra of linear transformations

in fi. Then a mappin g a: A --+ a(A) of 8 into the base field O is

called a weight of IJt for 8 if there exists a non-zero vector r such

ttult r('A _ a(A}I)mx,A :0 for all .A e 8. The set of vectors (zero

included) satisfying this condition is a subspace fllt" called the

wei.ght sface of llt corresponding to the weight a'

Theorem 6 specializes to the following result on weights and

weight spaces.

Tnronpu 7. Let I be a s|li,t nilpotent Lie algebra of linear

transformations in a rt,nite'd.imensional aector space. Then t has

only a fi,ni.te number of distinct weights, the weight spaces are

submodules, and. m l's a direct surn af these. Moreouer, let

m - !lt, o llt, o . . . o Tft, be any decomposition of tm into subspaces

llh ;e 0 inuariant under B such that (l) for each i, the restriction of

any A e g has onl! one characteri.stic root at(A) (with a certain mult-

ipilctty) in Tlti and (2) if i + i, then there erists an A e 8 such that

in@) + a{A). Then the mabfings A -. a;(A) are the wei'ghts and

the spaces Tft;, ara the weight spaces.

5. Lie algebras with semi'simple enueloping

assoeiatioe algebraa

Our main result in this section gives the structure of a Lie alge'

bra 8 of linear transformations whose enveloping associative alge'

bra 8* is semi-simple. In the next section the proof of Lie's theo'

rems will also be based on this result. For all of this we shall

have to assume that the characteristic is 0. we recall that the

trace of. a linear transformation A in a finite'dimensional vector

space, which is defined to be >ifau, for any matrix (a;) of .4, is

the sum of the characteristic roots Pb 'i' : !, ' ' ' t fr, of A' Also

tr Ak : Z?=rpt. lf. A is nitpotent all the p; "r€

0, so tr Ab :0,

h :1,2, . . .. If the characteristic of O is 0, the converse holds:

I f t r Ak :0, k :1,2, " ' , then A is ni lpotent ' Thus we have

The formulas (Newton's identitiesi cf.110) expressing Xpi in terms of the el-

1p t - 0 , k : L ,2 ,Jacobson, fzl, vol. I, p.

LIE ALGEBRAS

ementary symmetric functions of the p; show that [f the char-acteristic is 0 our conditions imply that all the elementdry symmet-ric functions of the p;

"te 0, so all the p; are 0. flence A is

nilpotent. We use this result in the following ]

Lslrul 4. Let C e 8,, the algebra of linear transforrh.ati,ons in artnile'dimensi.onal uectar sfuce oaer a field of characteri.stic 0 ands u p p o s e C : X i [ A i B r , l , A r , B i e O a n d l C A i l = 0 , i * L , 2 , . . . , r .Then C is nilpotent.

Proo f : We have lCk- 'A ; l :0 , k :L ,2 , . . . (Co - 1 ) . iHence Ck:ZiCk-'(AtBn - B;A): Xr(A;(r*-tB;) - 1g*-rg)A): )i lAi, ,*-tBil.Since the trace of any commutator is 0, this gives tr Ce - 0 forh: L,2, . . - . Hence C is ni lpotent.

The key result for the present considerations is the following

Tnponpu 8. Let 8, be a Lie algebra of linear transf$rmations ina finite-dimensional uector space oaer a fi.eld of char',acteristic 0.Assume that the enueloping associ.atiae algebra 8* is semi-simfle.Then 8 :8' @6 where 6, is the center of 8, and 8,, is an ideal of8 which is semi-simple (as a Lie algebra).

Proof: Let 6 be the radical of 8. We sho* first ithat 6 : 6the center of 8. Otherwis€, 6, : [89] is a non-zero solvable ideal.Suppose 6{u) : 0, 6lo-t) + 0 and set 6z: glu-t', 6s F [g;r8]. IfC € 6 ' , C : E l A i B ; . ] , A i € 6 2 , B i € 8 , a n d [ C A r ] : 0 s l n c e 9 ' t r € zand 6, is abelian. Hence, by Lemma 4, C is nilpofent. Thusevery element of the ideal €a of I is nilpotent; hencd, by Theo-rem 2,6g is in the radicat of 8*. Since 8n is semi-sinple, €s - Q.Since €s : [gr8] this shows that 6, tr 6. Since 6r E [86] g 8'every element C of. 6' has the form l,[A;B;1, Ai, Bi f 8, and wehave [CA;l: 0, since 6s g 6. The argument just used limplies that6z:0, which contradicts our original assumption that €[ : [89] + 0.Hence we have [89] - 0, and 6 : O. The argument wb have usedtwice can be applied to conclude also that 6 n 8' : 0. I Hence wecan find a subspace 81 3 ,8' such that I : I, O 6. Slnce 8, con-tains 8', 8, is an ideal. Also 8r:8/6 : 8/6 is semi-sitnple. Thisconcludes the proof.

Remark: In the next chapter we shall show that 8r e 8', so weshall have I : 8'€) G, 8' semi-simple.

Conorulnv 1. I-et I be as in the theorem, g* serni-s;npie. Then8, i,s solaable if and only if 8 ds abelian. More generflly, if 8 r's


soluable and ft is the rad.ical af g*, then ll*lm ls a commutatiae

algebra.Proof: If I is abelian it is solvable, and if 13 is solvable and 8*

is semi-simple, then I : G O 8, where 8, is semi'simple' Since the

only Lie atgebra which is solvable and semi-simple is 0 we have

8r :0 and I : O is abelian. To prove the second statement it is

convenient to change the point of view slightly and consider I as

a subalgebra of ?[r, U a finite'dimensional associative algebra over

a field of characteristic 0. since any such 1l can be considered as

a subalgebra of an associative algebra @ of tinear transformations

in a finite-dimensional vector space, Theorem 8 is applicable to I

and lI. We now assume I solvable, so (S + m)m, which is a

homomorphic image of 8, is solvable. Moreover, the enveloping

associative algebra of this Lie algebra is th.e semi-simple associative

algebra 8*/n, Hence (S + lt)/lt is abelian. This implies that

ta1 ml (b + n) - (b + S) (a + S) for any a,b e 8,. Since the cosets

a * ft generate 8*/n, it follows that 3x/S is commutative.

conou,.lny 2. Let g be a Li,e algebra of li,near transformati'ons i,n

a fi,nite-d,imensional uector space oaer a field 9f characteristi'c 0, let

6 be the radical of g and, ft the radical of 8*. Then I n S is the

totality of nit\otent elements of 6 and tgSl g ft'

Proof:, Since S is associative nilpotent it is Lie pilpotent' Hence

I n f tg6 . Moreover , thee lementso f Saren i lpo ten t so I nSt r9o

the set of nilpotent elements of g. If So denotes the radical of

the enveloping associative algebra @* then, by Corollary L,6*/So

is commutative. Now any nilpotent element of a commutative

algebra generates a nilqotent ideal and so belongs to the radical.

Since g,*/ffi0 is semi-simple, it has no non-zero nilpotent elements.

It follows that fro is the set of nilpotent elements of 6* and so

6o : fro 0 6 is a subspace of 8. Next consider (8 + S)/S. The

enveloping associative algebra of this Lie algebra is 8*/m, which is

semi-simple- Hence the radical of (8 + m)& is contained in its center'

Since (g + ft)ffi is a solvable ideat i. (8 * fr)/ft we must have

l(6 + S)/m, (s + mynl : 0, which means that t68l g S. Hence

tgsl g I n S E 60. Then [908] S 6o and 6o is an ideal in 8. Since

its elements are nilpotent, 6o S S by Theorem 2. Hence 6o I I n S

and 6o : I n S, which completes the proof.

There is a more useful formulation of Theorem 8 in which the

hypothesis on the structure of 8* is replaced by one on the action

46 LIE ALGEBRAS

of I on Dl. In order to give this we need to recall sdme standardnotions on sets of linear transformations. Let 2 be a set of lineartransformations in a finite-dimensional vector space IJI over a fieldO. We recall that the collection L(t) of subspace$ which areinvariant under 2 (nA=n,A eJ) is a sublattice of the lattice ofsubspaces of llt. We refer to the elements of. L(t) aE Z-subspacesof llt. If Il is a subspace then the collection of A e1E such thatnA E Tt is a subalgebra of @. It follows that if 9t e L(2), then 9tis invariant under every element of the envelopingi associativealgebra .E* of J and under every element of the enveloping alge-bra ^tt. Thus we see that L(E): L(2*) - L(lt).

The set J is called an irreducible set of linear transformationsand !]t is called J-irreducible it L(2) - {D[, 0] and IJt + 0. J isind,ecomposable and lll is J-indecomposable if there exisits no decom-position tlt - Dt, O IJtz where the IJL are non-zero elemEnts of L(2),Of course, irreducibility implies indecomposability. , J (and lytrelative to J) is called completely red,ucible if !n - > m[, Dto € Ut)and llt" irreducible. We recall the following well-knolwn result.

Tnronnu 9. 2 is completely reducible if and only if iL(Z) is com-plemented, that i,s, for eaery slte L(2) there exists anll'e L(E)suchthat Tft': ItOtt'. If the condition holds thenTft: IIlr OIJt, O . . . Ofi,where the Tfti, e L(2) and are ineducible.

Proof: Assume m - ! l)la, Tft, irreducible in L(J) and letIt e l,(J). If dim It : dim fi, Il : fi and IJf : It @'0. We nowsuppose dim It < dim !)l and we may assume the theoFem for sub-spaces !t, such that dim It, ) dim It. Since It c Dl : [ !]la there isan llt" such that Tft,EIt. Consider the subspace Dd" n 9t. Thisis a J-subspace of the irreducible J-space Tllr. Hence eitherI l l o f l I l =Dt " o r I J l "O I l :0 . I f m"OI l : ! I l r , f tS f i " con t ra ry toassumpt ion. Hence l l l ,n I l :0 and l t r : l t * I I fa : f te)Dt" . Wecan now apply the induction hypothesis to conclude that Un -

tt ,Oytl , Yt i e L(J). Then tJt -nel] t"Ott l -ytOTtl where Tt ' :!D1" O ni e L(z). Conversely, assume L(E) is completnented. LetItrt, be a minimal element + 0 of. L(2). (Such elemen{s exist sincedim l]t is finite.) Then we have m - Ilt' O It where ltle Z(t). Wenote now that the condition assumed for St carrie$ over to St.Thus let S be a J-subspace of !t. Then we can writelllt : POP"$' e l(g). Then, by Dedekind's modular [aw, $t : !]t rt 9t :

S + (S' n It) since lt = tS. If we set !p" - S' n !t, thbn $" n S:

il. soLvABLE AND NILPOTENTI LIE ALGEBRAS 47

S n S ' n I l - 0 . H e n c e ! t : $ O S " w h e r e S " € L ( 2 ) a n d $ " 9 T t ''we can now repeat for It the step taken for Ilt; that is, we can

w r i t e l t : f i r O $ w h e r e l J t s , P e L ( ' ) a n d l J t e i s i r r e d u c i b l e 'Continuing in this way we obtain, because of the finiteness of

dimensionality, that tn - m, O Dt, @ ' ' ' (E fi", Dti irreducible'

fir € L(r). This completes the Proof'we suppose next that J - lI is a subalgebra of the associative

algebra ri (possibty not containing 1), and we obtain the following

necessary condition for complete reducibility'

Tnponpu 10. If \I is an associatiae algebra of linear transforma'

tions i,n a f.nite-dimensi,onal aector space, then lI completely reducible

implies that \' is semi'simfle.-Proof: Let S be the radical of ?I and suppose tlt - ) Dlta, 'It" e L(U)

and irreducible. Consider the subspace Dtdn spanned by the vectors

of the form tN, t e IIlr, N e ffi. This is an ?I-subspace contained

in !Jt". Since ft&: 0 for some ft we must have tltd'n cIIl' (cf' the

proof of Theorem t). Since SJt" is irreducible we can conclude that

mrn -- 0 for every !Itr. since !n - xut" this implies that lltlt: 0,

that is, S :0. Hence l[ is semi'simple'Since ̂ I is completely reducible if and only if 2* and 2I are

completely reducible, we have the

Conor,r,lny. If t i,s completely reducible, then Ex and 2t are semi'

simPle.We shall say that a single linear transformation A is semi'simple

if the minimum polynomial p(],) of. A is a product of distinct prime

polynomials. This condition is equivalent io the condition that {A}l

has no nilpotent elements + 0. Thus if p(r) : n(l)ettr(l)" "'tr,(f,)"

then Z: n{A)rr(A) "' 78,(A) is nilpotent and Z + 0 if some et } L'

conversely, suppose the Tt, ?itf- distinct primes and all 0t: L. Let

Z : 0(A) Ue niipotent. If Z' :0, 6(4)' : 0 and d(,i)" is divisible

by pQ). Hencg d(,i) is divisible bv p(i) and Z - A(A) - 0' We

can now prove

Tnponnu 11. Let I be a completely reducible Lie algebra of

linear transformations in a fi.nite'dimensional uector space ouer a

field, of characteristic 0. Then I : G O I' where 6' ,s the center

and g, is a semi-sim\le ideal. Moreouer, the elements of E' are semi'

simflq.Proof: If t3 is completely reducible, 8* is Hence

48 LIE ALGEBRAS

the statement about 8 : G O 8, follows from TheorLm 8. Nowsuppose that 6 contains a C which is not semi-simple.J Then {C}tcontains a non-zero nilpotent N. Moreover, {C}t i$ the set ofpolynomials in C, so that N is a polynomial in C. Hence N is inthe center of 8t. Since this is the case, N8t : 8tN is an ideal in81. Moreover, (N8t)* g N&8t : 0 if ft is sufficiently llarge. SinceN e N81, N8t is a non-zero nilpotent ideal in 8t corltrary to thesemi-simplicity of 81.

We shall show later (Theorem 3.10) that the conYerse of thisresult holds, so the conditions given here are neces$ary and suf-ficient for complete reducibility of a Lie algebra of linpar transfor-mations in the characteristic zero case. Nothing like this can holdfor characteristic p + Q (cf.. S 6.3). We remark also that the con-verse of Theorem 10 is valid too. For the associative case onetherefore has a simple necessary and sufficient condition for com'plete reducibility, valid for any characteristic of the field. Theconverse of Theorem 10 is considerably deeper thani the theoremitself. This will not play an essential role in the sequel.

6. I'ie's theorems

We need to recall the notion of a composition serie$ for a set J

of linear transformations and its meaning in terms lof matrices.We recall that a chain ffi - 9Jt, ) [ft2) "' r Dt, r fir*, :0 of J-

subspaces is a composition series for IJt relative to 2 if for everyi there exists no llt' e L(t) such that Dti ) m' r lJtr+r. If It is a J-subspace of IJt, then J induces a set E of linear tralnsformationsin fi/It. As is well known (and easy to see) for groups with oper-ators the i-subspaces of !]t - Wtll,t have the form S/![ where S isa J-subspace of tlt containing Yt. It follows that lthere existsno P e Z(.x) with IJI > p = It if and only if llt/It is f-irreducible.Hence tJ t - I l t r )T f tz ) . . . - ) IJ lea l :0 , D t , eL( t ) , i s a compos i t ion

series if and only if every IJln/IJtr*, is i;-irreducible, 5; the set ofinduced transformations in lJtrllJti+r determined by the A e 2.

The finiteness of dimensionality of TJt assures the efistence of acomposition series. Thus let llts be a maximal J'subsface properly

contained in IJtr : IJt. Then IJt, ) IJlz and fi'/TJtz i$ irreducible.Next let IJt, be a maximal invariant subspace of fi2, * I)ts, €tc.

This leads to a composition series IJt, :IJt, r IJts r "' = F' : TJtr*' :0

for St.

I I . S O L V A B L E A N D N I L P O T E N T L I E A L G E B R A S 4 9

Let l l l -U f t , I1 l t r r I ] t s3 " ' : : I l t ' * r :0 bea descend ing cha ino f

J-subspaces and let (er, . . .,0r) be a basis for TJt such that (et, " ' , en')

is a basis for )ltr, (er, . . .,0n1+n2) is a basis for Tjte-r, etC. Then it

is clear that if A e 2, the matrix of A relative to (e;) is of the

form

M,

, M z(10)

(12)

0-ltII

* MrJIt

M _

Thus we have

( 1 1 ) € n 1 + . . . + n 1 - p e A : + u f ) , e n r * . . . * n 1 - 1 + t * x ,

k , l : L , " ' , t l i ,

where r is a vector in fir-ia2. Hence we have (10) with Mt:

Qt f l l . The cosets 7n1+. . .+nJ-r+r = znt+. . -+*J-1+r * l l ts - r+sr k : L , " ' ,? l i

form a basis for the factor space 1]tr-r+,/Ilts-r'+r zlod (11) gives

which shows that the matrix of the linear transformation 24. induced

in Ijts-r+r IAU-i+z is Mi.We shall now see what can be said about solvable Lie algebras

of linear transformations in a finite'dimensional vector space over

an algebraically closed field of characteristic zero. For this we

need the following

Lpuu.r, 5. Let g be an abelian Lie algebra of linear transformations

in the fi,nite-dimensional space tJt oaer O, O algebraically closed'

Suppose !)t ls g-irreducible. Then IJt ds one-dimensional.

Prod: If A e g, A has a non-zero characteristic vectot x. Thus

xA: a.x, d e O. Now let lft" be the set of vectors y satisfying

this equation. Then if Be8 and y€!It ' , (yB)A: yAB: eyB. Hence

yBe\!tr. This shows that ')to is invariant under eYery Be 8'-sitr." fi is irreducible, fi : Dt", which means that A: aL in fi.

Now this holds for every A e 8. It follows that any subspace of

Dt is an 8-subspace. Since TJt is 8-irreducible, l)t has no subspaces

other than itself and 0. Hence lJt is one-dimensional.

50 LIE ALGEBRAS

We can now prove

Lie's theorem. If 8 is a solaable Lie algebra of lintfor transfor-mations in a finite-dimensional uector space Tft ouer an filgebraicallyclosed feld of characteristic 0, then the matrices of 8, dan be takenin simultaneous triangular form.

Proof: Let llt - tlt' I IJtz 3 . -. I l)t,+r :0 be a composition seriesfor Dt relative to 8. Let E; denote the set of induced liinear trans-formations in the irreducible space TItJ)!l;*r. Then Er iS a solvableLie algebra of linear transformations since it is a hpmomorphicimage of 8. Also Er is irreducible, hence completely reducible inIJt,/IJln*,. Hence Er is abelian by Theorem 11. It follo*s from thelemma that dimlJtr/Ij?;*r - 1. This means that if we use a basiscorresponding to the composition series then the matrices -1lr in(10) are one-rowed. Hence every M corresponding to the A e 8 istriangular.

If 8 is nilpotent, it is solvable, so Lie's theorem is applicable.We observe also that Lemma 5 and, consequently, Lie's theoremare valid for any field of characteristic 0, provided the cl,aracteristicroots of the linear transformations belong to the field. We havecalled such Lie algebras of linear transformations split. If wecombine this extension of Lie's theorem with Theorem )7 we obtain

Tsponpu 12. Let L be a split nilpotent Lie algebra of [,inear trans'

formations in a finite-dimensional uector space IIt oue* a rteM ofcharacteristic 0. Then Wt is a direct sum of its weight spaces Wlaand the matrices in the weight space Dlln can be taken shnultaneouslyin the form

(13)

a(A)

Proof: The fact that Dt is a direct sum of the weight spacesWt, has been proved before. We have x'(A - a(A)l)* : g forevery ra in llt", which implies that the only characteriistic root ofA in IJl, is a(A). Since the diagonal elements in (10) are charac'teristic roots it follows that the Mi in (10) for A are al(A). Hencewe have the form (13) for the matrix of A in Tlt". I

:lA - : [ ' . '


The proof of this result (or the form of (13)) shows that in Dlo

there exists a non'zero vector .r such that rA: a(A)x, A e 8,- If

A, B € 8 we have x(A * B): (a(A) * a(B))x: a(A * B)x, x(pA) -

pu(A)x : a(p(A))x, p e A, and xlABl - (a(A)a(B) - a(B)a(A))x -

a(lAB))x. The first two of these equations imply that a is a linear

function on 8. Since every element of the derived algebra 8' is a

linear combination of elements [AB], the linearity and the last

condition imply that a(C): g, C e 8'. We therefore have the

following important consequence of Theorem 12.

Conorr,nnv. Under the same assumpti'ons as in Theorent' L2, the

weights a: A -, a(A) are linear functions on 8, which uanish on 8' -

7. Applications to abstraet Lie algebras.Some eounter examples

As usual, the results we have derived for Lie algebras of lineartransformations apply to abstract Lie algebras on considering theset of linear transformations ad 8. We state two of the resultswhich can be obtained in this way.

Tnponsu 13. Let I be a finite-dimensional Lie algebra auer a

field of characteristi,c 0,6 the radical, and Tt the nil radical. Then

[8s] g Yt.Proof: By Corollary 2 to Theorem 8, [ad I, ad 9] is contained

in the radical of (ad 8)*. This implies that there exists an integer.A/ such that for any N transformations of the form fad a;, ad srl,a ie 8 , s ; € 6 , we have ladar , ads t ] Ladar , adsz l . . . lad a1y ,?dsr ] :0 .

Hence ad [ars,] ad lazsz] .''ad [assn'] : 6. Thus for any r € 8 wehave 1... l lxlars,l l[azszll . . . ,Iars*]J : 0. This implies that [68]rv+r -

0, so [68] is nilpotent. Since this is an ideal, [98] g tt.

Conoruny 1. The deriued algebra of any finite-dimensional soluableLie algebra of characteri,stic 0 is nilpotent.

Proof: Take 8 : 6 in the theorem.

Conoruo.ny 2. Let 6 be solaable finite-dimensional of characteristic0, It the nil radical of 6. Then 6D = rft for any deriaation D ofTE\

Proof: Let I - gO O D the splitextension of. OD byG (cf. $1.5).Here g is an ideal and [s, DJ : sD for s e 6. Since 8/6 is one-

52 LIE ALGEBRAS

dimensional, it is solvable. Hence 8' is a nilpoten{ ideal in 8.

On the other hand, g' : @' + 6D, so g' + @D is a nllpotent idealin I and €/ + eD g tl. Hence gD = ,t.

Tnponpu L4 (Lie). Let I be a fi,nite.d,imensionali solaable Li,e

algebra oaer an algebraically closed fi.eld of characterktic 0. Then

there ex is ts a cha in o f idea ls 8 : ,8 , )8o- r r " ' : ,8 t ]0 such tha t

dim 8; : i.Proof: Since I is solvable, ad I is a solvable Lle algebra of

linear transformations acting in the vector space 8. Let I -

g, =,82 r . . . r,8r+r : 0 tle a composition series of I rel6tive to ad 8.

Then 8; ad 8 s 8i is equivalent to the statement that $; is an ideal.

By the proof of Lie's theorem we have that dim 8r/8r+l : 1. Hence

the composition series provides a chain of ideals of the type re-

quired.We shall now show that the assumption that the bharacteristic

is 0 is essential in the results of $$ 5 and 6. We begin our con-

struction of counter examples in the characteristic p + 0 case with

a P-dimensional vector space lJt ovel O of. characteristic 1. Let E

and F be the linear transformations in Dt whose effdct on a basis(er, er, " 'r op) is given bY

e ; E : € i + r t i < p - 1 , f

(14) gpE: €1 ,

o;F : (i _ L)e; .

T h e n e i E F : i z i + r , i < P - \ e p E F : A a n d e i F E : ( i - I Y i * t ,

erFE- -€r. Hence

e{EFl-_ €i+t , epLEFl: e, ,

so that we have tEFl - E. Hence 8solvable Lie algebra. We assertThus let It be an 8'subspace + 0Then

: OE + OF is a two'dimensionalthat I acts irreducibly in 1]t.and le t r :XEp l+0 be in f t .

x : E & r * E z e z + . . . + E * o e T , trF : IEzez * ... + (f - L)Epo € Tt

xFz : l"Erer+ " ' + (p - I)28*o eft

rFn-t : Le-tErez+ . .. + (p _ L)o-rEoeo € I{

The Vandermonde determinant

(15)


V _

10 +0 .

If we multiply xFi-t by the cofactor of the (i , i) element of v and

sum on -t we obtain that VE&t € Tt and so E;ei € \l' It follows

from this that Tt contains one of the e;. If we operate on this ei

by the powers of E we obtain avet! €i. Hence Tt = DJt, which

proves irreducibility and shows that Theorem 11 fails for charac'

ieristic p. Also since cornplete reducibility implies semi'simplicity

(actually !* : @ is simple), Theorem 8 also fails for characteristic p.

Next let € denote the two'dimensional non'abelian Lie algebra

oe *Af, Vf l : e, a of character ist ic p. Then e- '8, f -F, g iven

by (14), defines a representation of € acting in llt and llt is an €'

module wi th basis (7r,er, . . . ,€o). Let A be the spl i t extension

.8 : 6 O !lt. Then 9lt is an abelian ideal in A and S/llt =' € is

solvable. Hence S is solvable. The derived algebra .R' :

O e + 1 1 ) 1 9 J l : A e + m . S i n c e [ U l e ] - f r l E : m , ( $ ' ) ' : ( A ' ) ' : " ' - m '

Hence .R' is not nilpotent. This shows that also Theorem 13 fails

for characteristic P.It is still conceivable that Lie's theorem might hold for charac'

teristic p if one replaces the word "solvable" by "nilpotent."

This would imply that we have the nice canonical'form (13) for

nilpotent Lie algebras. However, this is not the case either.

Thus let E and F be as before. Then EF - FE: E implies that

E(FE- ' ) - (FE- ' )E - 1. Set G : FE- ' , H: l , Then we have

lEGl -- H, l,EHl : 0 : tGHl .

1

(16)

(17)

Hence \l: AE + oG * AH is a nilpotent Lie algebra. On the

other hand, Tl* : 8*(! : AE + AF) is irreducible even if the base

field is algebraically closed. Hence the matrices cannot be put in

simultaneous triangular form. It X : EE + IG + Cl one can verify

tha t (X - (E +C) l ) ' : 0 i f p # 2 . Hence the mapp ing X- ' { + Cis the only weight for !t. Note that this is l inear. (Cf. Exercise 24,

Chap. V.)

Exereises

In these exercises all algebras and vector spaces are finite'dimensional.

l. Let O be algebraically closed of characteristic 0 and let I be a solvable

54 LIE ALGEBRAS

subalgebra of nr. dimensions of the Lie algebra I properly containing E.

that E is contained in a subalgebra of. m * I dimensions.

2. @ as in Exercise 1, I solvable. Show that 8 has a basis (er, ee,

Show

. , . r 2 n )

with a multiplication table [epi = Z'*:rmlrer, i < i.3. @ as in Exercise l. Let 8 be a solvable subalgebta of Qnr,' Prove

that dim I S n(n + L)12 and that I can be imbedded in a solvable subalgebra

of n(n + l)12 dimensions. Show that if 8r and 82 ar€ solvable slrbalgebras of

n(n * l)/2 dimensions, then there exists a non-singular matri:t A such that

8z = A-18rr4..4. Show that if 8 is not nilpotent over an algebraically cloped field then

I contains a two-dimensional non-abelian subalgebra.

5. @ algebraically closed, any characteristic. If a is a characteristic root

of a linear transformation 4, the characteristic space W6 = {n"l ryr(A - al)t ='

0 for some &1. We have ![It: X@Ut". Let A be an automorphism in a

non-associative algebra 2I and let 2t : I @ 2[r be the decorn]position of 2[

into characteristic spaces relative to A. Show that

2;r21p - 10 if ap is not a characteristic root;- I S llap if ap is a characteristic root.

Let S be the linear transformation in 2I such that S : al

ft. Show that S is an automorphism which commutes with

S-tA is an automorphism of the form I + Z, Z nilpotent.

6. O as in Exercise 5, I a Lie algebra over A, A an automlorphism in 8.

Let 8 : I e 8" be the decomposition of I iqto characteriStic subspaces

relative to A. Let s3 : U ad 8a where ad gc - {ad nalfra € 8o}.i Show that EB

is weakly closed relative to l4, x A - IABI.7. Q, g, A,8 as in Exercise 6. Assume A is of prime lorder. Show

t h a t i f t h e o n l y o € 8 s u c h t h a t r A : c i s r = 0 t h e n e v e r y l e l e m e n t o f [ B

is nilpotent. Hence prove that if a Lie algebra over an arbi{rary field has

an automorphism of prime order without fixed points except 0, then I is

ni lpotent.8. Let D be a derivation in a non-associative algebra ?I and let 2I: IO2Ic

be the decomposition of !I into characteristic spaces relative to D. Show

that2 1 " 2 1 9 - t o i f ' a + p i s n o t a c h a r a c t e r i s t i c r o o t ;-

I g Ua+o if a * I is a characteristic root.'

Let S be the linear transformation in 2l such that S : al in A, for every(\. Show that S is a derivation which commutes with D and that D - S =

Z is a nilpotent derivation.g. Prove that if I is a Lie algebra over a field of charactgristic 0 and I

has a derivation without non-zero constants, then 8 is nilpote4t.

10. (Dixmier-Lister). show that if 8 is the nilpotent Lie algebra of

Exercise 1.7 and the characteristic is not 2 or 3, then ever$ derivation D

satisfies g,D = 8' and hence is nilpotent. Show that this Lie blgebra cannot

be the derived algebra of any Lie algebra.

inlU, for everyA and, that U=

U. SOLVABLE AND NILPOTENT LIE ALGEBRAS 55

ll. Let 8 be a Lie algebra of linear transformations such that every

4 € t has the form a(.A)L + Z where a(A) € t0 and. z is nilpotent. Prove

that 8 is nilpotent.

The next group of exercises is designed to prove the finiteness of the

derivation tower of any Lie algebra with 0 center. The corresponding result

for finite groups is due to Wielandt. The proof in the Lie algebra case is

due to Schenkman [11. It follows fairly closely Wielandt's proof of the

group result but makes use of several results which are peculiar to the Lie

algebra case. These effect a substantiat simplification and give a final result

which is sharper than that of the group case.

Let I be a Lie algebra with 0 center. Then 8 is isomorphic to the ideal

8r of inner derivations in the derivation algebra 8z : D(8)' We can identify

8 with 8r and so consider I as an ideal in 82. We have seen (Exercise 1.4)

that 8z has 0 center. Hence the process can be repeated and we obtain

8r G 8r :c 8s where 8r is the derivation algebra of S(8), D(D(8)), and 8e is

identified with the ideal of inner derivations of 82. This process leads to a

chain 8=8r ggzgSs I " ' , where each 86 is invariant in 8c+r. Hence

8r :8 is sub.invariant in 8e (Exercise 1.8) and every 8t has 0 center. The

tower theorem states that from a certain point on we must have 8r,.= 8r+r= ' ' '.

This means that if I has 0 center, then for a suitable K O(O("'D(8)"'))

is a complete Lie algebra.Exercise 1.4, 1.8, and 1.9 will be needed in the present discussion. Besides

these all that is required are elementary results and Engel's theorem that if

ad o is nilpotent for every o, then 8 is nilpotent.

12. Prove that every Lie algebra 8 has a decomposition I : 8'* 0 where

8o, = 1i_ir8{ and S is a nilpotent subalgebra.

13. Let g be a Lie algebra and let I be the centralizer of 8'. Show that

if 8 has 0 center, then $ g 8'.

sketch of proof: $ is an ideal. write 8: 8, + s, 0 a nilpotent sub-

algebra. Set 8r :8 + 0, which is a subalgebra since $ is an ideal. Write

8r : 6r + 8f, Sr a nilpotent subalgebra. Then gi E .S and 8 : 8'* 0r. If

Gr is the center of 0r, I n Gt is contained in the center of 8 so that 3 fl Gr =

0. This implies that Sr O I : 0 since 6r O ,8 is an ideal in Sr. Then S =

8fg 9,.

14. Let ?I be subinvariant in I and assume that the centralizer of ?I in

8 is 0. Prove that the centralizer I of ?I' in 8 is contained in ?I'.

Slcetch of proof: Assume I g 2I'. Then, bv Exercise 13,,3 g 2I. I is an

ideal and S=8*?I is a subalgebra; nr?I . The normal izet o f ? I inS

properly contains 2l and contains a ze8, eA. Then p, - @? *?t is a sub'

algebra such that E, = 2Ir". The centralizer of E in E contains z f E-.

Hence, by Exercise 13, E has a non-zero center and so ?I has non'zero

centralizer in I contrary to assumption.

b b LIE ALGEBRAS

15. Let 8 be a Lie algebra, E a subinvariant subalgebra, 9f an ideal in

E. Show that if the centralizer of ?I in E is 0 and the cent(alizer of E in

8 is 0, then the centralizer of ?I in 8 is 0.1 6 . L e t S : 8 r E f J z G 8 s g . . . , 8 t : D ( & - t ) b e t h e t o w e r i o f d e r i v a t i o n

algebras for a Lie algebra with 0 center. Show that the cefitralizer of 8r

in 8.r is 0 and that the centralizer of Si in 8c is contained in Pi. Let C(8i)

be the center of Sfl and D(Sf) the derivation algebra of 8r'. frove that

dim 81 5 dim c(!r) + dim !D(8i)

Hence prove that there exists a rn such thatSclwnkman's d,eriuat'ipn tower theorem.

9 * : 9 r n + r

17. Let I be a Lie algebra of linear transformations in a vector space

over a field of characteristic 0. Let 0 be a subinvariant srhbalgebra of I

such that every K € S is nilpotent. Prove that S is contained in the radical

of 8* (Hint: Use Exercise L.22.)18. Let I be a nilpotent Lie algebra of linear transformatiohs in a vector

space Dl such that lll is a sum of finite-dimensional subsflaces invariantunder 8. Show that IIt : I @ fio where the IJlo are the primary com-ponents corresponding to the primary functions n: ,4 '> r(A). lShow that thefito are invariant. Show that if il is any invariant subspdce, then Il:

I O ft" where Xto = Xt fi 9[,'.

CHAPTER III

Cartan's Criterion and lts Consequences

In this chapter we shall get at the heart of the structure theory

of finite-dimensional Lie algebras of characteristic zero. We shall

obtain the structure of semi-simple algebras of this type' prove

complete reducibility of finite-dimensional representations and prove

the Levi radical splitting theorem. All these results are conse-quences of certain trace criteria for solvability and semi-simplicity.One of these is Cartan's criterion that a finite-dimensional Lie alge-

bra of characteristic 0 is solvable if and only if tr (ad a)' :0 for

every a in g'. Clearly, this is a weakening of Engel's condition-

that ad a is nilpotent for every a e 8/. The method which we

employ to establish this result and others of the same type is

classical and is based on the study of certain nilpotent subalgebras,called Cartan subalgebras. We shall pursue this method further in

the next chapter to obtain the classification of simple Lie algebras

over algebraically closed fields of characteristic 0.

1. Cartan aubalgebras

If E is a subalgebra of a Lie algebra I then the normalizer \t ot

E is the set of r € 8 such that [rE] g E, that is, [rD] € E for every

D e E. It is immediate that Tt is a subalgebra containing E, and E

is an ideal in It. In fact, as in group theory, It is the largestsubalgebra in which E is contained as an ideal. We now give the

following

Dprrxrrror.i 1. A subalgebra $ of. a Lie algebra I is called a

Cartan subalgebra if (1) 6 is nilpotent and (2) 6 is its own normalizerin 8.

Let 6 be a nilpotent subalgebra of a finite-dimensional Lie algebraI and let I : llo O 8, be the Fitting decomposition of 8 relative toadgO. We recall that IJo : {r I r(ad h)o :0, h e $, for some integerhj.

'We can now establish the following criterion.

Pnorosrrrox 1.. Let sl be a nilbotent subalgebra of a fi,nite-dimen'

1 , 5 7 l

LIE ALGEBRAS

sional Lie algebra 8. Then 6 is a Cartan subalgehra:if and onlyif $ coincides with the Fitting component 8o of 8 relatiUe to ad b.

Proof: 'We

note first that .80 3 It, the normalizer of ,b. Thus ifr e ft then lrh) e 6 for any h e 6. Since S is nilpot€nt,

l ' ' ' l rh lh l ' ' ' h l : x(ad h)r : 0

for some ft. Hence r e 80. Hence if tt r 6, then 8o > 6. Nextassume 8o > 6. Now 8o is invariant under ad O and every restric-tion adgoh, h e 6, is nilpotent. Also $ is an invariant subspag: of8o relative to ad O. Hence we obtain an induced Lie Algebra 0 oflinear transformations acting in the non-zero space 8J6. Sincethese transformations are nilpotent, one of the versionB of Engel'stheorem implies that there exists a non-zero vector x *\ 6 such that(r + 6)F : 0. This means that we have txhl e S {or every h;h e n c e r € ! t a n d r F 6 s o t h a t ! t = S . T h u s ! t : O i f a n d o n l yif .80 > S, which is what we wished to prove

PnoposnroN 2. Izt 6 be a nilpotent subalgebra of the finite-dimensional Lie algebra 8 and let 8: ,80 O 8, be the Fltting decom-position of 8, relatiue to ad 6. Then ,80 is a subalgebra bnd [8180] ga, . ;1 .

k

Proofz Leth e Oanda € 80. Then t...noffi lJ *0forsomeh. Hence

[ . . - [ a d a a d h ] a d h l . . - a d h l - 0 .

This relation and Lemma 1 of g 2.4 implies that the Fitting spaces8o

"c o and l]r a<r n, of I relative to ad h are invariant undet' ad a. Since

8o: 0oe6,8o"aa and 8r: Xr,e68r"on, it follows that 8o4da S 8o andSrad asV,-r. Since a is any elLment of 80,[8080]S8o artd [8,80]S8r.

DprrNrrroN 2. An elemeni h e IJ is called, regular if the dimen'sionality of the Fitting null component of I relative to ad h isminimal. If this dimensionality is /, then n - l, where n - dim 8, iscalled the rank of 8.

We have seen that the dimensionality of the Fitting null com'ponent of a linear transformation / is the multiplicity of the root0 of the characteristic polynomial f(l) : det(tl - A) of A. Hence& is regular if and only if the multiplicity of the characteristicroot 0 of. ad h is minimal. Since lhnl - 0 for every ft it is cleartlrr;t ad h is singular for every h. Hence the numtfer / of the

III. CARTAN'S CRITERION AND ITS CONSEQUENCES 59

above definition is > 0. We remark also that I is of rank 0 if

and only if every ad h is nilpotent. By Engel's theorem this holds

if and only if 8 is a nilpotent Lie algebra. Regular elements can

be used to construct Cartan subalgebras, for we have the following

Tnnonnru 1. If g is a finite-dimensionat Lie algebra ot)er an in'

fi,nite field 0 and, a is a regular element of 8, then the Fitting null

cornbonent 6 of g relatiae to ad a is a cartan subalgebra.

Praof: Let I - 6 O S be the Fitting decomposition of I relative

to ad. a. Then, by Proposition 2, 6 is a subalgebra and t6nl g S.

We assert that every ad6b, b e 6, is nilpotent. Otherwise let 6 be

an element of 6 such that ad5} is not nilpotent. We choose a

basis for I which consists of a basis for 6 and a basis for S. Then

the matrix of any ad h, h e S, relative to this basis has the form

(1) $) ,9)\ o (p)l

where (p,) is a matrix of. ad,6h and (pz) is a matrix of adgh. Let

, : (t''J (-,3) , B: (u'; .u,3) 'respectively be the matrices lor ad a and ad D. Then we know

that (az) is non-singular; hence det(ar) + 0. Also, by assumption'

(F,) is not nilpotent. Hence if n - / is the rank then dim 6 - I and

the characteristic polynomial of (F,) is not divisible by,lz. Now let

f,, il, I be (algebraically independent) indeterminates and let F(1, p,v)

be the characteristic polynomial F(f,, pt, y) : det(,ll - pA - vB). We

have F(1, pt, u) - F1(i, 1t,v)Fz(],p, u) where

Fi(f,, P, v) : det(,il - P(e) - v(Fil '

We have seen that Fz(l, L,0) : det (r1 - (as)) is not divisible by A

and R(,t,g, 1) - det(tl - (0,)) is not divisible by 'lt' Hence the

highest power of t dividing F(1, p, r,) is l.t', l' < l. Since dl is in-

finite we can choose Fo, uo in @ such that F(t, po, vo) is not divisible

by i,'*t. Set c : poa + vab. Then the characteristic polynomial

det (il - ad c) : det(,il - thA - voB) - F(A, Po, vo) is not divisible by

S,r'+t. Hence the multiplicity of the characteristic root 0 of ad-c

is l' < l. This contradicts the regularity of. a. We have therefore

proved that for every b e 6, adgb is nilpotent. Consequently, by

Engel's theorem, O is a nilpoteni Lie algebra. Let 8o be the Fitting

( 2 )

LIE ALGEBRAS

null component of I relative to ad 6. Then 8o S 6 sirilce the latteris the Fitting null component of ad a and a e6. Qn the otherhand, we always have that 8o 2 S for a nilpotent subalgebra.Hence 8o : 6, and O is a Cartan subalgebra by Propdsition 1.

Another useful remark about regular elements and Cartan sub-algebras is that if a Cartan subalgebra b contains a regular elementa, then 6 is uniquely determined by a as the Fitting null componentof ad a. Thus if ft is this component then it is clear that tt = 6since 6 is nilpotent. On the other hand, we have jrfst seen thatTt is nilpotent so that if !t : O, then It contains an ellement z 6 6such that [e0] g 6 (cf. Exercise 1). This contradicts the assump-tion that S is a Cartan subalgebra. An immediate consequence ofour result is that if two Cartan subalgebras have a regular elementin common then they coincide. We shall see later [Chapter IX)that if @ is algebraically closed of characteristic zerQ, then everyCartan subalgebra contains a regular element. We ndw indicate afairly concrete way of determining the regular elemedts assumingagain that O is infinite. For this purpose we need to introduce thenotion of a generic element and the characteristic polynomial of a

Lie algebra.Let 8 be a Lie algebra with basis (er, er, ' ' ' , 0o) over the field O.

Let. tr, Er, " ' , En be indeterminates and let P : CI(Er, Ez " ', fr), thefield of rational expressions in the tr. We form the extension8.p : Pe, * Pe, + . -. * Pen. The element x: LTE;e; o[ 8r is calleda generic element of 8 and the characteristic polyndmial f,(l) of.ad r (in 8") is called the characteristic polynomi.al of the Lie algebra

8. If we use the basis (er,er, " ',0*) for 8r', then welcan write

( 3 ) L e ; x 7 : i . p o i e i , i : L , 2 , " ' , / t ,

where the p;l are homo;";""". expressions of degree one in the fr.It follows that

( 4 ) f , ( l ) : d e t ( , 1 1 - ( P ) ) I

: f,n * r,(€)1"-' * +(E)]"-z - ... + (- L)rr*1,(€)]r ,

where rr is a homogeneous polynomial of degree I in i the f's and

r * - t ( € ) + 0 b u t t n 4 + n : Q r i f . k > 0 . S i n c e x a d x : A a n d r * 4 ,

det(p) : 0 and , > 0. The characteristic polynomial I of any (t :

I- le.ier e I is obtained by specializing Er: a; i:1,2,t ' ,n, in(4).

Hence it is clear that the multiplicity of the root 0 for the char'

I I I . C A R T A N ' S C R I T E R I O N A N D I T S C O N S E Q U E N C E S 6 I

acteristic polynomial of ad a is at least /. On the other hand, if @

is an infinite field then, since the polynomial r"-r(E) + 0 in the

polynomial a lgebra OlEr,Er, " ' ,€ol , we can choose Et: et so that

,*-i@) + O. Then ad a for a - \aih has exactly / characteristic

roots 0, and so a is regular. Thus we see that for an infinite field,

a is regular if and only if

r,-r(a) + 0 .( 5 )

( 6 )

In this sense "almost all" the elements of 8 are regular. (In the

sense of algebraic geometry the regular elements form an open set')

It is also clear that n - / is the rank of 8.

All of this depends on the choice of the basis (e). However, it

is easy to see what happens if we change to another basis

( f r , f r , . . . , f * ) where f r . :Lk f l i . Thus i f ,1 r ,42 , " ' rQn arg indeter '

minates, then y : );tl+f+: Zrlil-tti€i. Hence the characteristic poly'

nomial /"(,1) is obtained from /"(r) bv the substitutions Ei->,;tlitt;t

in its coefficients (of the powers of ,i).

If I is any extension field of O, then (e) is a basis for 8o ov€r

P. Hence x - 2E& csll be considered also as a generic element

of ,8p and the characteristic polynomial f.()') is unchanged on ex'

tending the base field O to 9. It is clear from this also that if @

is infinite, a € 8 is regular in 8 if and only if a is regular as an

element in 8o. (In either case, (5) is the condition for regularity')

We have seen that the Fitting null component 6 of ada, aregular

is a Cartan subalgebra. {he dimensionality of O is /, which is the

multiplicity of the characteristic root 0. It follows that the Cartan

subalgebra determined by a in 8a is 6r.

2. Prodacts of weight sPaces

It is convenient to carry over the notion of weights and weight

spaces for a Lie algebra of linear transformations to an abstract

Lie algebra ,8 and a representation rR of 8. Let IJt be the module

for 8. A mapping a--a(a) of I into @ is called a weight of. IIt if'

there exists a non'zero vector x in TIt such that

x(a* - a(a)L)k -g

for a suitable &. The set of vectors satisfying this condition to'

gether with 0 is a subspace IJt, called the weight space corresponding

to the weight a. If t3 is nilpotent, then Lemma 2.1 shows that

!Jl, is a submodule. If tlt : Ift,, then we shall say that lJt is a

LIE ALGEBRAS

weight module for 8 corresponding to the weight a.Let 8 be a Lie algebra and let tn be a finite-dimenslonal weight

module for I relative to the weight d. Then for ,2ny x e Tft,x(a* - a(a)L)h: 0 if ft is sufficiently large. Moreover, if dim T!t: ?t,then (,1 - a(a))" is the characteristic polynomial of an.i Hence wehave x(a* - a(a)l)" - g for all r e tDt. We consider I the contra-gredient module Dt* which carries the representation rt* satisfying

( 7 ) 1 * a o , y * ) + ( x , y a * ' ) : O , l

r e Tft, J* e Dt*. 'We

have

( S ) - ( a ( a ) x , y * ) + ( x , a ( a ) y * ) : 0 ,

which we can add to (7) to obtain

(x(ao - a(a)L),y*) + (x, y*(ao* + a(a)I)): 0 .l(e)Iteration of this gives j

(10) (r(a* - a(a)l)h,J*) + (r,y*(a*'+(-1)'- ' o(a)l)*) - 0 .

If h,: ,rt, r(ao - a(a)I)" - g for all r and consequently, by (10),(x, !*(a*' * a(a)l)") : 0. Hence J*(a"' * a(a)l)":0 for all y* e llt*.This shows that lltx is a weight module utith the weight - a.

Pnorosruor.r 3. If Wt is a fi,nite-di,mensional weight mbdule for 8with the weight a, then the contragredient module llt*iis a uteightmodule wi,th the wei.ght -a. l

We consider next what happens if we take the tensor productof two weight spaces. Thus let fi,It be weight mddules of Irelative to the weights a and 9. Let .R and S denote the repre'sentations in llt and Tl, respectively. Then any rc e Iftr satisfies

(11) r(a* - a(a)I)h - g

f o r s o m e p o s i t i v e i n t e g e r k , a n d ' e v e r y J e y t s a t i s f i e s

(rz) t(as - g@)l)k' :0

for some positive integer &'. Let !p.: lDt 8It and denote the repre'sentation of I in S by

". Then we have

( 1 3 ) ( x @ y ) a ' : x a * Q y + x & y a u

or a* : aR 81 + 1 I a". Hence


ar -@@)+ 9@DL: (a *8 1 - a (a )L€ ) 1 )+ (18au -18 9@) I ) .

Since the two transformations in the parentheses commute

can apply the binomial theorem to obtain

(15) (a' - (a(a) + p@)lL)*

- a(a)€) 1)t(1 @au - 18 9@))^-t

If we apply this to rOY, we obtain

(x89@'-(a(a)+ F@DL)**-.(T\r<oR - a(a)L)'8 v@u - F@)L)^-': L, , ,

(14)

(16)

(17)

and

(18)

where p,6, . . -tco(h* - p(h)L)^

tjt - Do e) yJt" o

8 :8 ,O8e@

:E(T)(a"8 r

If we take m: k * k' - !, then for every l, either x(a* - a(a)l)i :0

or t(as - F@)L)^-d:0. Hence (r&y)(a* -(a(a) + g(a))l)-:0 and

we have the following

PnoposrrroN 4. If n and Tt are weight modules for 8 for tlu

we igh tsa ,and 'B , res \ec t iue ly , then* : I l t& I t l ' sawe igh tmodu lewith tlw weight a * F.

We suppose now that I is a finite-dimensional Lie algebra, 6 a

nilpotent subalgebra, and !]l is a finite-dimensional module for 8,

hence for o. If R denotes the representation it 4 and adg the

adjoint representation in 8, then we assume that 08 and adg$ are

split Lie algebras of linear transformations, that is, the character'

istic roots of lz8 and adgft , h e 0, are in the base field o. This

witl be automaticalty satisfied if O is algebraically closed. If

hn -, a(h") is a weight on 08, then h -', a(h) = a(ha) is a weight

for 6 in the module Dl. The result on weight spaces for a split

nilpotent Lie algebra of linear transformations (Theorem 2.7) im'plies that llt is a direct sum of weight modules Dto. Similarly, w€

h"u" a decomposition of 8 into weight modules 8r. Thus we have

OUI.'

@8r

are mappings of O into @ such that if roeUlto then- 0 for some m, etc. The weights a, P, "', associat'

64 LIE ALGEBRAS

ed with adg$ will be called the roots of .b in 8. SincQ h -, hR andh -, adgh are linear, it is clear that in the characteristic 0 case theweights p, ..., r and the roots &t .. ., D are l inear funbtions on bwhich vanish on O'.

The following important result relates the decompositions (17)

and (18) relative to 6.

PnorosrrroN 5. tJlp8, E8; otherwise Tllo8r:0.

ffio*" if p + a. is a weight of V:ft relatiae to

Proof: The elements of Dlp8" are ofat) e 8,r. The characteristic property

spaces shows that we have a linearfiplla such that

,F"1" @aS\),, - 2x[1)at) .

We shall now show that T is in fact a homomorphism for the

S-modules. Thus let h e $. Then we have

(roB ar)h: xphSau* )cp&la"hl (xph)a"* xolarhl:(xoad)h .

On the other hand, the image of xo$a, under n is Aear. Follow-

ing this with the module product by h we again optain (roa")h.

We have therefore proved that Dlo8" is a homomorpfiic image of

lltp @ 8r. Moreover, the latter is a weight module fdr the weightp + e. Now it is clear from the definition that the [romomorphicimage of a weight module with weight P is either 0 or it is a

weight module with weight 0. The result stated follqws from this.

If we apply the last result to the case in which ffii:8 and the

representation is the adjoint representation of 8, we Obtain the

Conor,utny. [8"8B] E !a+F if a* p is a root, and [8,i8p]:0 other'

wi,se.

the form 20"[t'qt', x[" eUfto,of the tensor prbduct of twomapping r of Wp I 8a onto

3. An example

Before plunging into the structure theory it will bb well to look

at an example. Let Dt be an a-dimensional vector fOace over an

algebraically closed field @, @, the associative algehra of linear

transformations in fi, 8 : Oz, the corresponding Lielalgebra. We

wish to determine the regular elements of I and ttie correspond-

ing Cartan subalgebras.


Let Aeg and le t l l t : \ f t , r9T f t , r@" '@f f i r " be the decom'

position of llt into the weight spaces relative to OA. The 4; are

distinct elements of O and these are yust the characteristic roots

of the linear transformation A and the lJton are the corresponding

characteristic spaces. It is well known that a decomposition of a

space as a direct sum leads to a decomposition of the dual space

into a direct sum of subspaces which can be considered as the

conjugate spaces of the components. Hence we have

IJ r * :gn lo tn fe . . .oDt Iwhere lltl can be identified with the conjugate space of l)tr,. gnf

is invariant under - A* and Proposition 3 shows that this is a

weight module for the Lie algebra oA with the weight - dr.

Accordingly, we write *!or. for IJII and we have

TJt* : *1or @rl t l r rO - . . ODt1r, .

We now consider the module Dt I tJt* relative b AA. As is well

known, we have m I'Jl* : X?,r=, IJl"o I *!ot By Proposition 4,

fir;@ *!at is a weight module for OA for the weight ar- di.

lf. Ari denotes the linear transformation in llt"n 8ln:"J correspond'

ing to A, then (Au - (ai * a)I)kti: 0 for suitable &rr. Hence Air'

is non-singular if ar - aj + 0 and Anr' is nilpotent if. ar - di. This

implies that the Fitting null component of IJt I Tn* relative to the

linear transformation .A corresponding to A (A'- A is the repre-

sentation of 8 in ut I tn*) is >,T=,IJt,o8 TJt1"o. If dim !Jt,, -- /tit

then the dimensionality of this Fitting space is l,lTnlt. Since Dn;:rt,}nl is minimal if and only if every /t;,: L, which is equivalent to

saying that A has n distinct characteristic roots. Now, we recall

that the module tJt I SJt* is isomorphic to the module I : Oz rela-

tive to the adjoint mapping X--+ lXAl. Hence we see that the di''

mensi,onali,ty of the Fitting null component of E, relatiue to ad A is

minimal if and only if A has n distinct characteristic roots. Thus.4 is regular in I : Er if. and only if A has z distinct characteristicroots. The corresponding Cartan subalgebra O is the Fitting nullcomponent of 8 relative to ad A and dim $ : le.

Since A has z distinct characteristic roots we can choose a basisfor IJt such that the matrix of. A relative to this basis is diagonal.Let Or be the set of linear transformations whose matrices relativeto this basis are diagonal. Then dim S' - n and, LHA):0 for everyH e 8'. Hence 0' g O. On the other hand, dim $ - n. Hence

66 LIE ALGEBRAS

6r:O and the Cartan subalgebra determined by 4l is just thecentralizer of the linear transformation A.

Let (e* €2, . . . ,0o) be a basis such that eiA : d;e;. Then we haveseen that the Cartan subalgebra O is the set of lineari transforma-tions which have diagonal matrices relative to the basis (er). Henceif H e 6 then we have e;H : li(HV;, l,{H) e o, aFd we mayassume the notation is such that l,{A) : a* The spaces Oe;1: I}t';)are weight spaces and Oei corresponds to the weighti f(n. Weshall therefore write Wtx, for Oei. As for the single linear trans'formation A we can now write l l t*:"1lrrOll t1^r$"'O!l t l^"and IIt S IIt* : Xi,i O (fi^,8 tn:rr. fir,8 filry is a weight spacefor the Cartan subalgebra S corresponding to the wdight il - h.We can summarize our results in the following

Tnponpu 2. Let 8, :8", ihe Lie algebra of linear tra4sformationsin an n-dimensional uector space Tft oaer an algebraically closed fieldO, Then .4 e I is a regular element i,f and only if the &haracteristicpolynomial det (fl - A) has n distinct roots. Th.e Cartan subalgebra$ determined by A is th.e set of linear transformations H such thatIHAI - 0. If the weights af H acti.ng in Ift are tr(I{), '" r l"(H)then the roots (weights of the adioint representatio* i'n 8) are^t(H) - l/H), i, i : t, "' , /t. l

If @ is not algebraically closed but is infinite then dhe extensionfield argument of $ 1 shows that again A is regular ih Gz if andonly if A has n distinct characteristic roots (in the algebraicclosure A of. 0. It is well known that the centralizer of such anA is the algebraAlAl of polynomials in A and dim0[Al - n. Sincethe dimensionality of the Cartan subalgebra determindd by A is n,it follows that this subalgebra is OtAl.

If 8 is the orthogonal or symplectic Lie algebra in\ a Zt'dimen'sional space over an infinite field one can show that the rank isN - l, N - dim I (cf. Exercises 4 and 5). It is notl difficult todetermine Cartan subalgebras for the other important examples ofLie algebras which we have encountered (cf, Exercise 3).

4. Cartan's eriteria

We now consider a finite'dimensional Lie algebra over an alge-

braically closed field O, a nilpotent subalgebra 6 of i I and TJt a

module for 8, which is finite-dimensional over {D. Le!


l l t - u toot l t ,o . . . @Dtn,8 :8 ,@8eO" 'O8 r

(1e)

(20)

be the decompositions of IJt and I into weight modules relative to

8. We have seen that

rm a - p if P + a is not a weight of mil|P'.,6 -

lg TItp+o if p + a is a weight

r8"8pr: tt=ltrL*rru )'.TiJ jT;, .Now suppose 6 is a cartan subalgebra. Then 6: 80, the root

module corresponding to the root 0. Also we have !/ - [88] :

5,[8,8e1 where the sum is taken over all the roots a,9. The

formula for [8"8p] shows that

(2t) O n 8'- ![8"8-"1 ,

where the summation is taken over all the a such that - a is also

a root (e.g., a: 0).We now prove the following

Lsuu.A 1. Igt o be algebraically closed of characteristic 0, 6 a

Cartan subalgebra of 8 ouer O, YIt a module for 8,. Suppose a i,s a

roat such that - a is also a root. Let eo € 8", €-o Q 8-n, h, : le"e-rl.Then p(h,) is a rati,onal multifle of d(h,) for euery weight p of 6in Wt.

Proof: Consider the functions of the form p(h) + id(h), i' :0,-r- 1, = 2, ..., which are weights, and form the subspace tt -

X;Iltp+io summed over the corresponding weight spaces. This space

is invariant relative to O and, by (20), it is also invariant relative

to the linear transformations e8, e\r, where R is the representation

of I determined by Slt. Hence, if tru' denotes the trace of an

induced mapping in It, then tr/tr:trnlefl,e!,1 -9. On the other

hand, the restriction of h! to I)t' has the single characteristic root

o(h,). Hence

0 - trnh} : lnr*;,(p

* ia)(h,)

lta : dim lll'. Thus we have

(2no*r")p(h,) + (2no*r,i)a(h,) : Q .t {

68 LIE ALGEBRAS

Since Zr/tp+;, is a positive integer this shows that p(h") [s a rational

multiple of. a(h").We can now prove

Cartan's criterion for soluable Lie algebras. Let I be a fi'nite'dimensional Lie algebra ouer a rteM of characteristic A.) Suppose 8'has a fi.nite-dimensional module TIt such that (l) the kerrl'el R of the

associated representation R is soluable and (2) tr (aR)z a A for euer!

a e 8t. Then 8, is soluable.Proof: Assume first that the base field O is algebraibally closed.

It suffices to prove that 8' c 8; for, conditions (1) aqd (2) catty

over to 8' and Dt as 8'-module. Hence we shall have [hat

8 : 8 ' : 8 " : r . . . r 8 ( / ' ) - 0 . ,

We therefore suppose that 8' : 8. Let 6 be a Cartanj subalgebra

and let the decomposition of lJt and I relative to O be] as in (19).

Then (21) implies that 6 - l[8"8-"] summed on the s such that- a is also a root. Choose such an &, let e, € 8r, e*q, € 8-a, and

consider the element h, - le,e-,). The formula O - >[848-"] implies

that every element of 6 is a sum of terms of the {orm lere-"1.The restriction of. hI to )Jto has the single characteristic root p(h,).

Hence the restriction of, (hI)' has the single characteristic rootp(hr)', and if np is the dimensionality of fi', then we have

0 : tr (hI), : \npp(h,)' . l

By the lemma, p(h,): rpa(hr), ro rational. Hence a(h)4,(2nori) - 0.

Since the n, are positive integers, this implies that a(h') : 0 and

hence p(h,) - 0. Since the p are linear functions and Every h e 0

is a sum of elements of the form ltr, hF, ' ' ' , etc., We see that

p(h) -- o. Thus 0 is the only weight for !lt; that ils, we have

$Jt : TJto. If a is a root then (20) now implies that Plt8" :0 for

every a * 0. This means the kernel S of R containsi all the 8",

d + 0. Hence 8* : 8/S is a homomorphic image of &. Thus 88

is nilpotent and 8 is solvable contrary to 8' : 8. l

If the base field @ is not algebraically closed, then llet J? be its

algebraic closure. Then fip is a module for 8o and So iis the kernel

of the corresponding representation. Since n is solyable, So is

solvable' Next we note that the condition tr(ao)'= 0 and tr aRbB :

tr bRaR imply that

III. CARTAN'S CRITERION AND IT5 CONSEQUENCES 69

tr anbn : +ff(aLbn

+ bR an)

: + ftr@R + bo)' - tr(aR)z - tr(bn)'z] - o .2 -

Hence if the ai e g and arr e g, then tr(}o'iaf)' - Zrn,itt a?af -0.

Hence the condition (2) holds also in 8p. The first part of the

proof therefore implies that ,8o is solvable. Hence I is solvable

and the proof is comPlete.

Conor,ur,ny. If 0 is' of characteristic 0 then I is soluable if and

anly if tr (ad a)' : 0 for eaerY a e 8,t .Proof: The sufficiency of the condition is a consequence of

Cartan's criterion since the kernel of the adjoint representation is

the center. Conversely] assume I solvable. Then, by Corollaty 2

to Theorem 2.8, applied to ad 8, the elements adga, a e 8' , are in

the radical of (ad8)*. Hence adga is nilpotent and tr(adga)z - Q.

Let R be a representation of a Lie algebra in a finite-dimensional

space llt. Then the function

(22) f(a, b) = tr aRbn

is evidently a symmetric bilinear form on Dt with values in O.

Such a form will be catted a trace forrn for 13. In particular, if I

is finite-dimensional, then we have the trace form tr(ad a)(adb),

which we shall call the KiUing form of 8. If / is the trace form

determined by the representation R then

f(lacl, b) + f(a, [Dc]) ::

form f(a, b) on I

tr(laclRbR + a*Lbcl*)

tr (laBc*lbo + a*lb* c*11

trLanbR,c" l :0 .

which satisfies this conditionA bilinear

(23) f(tacl, b) * f(a, [Dc]) - g

is called an inuaria,nt form on 8. Hence we have verified thattrace forms are invariant. We note next that if. f(a,&) is anysymmetric invariant form on 8, then the radical 8a of the form;that is, the set of elements e such that f(a, z) : 0 for all a e 8, isan ideal. This is clear since f(a,lzbl) - - f([ab], e) : 0.

We can now derive

Cartan's criteri.on for semi,-sirnplicity. If 8, is a finite'dimensional

70 LIE ALGEBRAS

semi'simple Lie algebra ouer a fi.etd of characteristic 0, then the traceforrn of any l:l representation of g is non-d.egenerate. If the Kittingform is non-degenerate, then g is semi-simple.

Proof: Let R be a 1: 1 representation of g in a finite,dimensionalspace llt and let f(a,D) be the associated trace form. Then gr isan ideal of I and f(a, a) : tr(a')': 0 for every a € ga, Hence gris, solvable by the first cartan criterion. since u is semi-simple,8a : 0 and f(a, b) is non-degenerate. Next suppose that g is notsemi'simple. Then I has an abelian ideal t + b. If #e choose abasis for I such that the first vectors form a basis, for E, thenthe matrices of ad a, a e 8, and adb, b € E, are, respectively, ofthe forms

(l ), (:3)This implies that tr(ad DXad a) : 0. Hence E g ga an{ the Killingform is degenerate.

lf. f(a, b) is a symmetric bilinear form in a finitefdimensionalspace and (er, e", ' ' ' , €n) is a basis for the space, then it is wellknown that f is non-degenerate if and only if det(f(e;,e)) +0. If8 is a finite'dimensional Lie algebra of characteristic zero withbas is (e r ,e r , . . . ,e* ) and we se t 0u : t rad erxdr t , then g is semi -simple if and only if det (B;r.) + 0. This is the determinant formof Cartan's criterion which we have just proved. If J? is an ex.tension of the base field of 8 then (er, er, . . ., €n) is a lbasis for goover g. Hence it is clear that we have the following iconsequenceof our criterion.

conou,rnv. A finite-dirnensional Lie algebra g ouer a fietd a ofcharacteristic zero is semi-simple if and only if go is setni-simple foreuer! extension rteM I of A.

5. Structure of serni-simple algebras

we are now in a position to obtain the main structure theoremon semi-simple Lie algebras. The proof of this resullt which weshall give is a simplification, due to Dieudonnd, of Cartan's originalproof. The argument is actually applicable to arbitraryinon-associa-tive algebras and we shall give it in this generar forrd.

Let lI be a non-associative algebra over a field @.t A bilinear

III. CARTAN'S CRITERION AND ITS CONSEQUENCES 7I

form f(a, b) on ?l (to O) is called associ'atiue if

(24) f(ac,b) : f(a, cb) .

If / is an invariant form in a Lie algebra then f(tacl, b) + f(a, [Dc]) - g-

Hence f(lac|,b) - f(a, [c&]) - 0 and / is associative. Now let f(a,b)be a symmetric associative bilinear form on l[ and let E be an

ideal in lI. Let a e Ea so that f(a, b) : 0 for all b e s' Then for

any c in 21, f(ac,b) -f(a,cD) - g since cD e E. Also f(ca,b) -

f(b, ca) - f(bc, a) :0 since bc e E. Hence Er is an ideal.

The importance of associative forms is indicated in the follow'

ing result.

THponsM 3. Let \ be a finite-dimensional non'a,ssociatiue algebra

aaer a fi.eld A such that (L) V, has a non'degenerate syrnmetric as'

sociatiue form f and (2) lI has no ideals E with Ez : A. Tlwn lI is

a direct sum of i.deals which are simple algebras.(We recall that lI simple means that lt has no ideals + 0, ?I, and

?12 + 0.)Proof: Let E be a minimal ideal ( + 0) in lI. Then E n Er is

an ideal contained in E. Hence either E n Ea : E or E n Er :0.

Suppose the first case holds and let bs bz e E, a e 2I. Then

f(brbr, q) - f(bv b2a) : g. ' Since f is non"degenerate b'b, - g and

E2 : 0 contrary to hypothesis. Hence E n Er :0. It is well

known that this implies that lI - E O El and Ef is an ideal. This

decomposition implies that EEr : 0 : EtE; hence every E'ideal is

an ideal. Consequently, E is simple. Moreover, St satisfies the

same conditions as ?I since the restriction of, f to Er is non'

degenerate and any Er-ideal is an ideal. Hence, induction on dim ?I

implies that Ea : ?I, (E '.' @ U" where the ?Ie are ideals and are

s imptea lgebras . Thenfor l l r :E we have 2 [ :? I r@?t rO " 'OU" ,lli simple and ideals.

This result and the non-degeneracy of the Killing form for a

semi-simple Lie algebra of characteristic zero imply the difficulthalf of the fundamental

Structure theorem. A finite-di.mensional Lie algebra oaer a rteldof characteristic 0 is semi'simgle if and only if 8:8'@8rO "'@8'where the 8i are ideals which are simple algebras.

Proof: If I is semi-simple, then I has the structure indicated.

Conversely, suppose 8=8'O8rO"'@8', 8 i ideals and simple.'l{e

consider the set of linear transformations ad I : {ad a I a e 8}

72 LIE ALGEBRAS

acting in 8. The invariant subspaces relative to this set are theidea ls o f 8 . S ince 8 :8 rO8rO. . . 08 , where the g ; la re i r reduc-ible, we see that the set ad I is completely reducible. jHence if Eis any ideal + 0 in 8, then 8: E@o where o is an idehl (Theorem2.9). Moreover, the proof of the theorem referred to shows thatw e c a n t a k e E t o h a v e t h e f o r m D : I i r O 8 , r O . . . g ; * f o r a s u b s e t{ 8 r , } o f t h e 8 i . T h e n E = 8 / ( l l ; r O . . . @ 8 r e ) = . 8 r r 6 8 i l e . . . 6 8 r , rwhere the 81o are the remaining 8;. Since 8r is simple, 8? : ,8r.Hence (8a O 8ir@ ... @ 8r,)' : 8r.r O 8r.2 O . .. O 8i, and| consequent-ly E' - A. Thus E is not solvable. We have iherefore provedthat 8 has no non-zero solvable ideals; so 8 is semi-sirnple.

The argument jus t g iven tha t * - .8 r r@8rr@. . .88 i , has thefollowing consequence.

Conor,r,.r,ny 1. Any ideal in a semi-simple Li.e algebra of charac-teristic 0 is semi-si,mple.

If 8r is simple then the derived algebra 8{ : 8;i hence the struc-ture theorem implies the following

Conor,ulny 2. If 8, is semi-simple of characteristic 0,ithen 8' : 8.Remark. We have proved in Chapter II that if I isla completely

reducible Lie algebra of linear transformations in a ]finite-dimen-sional vector space over a field of characteristic 0, thdn 8 : 6 O,8,where G is the center and I, is a semi-simple ideal.I Then 8/ -8i : 8,. Hence 8 : 6 e 8', 8' semi-simple.

We prove next the following general uniqueness thborem.

THponpu 4. If 2I is a algebra and

l I : ? [ r O l t r O . . . O U , : E r O E , O . . . O $ ,

where the \Ii and Et are tdeals and are simple, thenEis coincide with the W";'s (excebt for order).

Proof: Consider 2Ir 0 Ei, j : L,2, . .., s. This is An ideal con-tained in llr and 81. Hence if lI' fi Er' + 0, then lI, :illr n Ei - E;since lI' and E; are simple. It follows that Ur o Ej' * 0 for at mostone Er'. On the other hand, if ?I' n E, :0 for all j then llr8r S2 [ ' n E i : 0 f o r a l l j . S i n c e l [ : E ' O E ' O . . . O E , n t h i s i m p l i e sthat lIr?I : 0 contrary to the assumption that lli + 0. r Hence thereis a i such that ?Ir - fl51. Similarly, we have that every lI; coincideswith one of the El and every E1 coincides with cjne of the ?Ii.

s and the

III. CARTAN'S CRITERION AND ITS CONSEQUENCES

The result follows from this.

It is easy to see also that if ?I is as in Theorem 4, then lI hasjust 2" ideals, namely, the ideals 2Ir, O ' . ' O \r*, {ir, "', ir} a subsetof {1, 2, . - -, r}. We omit the proof .

The main structure theorem fails if the characteristic is y' + 0.To obtain a counter example we consider the Lie algebra Et of

linear transformations in a vector space lJt whose dimensionalityz is divisible by f. It is easy to prove (Exercise 1.20) that the onlyideals in Or are @L and oL, the set of multiples of 1. Since @i isthe set of linear transformations of trace 0 and tr 1 : n:0, AL g VL.Hence I : g,ilO| has only one ideal, namely, @ilAt, and the latteris simple. This implies that 62101 is semi'simple, but since ELIOLis the only ideal in Ezl0l, E"IOL is not a direct sum of simpleideals. This and Theorem 3 imply that Erl01 possesses no non-

degenerate symmetric associative bilinear form.

We conclude this section with the following characterization ofthe radical in the characteristic 0 case.

THponpu 5. If 8 is a fi,nite-dimensional Lie algebra ouer a fieldof characteristic 0, then the radical 6 of 8, is the orthogonal com'plement g't of gt relatiue to the Kitting form f(a,b).

Proof: E : 8'a is an ideal and if 6 € E', then tr(ad gb)'z :f(b,D) :0.

The kernel of the representation a + adga, a e E, is abelian. HenceE is solvable, by Cartan's criterion, and E s @. Next let s € 6,a,b e 8,. Then f(s,[ab]) - f([sa],b). We have seen (Corollary 2 toTheorem 2.8) that adtsal is contained in the radical of the envelop'ing associative algebra (ad 8)*. Consequently, ad [sa] ad D is nilpotentfor every b andhence f(lsal,b):0. Thus f(s,labl):0 and s € 8'r.T h u s O g S ' r a n d s o 6 : 8 ' 4 .

6. Deriuations

We recall that ad a is a derivation called innzr and the set ad 8of these derivations is an ideal in the derivation algebra O(8). Infact, we have the formula fad a, Dl : ad aD f.or D a derivation.Hence

ladaad'b, Dl: ad aad(bD) * ad (aD)adb

which implies that

74 LIE ALGEBRAS

0 : tr [ad a ad b, D) - tr ad a ad (bD) * tr ad (aD)tad b -

Thus, for the Killing form f(a,b): trad aadb we have

f(a, bD) + f(aD, b) : 0 ;(25)

that is, every derivation is a skew-symmetric transformaltion relative

to the Killing form.We prove next the following theorem which is due toiZassenhaus.

Tnponnu 6. If 8, i,s a finite-di,mensional Lie algebra which has a

non-degenerate Kilti.ng form, then euery deriuation D of'8 is inner.

Proof: The mapping r--tr(adr)D is a linear mapping of I into

A; that is, it is an element of the coniugate space 8* of 8. Since

J@, b) is non-degenerate it follows that there exists i an element

d e g such that f(d', x): tr(ad x)D for all .r e 8'Let E be the derivation D - ad d- Thentr (ad x)E : tr (ad x)D - tr (ad r)(ad d) : tr (ad r)D - f(d, r) : 0 '

Thus

(26)

Now consider

t r (ad x )E -0 .

f(xE''y) : tr (ad xE)ad Y: tr lad x, E)ad' Y: t r ( ( a d x ) E a d Y - E a d r a d Y ) ,: t r (Eady ad x - Ead xadY) i- tr E[ad y, ad x)- tr E adlYxl: 0 ,

by (26). Since / is non-degenerate, this implies that.0 - 0. Hence

D : ad d is inner.This result implies that the derivations of any finitb dimensional

semi-simple Lie algebra over a field of characteristicr zero are all

inner. We recall also that if 6 is solvable, finite'dimensional of

characteristic 0, then 6 is mapped into the nil radical Il by everyderivation of 6 (Corollary 2 to Theorem 2.1-3). We cdn now proYe

Tnuonpu 7. (l) Int 8 be a finite-dimensi.onal Lie Algebra ouer a

field of characteri,stic 0, g the radical, It the ni,l radics'|. Then any

deriuation D of g maps 6 into Tt. (2) Let 8 be an ideal in a finite'd.imensional algebra gr, @r, 9tr the radical and nil ladical of 8r


T h e n 6 : 8 O 6 r , I t : 8 O l l r .Proof: We first prove (2) f,or the radical. Thus it is clear that

8 n g, is a solvable ideal in l], hence I n 6r s g. Then g(8 n gr)

is a solvable ideal in 81(8 n 6r). On the other hand, 8(8 n @') =(,8 * 6')/6,, which is an ideal in 8'/€'. Hence (8 + gr)/gr and8/(8 n €,) are semi-simple. Hence 6(8 n 6,):0 and €:8 fl 6r.Now let 8r be the holomorph of 8, €,, 9t, the radical and nil radicalof 8r. Then we know that [8rgr] 5 ltr (Theorem 2.13). Since6 s 6' by the first part of the argument, [8'g] I (!t' n 8) E It.This implies that every derivation of I maps 6 into It, which proves(1). Now let 8r be any finite'dimensional Lie algebra containingI as an ideal. If ar € 8r then ad ar induces a derivation in 8.Hence Ttad,a, gIt by (1). This means that 9t is an ideal in ll sothat It S Tlr fl 8, 9tr the nil radical of ,8'. Since the reverse in-equality is clear, lt: Itr fl ,8.

This result fails for characteristic p + 0. To construct a counterexample we consider first the commutative associative algebra 3with the basis (L,2,22,. . . ,2o- ' ) wi th zp:A. The radical f t of 3has the basis (2, z', . . . , z'-') and 3/ft : AL. It is easy to prove

that if ar is any element of 3, then there exists a derivation of 3mapping z into w. In particular, there is a derivation D such thatzD: !. Now let E be any simple Lie algebra and let I be theLie algebra E 13. The elements of this algebra have the form

)}i8 zi, bi € E, zi Q g, and [(D &z)(b'&z')l - lbbt]@zz'. Then Iis a Lie algebra (Exercise 1.23) and E I ft is a nilpotent ideal in 8.Moreover, S(E8n)=E8OI:E is s imple. Hence EOn is theradical and the nil radical of 8. lt. D is any derivation in the as'sociative algebra S, then the mapping )Dc I z; + 2h I eiD is aderivation in 8. If we take D so that zD :1 and let b * 0 in Ethen Df i -z- t$ lElS8S. Hence we have a der ivat ion whichdoes not leave the radical invariant.

7. Complete red,ucibilitA of the representationsof semi-aimple algebras

In this section we shall prove the main structure theorem formodules of a semi-simple Lie algebra of characteristic zero and weshall obtain its most important consequences. The main theoremis due to Weyl and was proved by him by transcendental methodsbased on the connection between Lie algebras and compact groups.

LIE ALGEBRAS

The first algebraic proof of the result was given by Casimir and

van der Waerden. The proof we shall give is in es$ence due to

Whitehead. It should be mentioned that Whitehead'$ proof was

one of the stepping stones to the cohomology theory of Lie algebras

which we shall consider in $ 10. We note also that in the char-

acteristic p case there appears to be little connection hetween the

structure of a Lie algebra and the structure of its modt{les since, as

wilt be shown later, every finite'dimensional Lie algdbra of char-

acteristic P +0 has faithful representations which are ndt completely

reducible and also faithful representations which ard completely

reducibleWe obtain first a criterion that a set ^5 of linear trafisformations

in a finite-dimensional vector space l}t be completely reducible. We

have seen (Theorem 2.9) that J is completety reducibld if and only

if every invariant subspace Tt of IJt has a complement S which is

invariant relative to J. Now let Tt' be any complerhentary sub-

space to Il: Un - n e n'. Such a decomposition is as{ociated with

a projection .E of lJt onto It. Thus if. r e T!1, then v'1e can write

r in one and only one way as .r : y + !', ! € It, J' le ttt, and -E

is the linear mapping x-+y. Conversely, if ,E is any idempotent

linear mapping such that Tt - !It.E then It/ : IJt(1 - E)'is a comple'

ment of Tt in IJt. Now let A e 2 and consider the llinear trans-

formation LAE| = AE - EA. If r e [It, xAE e It and xEA e Tt;

hence tAEl maps llt into Tt. If y e Tt then yAE = yA. Hence

tAEl maps It into 0. Then if I denotes the set of lineAr tranforma-

tions of Dt which map IJI into It and It into 0, [AE] e f. It is

clear that t is a subspace of the space G of linear trNnsformations

in !Jt. We now prove the following

Lnuu.r, 2. Tt has a complement S which is inuarianl if and only

if there exists a D e X. such that lAEl - lADl for all A e t. Here

E is any froiection onto [t.Proof: Let $ be a complement of Tt which is invafiant relative

to J and let F be the projection of St onto Tt deterrhined by the

decomposition tn - n CI S. Since $ is invariant, F cQmmutes with

every Ae 2, that is, [AF] -0. Hence LADI-lAEl fbr D: E- F'

Also, since ^E and F are projections on It, E - F maps !ft into It

and !l into 0. Hence D e f" as required. Conver5ely, suppose

there ex is ts a DeX, such tha t tAE l - lADl - Thdn F :E-D

commutes with every Ae t. If r e llt then xF: tc(E- D) e It


and i f y e t t then yF - yE:y. Hence F": Fandl t : I l tF. Then

S: !lt(l - F") is a complement of !t and $ is invariant under J

since F commutes with every A e 2.

Suppose now that I is a Lie algebra and !)t is an !-module, It a

submodule. We can apply our considerations to the set 8i of re'

presenting transformations a' determined by !]t. Let E be a pro-

jection of.Ift onto !t. If a e 8, set f(a): la*, El. Then a-+f(a) is

a linear mapping of I into the space fr of linear transformations

of fiwhichmap llt into !t and !t into 0. If Xe f and a € 8 then

IXaRJ e X,. Thus if r e tlt then x[Xa"] e sIt and if. y e It then

yXa*: 0 and ya*X: O. We denote the mapping y-rlXanl by oi.

It is immediate that a -, a* is a representation of 8 whose associat-

ed module is the space l. We have

f(labl) - flablB, El : IIaRb")Ol- l[anElb"l + la"[b"Ell- If@),6"1 + Iao,f(b)l- f(a)bfr - f(b)ai .

We are now led to consider the following situation: We have a

module f for 8 and a linear mapping a-.f(a) of I into f such

that

(27) f(tabl) - f(a)b - f(b)a .

A "trivial" example of such a mapping is obtained by taking

f(a): da, where d is an element of f. For, we have

f(abl) - dtabl - (da)b - (db)a : f(a)b - f(b)a .

The key result for the proof of complete reducibility of the modulesfor a semi-simple Lie algebra of characteristic zero is the following

Lnuu.e, 3 (Whitehead). Let 8, be fini'te-dimensional semi'simple of

characteristic zero and let X, be a fini,te-dimensional module for 8,

and a-+f(a) a linear mafbing af 8. into X, satisfying (27). Thenthere exi.sts a d e *" such that f(a) - da.

Proof: The proof will be based on the important notion of a

Casi,mir operatm. First, suppose that 8 is a Lie algebra and Er

and Ez are ideals in I such that the representations of I in E' andEz Zlr€ contragredient. Thus we are assuming that the spaces Er

andEgareconnected by a bil inear form (bt,br), D; € Ei, (bubz) e A,

which is non-degenerate and that for any a e 8, we have

78 LIE ALGEBRAS

(28) (lb,al, b") + (b,,lbral) - 0 .

lf (ur, . . . , il^) is a basis for Er then we can choose a co{nplementaryor dual basis (It', u' , . ., u^) for Es satisfying (ur, ui'\ - d;.r. Let

lu;al : !,ia,;iui and fuka): },r}*ilt. Then (Lu;a), uk) -- (Eatiut, uk) :

2au6i*: dik and (u;,luoal) : (?h,I,,lr,pt) : B*;. Hencel (28) impliesthat air : - 9u, that is, the matrices (a) and (p) determined bydual bases satisfy (F) : - (a)' ((a)/ the transpose of (a)). Now let

R be a representation of 8. Then the element

7 - iufu'"t : L

is called a Casimir operator oI R. We have

LI-, anl: \lufa"fu'* + \utlu''anl

: \aa1ufu'* + Zl;iufuiRt , J r , J

- \aipfu,'o - \a1sf;ui*

t , , t , J

: \a;iufrai, - \ariufu'n

- 0 .

Hence we have the important property that /' commuJtes with all

the representing transformations aa.Now let I satisfy the hypotheses of the lemma. ILet S be the

kernel of the representation R determined by I . Then we can

write I : F O 8, where I' is an ideal. Then the restriction of R

to 8r is 1 : I and ,8, is semi'simple. Hence the trace fOrm (br, br) :

trblbf , b; in 8r, is non'degenerate on ,8r. Also we knbw that the

trace form of. a representation is invariant. Hence lthe equation(28) holds for Di e 8r and a e 8,. Thus the representation of 8 in

8r coincides with its contragredient and if (uu ''', u^)i (ut, " ', il^)

are bases for 8, satisfying (ur ui) - dr., then /' - I,,Lruf;u'R is a

Casimir operator which commutes with every a*. 't|Ve note also

that tr l- : X; tr ufuio - 2 (ui, u') : re : dim 8r.We now decompose I into its Fitting components Ii and fr rela-

tive to l- so that I induces a nilpotent linear transformation in Io

and a non-singular one in f,. Since TaR --aRl, friani9I1 so thatthe f; are submodules. we can write .f(a): fo@) * f'(a) where

fi@) e I; and it is immediate tbat a-fi@) is a lineaf mappingof


8 into 17 satisfying (27). Now if both spaces f, are * 0, then

diml i< diml for j :0, t . Hence we can use induct ion on dimf

to conclude that there is a d1 € X,i such that f/a) : dP. Then

d.: do * d, satisfies f(a) * da as required. Thus it remains to

consider the following two cases: X : Xo and I : Xr.X, : fro: In this case /' is nilpotent. Hence m : tt f : 0. This

means that the kernel of R is the whole of 8, that is, ao :0 foral| a. Then the condition (27) is that f(labl) : 0, a, b e 8,. Thus

f(a ' ) - 0 for a l l a '€ 8 ' . Since 8' :8, th is impl ies that . f (a) :0 sothat d: 0 satisfies the condition.

f : Ir: Set y :}1=rf(u;)ui where the (ai) and (ud) are dual basesfor It as before. Then

ya: ZU@t)ut)a

= \U@t)a)u" * |,f(u)lu"al

: \(f(u;)a)ur + \giif@r)uii . J

= \(f(ut)a)u' - \a6f@)u'

- \(f(u)a)u' - Z(f[uia))ui

= \(f(u;)a)u'

- ZU@;)a)u' + l(f(a)u)u'

- f(a)r

Since /- is non-singular, d: yT-' satisfies the required condition

f(a) - da. This completes the proof of Whitehead's lemma.We can now prove the following fundamental theorem:

Tnponnu 8. If 8, is fi,nite-dimensional semi-simple of characteristic0, then eaery finite-dimensional module for 8, is completely redacible.

Proof: Let llt be a finite'dimensional 8-module, !t a submodule.Let t be the space of linear transformations of llt which map llt

into lt, !t into 0, and consider f as 8'module relative to the com-position Xa = IX, a*1, R the representation of St. Let E be anyprojection of !t onto It and set f(a) - la*, El. Then /(a) satisfiesthe conditions of Whitehead's lemma. Hence there exists a D e X,such that f(a) : Da - lD, a*1. As we saw before, this implies thatIt has a complementary subspace which is invariant under 8. Sincethis applies to every submodule It, llt is completely reducible.

If 8 is a subalgebra of a Lie algebra E, then a deriuation D of

80 LIE ALGEBRAS

g into E is a linear mapping of I into E such that l

(29) lt,l,lD - u,D,l,l + u,,l,D\ '

for every lr,lz e g. It is immediate that the set O(8jl8) of deriva'

tions of I into E is a subspace of the space of lineaf transforma'

tions of 8 into E. Whitehead's lemma to the theorerh on complete

reducibility has the following important consequence on derivations:

Tuponpru 9. Let E be a fi.nite'dim,ensional Lie algebra of char'

acteri,stic A and. let g be a semi'simple suhalgebra of E, Then eaery

deriuati.on of g into E can be extended to an inner deniuation of sA'

Proaf: Consider E as 8-module relative to the multflplication [6/]'/e8 ,6e13. Thena der iva t ion D o f - 8 in to E de f ines , f ( ' ) : lD

satisfying the condition

f(llrlrl) - U,lrlD : fl r, lrDl + [l'D,li- lf(t), t,] - [f(tz), t,]

of Whitehead's lemma. Hence there exists a d e E such that lD:

f(t) : ld,IJ. Then D can be extended to the inner derivation deter'

mined by the element - d.We recalt that we have shown in Chapter II (Theqrem 2.11) that

if 8 is a completely reducibte Lie algebra of linear trlansformations

in a finite-dimensional vector space llt over a field of ] characteristic

0, then 8 : 8, O 6 where [!' is a semi'simple ideal and 6 is the

center. Moreover, the elements C e G are semi'simplb in the sense

that their minimum polynomials are products of distipct irreducible

potynomiats. we are now in a position to establish the converse

of this result. Our proof will be based on a field extension argu-

ment of the following type: Suppose we have a sdt J of linear

transformations in lJt ovet o. If g is an extension fileld of o, every

A e Z has a unique extension to a linear transformhtion, denoted

again by A, in Dto. In this way we get a set J - {A} bf linear trans-

formations in llto ovet 9. We shall now prove the ifollowing

Lpuu.o, 4. I-et 2 be a set of linear transformatioths in a finite'

dimensional aector space Ift ouer O and let 2 be the set of extensions

of these transformations to Wto ouer !), Q an extertsion field of A'

Su\fose the'set 2 i.n \fto is completely reducible. Then the original

X is comPletely ,reducible in Tft.Proof: Let Tt be a subspace of SJt which is invdria

l

nt under J


and let E be a projection on It. Then our criterion for comple-

mentation (Lemma 2) shows that Tt will have a complement which

is invariant relative to -I if and only if there exists a linear trans-

formation D of !)t mapping tn into It, It into 0 such that IAE) -

tAD) for all A e Z. lf Ar, Az, "', A* is a maximal set of linearly

independent elements in .X, and we set Bi: lArEl, then it suffices

to find a D such that IAD\: Br i : t,2, " ', k. This is a system

of k linear equations f.or D in the finite'dimensional space f of

linear transformations of TJt mapping Dt into yt, [t into 0. Thus if

we have the basis (Ur, Ur, " ', U,) for X, we can write Bi:

Ztr=r\rrtl,, lAr(Iuf :ZrTnrU,, D * li|lUi, then our equations are

equivalent to the ordinary system: DnTu"rdn: Ftu, i : L,2, '", k,

s : 1, 2, . ", r, for the d1 in O. Hence It has a J'invariant com-plement if and only if this system has a solution. We now pass

to !)to and the invariant subspace lfo relative to the set J of exten-

sions of the A e 2. Then our hypothesis is that lto has a J-invariant

complement in llto. Now the extension E of. E is a projection of

l]to onto !tp. Hence we have a linear mapping D of. IJto mappingl}to into lto, Tlo into 0, such that IA;DI : Bu = lA&), i : !,2, ' '', h-

The extensions Ur, Ur, ''', U, form a basis for the space of linear

transformations of fio mapping IJto into \lr, Ilo into 0' Hence if

D : Xid i.Jt then the 8 satisfy the system Xarie,be : Fa. Since the

rinc ?1d Fa belong to O, it follows that this system has a solution(d,, ..., d"), d's in O. Hence there exists a D e f such that[AD]:

[AE], A e 2, and so It has a J'invariant complement in lft.

We can now prove the following

Trrponpu 10. Let 8, be a Lie algebra of linear transformatians in

a fi.nite-dimensional uector space Wl oaer a field of characteristic zero.Then 8 fs completely reducible in Wt i,f and only if the followingcondi.tions hold: (1) I : Ir e 6, 8, a serni,-simble ideal and E thecenter and (2) the elements of 6 are semi'simple.

Proof: The necessity has been proved before. Now assume (l)

and (2) and let g be the algebraic closure of the base field. Then

the lemma shows that it suffices to prove that the set I of exten-sions of the elements of 8 is completely reducible in llto. The setof. !}-linear combinations of the elements of I can be identifiedwith 8c and similar statements hold for I' and O. Now let Ce6.Since the minimum polynomial of C in llt has distinct irreduciblefactors and since the field is of characteristic 0, the minimum

82 LIE ALGEBRAS

polynomial of C in lllo has distinct linear factors in A. Conse-quently, we can decompose fis as !J1", O ''' O IJt"* where

Tltrn : {x; I x; € fio, xiC : aix;}

and, ar, dzt . . ., ar are the different characteristic rootsi of' C. Since

AC : CA for A e g, ySl,tA c Y)ldi We can apply the same pro-

cedure to the Tft,, relative to any other D e G. This leads to a

decomposi t ion of IJto: IJI ,OIJI ,O.. .e) tn ' into f i r ivar iant sub-

spaces such that the transformation induced in the'IJtr by every

C e O is a scalar multiplication. To prove I completbly reducible

in TJto it suffices to show that the sets of induced transformations

in the TJtr are completely reducible and since the elenlents of G are

scalars in IJtr it suffices to show that 8r is completely reducible in

every !]t;. The invariant subspaces of IJtr relative to lJ' are in-

variant relative to J?8r, the set of g-linear combinatiofirs of the ele-

ments of .8,. Now J?8, is a homomorphic (actually isomprphic) image

of the extension aigebra 8ro, which is semi-simple. flence 9.8.- is

semi-simple and consequently this Lie algebta of linegr transform-

ations is completely reducible by Theorem 8. Thus wd have proved

that 8 is completely reducible in IJls and hence in 'lt;

We now shift our point of view and consider a finite dimensional

Lie algebra 8 of characteristic 0 and two finite-dimdnsional com-

pletely reducible modules lJt and yt for 8. We shhll show that

lft O Tl is completely reducible. Now the space p + :Ut @ Tt is a

module relative to the product (x + y)l: xl * !1, i e TJt, y e tt.

Evidently IS is completely reducible and 9Jt @ It is a Fubmodule of

S I $. Hence it suffices to prove that S I S is conlpletely redu-

cible. If we replace ti by 8/S where R is the kernelrof the repre-

sentation in $, then we may assume that the associalted represent-

a t ion R in S is 1 :1 . Then we know tha t 8 :8 ,@0 where f , i s

a semi-simple ideal and 6 is the center. Moreover, the elements

C*, C e O, are semi-simple. Now, in general, if ,R is a faithful

representation of a Lie algebra lJ, then the represerftation l? I R

in sap is also fa i thful . Thus, i f ae I and a' iF not a scalar

mUltiplication, then, since the algebra of linear tran$formations in

S A S is the tensor product of the algebras of linqar transform-

at ions in S, ao&a*, cf t81, I@aR and 181 arel l inear ly inde'

pendent, so d* I 1 + | & aR + 0. Hence if a@o :0, a* must be a

scalar, say a' : a. Then a@* :2a (in S I S) and ia : 0. Since

R is 1 : 1 this implies that a :0. We can now conclude that


I]san - Sfos * 6nsn where 8i'8" - [3, is semi-simple and Onso isthe center. Our result will therefore follow from the criterion ofTh. 10 provided that we can prove that every

g n s n : c e 8 1 + 1 8 c " ,

C e O, is semi-simple.Let I be the algebraic closure of the base field and let dr, dz, . . . r dk

be the different characteristic roots of C*. Then the proof ofTheorem 10 shows that lpo : Srr 0 Sr, O . .. O !$"* where xr.QR :

aixo,r, for rr, e E r. Hence (S @r$)p : To Sot$o : .Xt[tr; I Sr1 and

!C^@o : (a; + ar)l for every -/ € S"r 8 S"j. It follows that theminimum polynomial of 6aan has distinct ioots in (S I ![')o. Sincethis is also the minimum polynomial of 6nan in S I $, it followsthat this polynomial is a product of distinct irreducible factors.Thus C@* is semi-simple and we have proved

THoonpu 11. Izt 8, be a fi.nite-dimensional Lie algebra ouer a fieldof characteristic zero and let Tft and Tt be finite-dimensianal com-pletely reducible rnodules for 8. Then IX I tt ls completely re-ducible.

8. Representations of the aplit three-dimetuionalsimple Lie algebra

In $ 1.4 we called a three-dimensional simple Lie algebra S splitif ft' contains an element h such that ad /a has a non-zero char-acteristic root p belonging to the base field. We showed that anysuch algebra has a basis (e,f ,h) with the multiplication table

(30) [ehl - 2u, [fh]: lefl - h

The representation theory of this algebra is the key for unlockingthe deeper parts of the structure and representation theory of semi-simple Lie algebras (Chapters IV, VII, and VIID. We considerthis now for the case of a field CI of. characteristic 0. We supposefirst that @ is algebraically closed and that IJt is a finite-dimension-al module for S. The representation in Dt is determined by theimages E,F, H of. the base elements e,.f,h and we have

[E, H] [F, HI _ IE ,FJ -H .(31)

any three linear transformations E, F, H satisfying

84 LIE ALGEBRAS

these relations determine a representation of S and hence a S''

module. Let a be a characteristic root of. HandraOorresponding

characteristic vector: r + O, xH : ax. Then

(32) @E)H - x(HE + 2E) - (xD)(a + 2) '

lL xE + 0 then (32) shows that a * 2 is a characteristic root for

H and xE a corresponding characteristic vector. We can replace t

by xE and repeat the process. This leads to a seqtrence of non-

zero vectors t, xE, JcEz, "', belonging to the charaqteristic roots

a, a * 2, a * 4, . .., respectively, f.or H. Now f/ ha$ only a finite

number of distinct characteristic roots; hence, our se{uence breaks

off and this means that we obtain a k such thatt: xEk * 0 and

x E k * ' : 0 .

If we replace r by xEh we may suppose at the start that r + 0

and

(33) xH: ax , rE :O '

Now set ro : r and let ri: x;-tF. Then, analogous to (32)' we

obtain

34) xtH : (a - 2i)x; ,

and the argument used for the vectors rEd shows that there exists

a non-negative integer zz suCh that ro, )r,t, '''I Xvl vtu + 0 but Xrn+r:

0. Thus xF^+r - 0, xF^ + 0.

Then ri, o = i = m, is a characteristic vector oL H belonging to

the charac ter is t i c roo t a -Z i . S ince d ,d -Z ,d- '4 " " 'a -2m

are all different it follows that the n; ?fE linearly indEpendent. Let

Tt : xto oxi so that Tt is an (m * l)-dimensional subspace of 'lt-

We shall now show that It is invariant and irreduclble relative to

n. We first establish the formula

(35) x;E: (- ia + i(i - 1))rr-'

Thus we have )hE:0 as given in (35)' Assumei (35) for i - t'

Then

x; E -- r;-rFE - r;-t(EF - H)

- (- (i - L)a + (i - lxt - 2))ri-zF

- (a - 2Q - 1))rr-r

- (- ia + i(i - l))ri-r


as required. It is now clear from (34), (35), and xtF: )ci+r that 9ti s a S-subspace of llt. Since 11 - [EF] we must have tryTH : 0.This, using (34), gives (m * l)a - m(m * 1) : 9. Hence we obtainthe result that a,: rn. Our formulas now read t

!c;H - (m - 2i ')n , i :0, " ', rtt

( 3 6 ) h F : r i + t , i - - 0 , . . . , r n - 1 , x ^ F : 0

x s E : 0 , x ; E - - ( - m i + i ( i - l ) ) r , i - r , i - 1 , . . . , r n

and we note that in the last equation

-m i+ t ( l - 1 )+0 .

Now let llr be a non-zero invariant subspace of !t and let

! : F;x; * 0i+rri+r + "' * F^x* ,

Bi. * 0, be in Tlr. Then xrn: Fr'lF*-o € ltr. Hence by the lastequation of (36) every h € \\ and Itr : ll. Hence if ![l is S-irre'ducible to begin with, then lll - tl. In general, the theorem on com-plete reducibility shows that IIt is a direct sum of irreducible in-variant subspaces which are like the space !t.

We can now drop the hypothesis that A is algebraically closed,assuming only that @ is of characteristic 0. We note first thefollowing

LsMuA. 5. Let R be the spli,t three-dimensional simple Lie algebraouer a fi.eH A of characteristic zero and let e-sE, f -F, h-Hdefi,ne a fi,nite-dimensional representation of fi. Then the charac'teristic roots of H are integers.

Proof: If tJt is the module of the representation and .? is thealgebraic closure of O then llto is a module for Ro which satisfiesthe same conditions over Q as S over @. Then !]to is a direct sumof irreducible subspaces It with bases (No, Nr, ..., x*) satisfying (36).Hence if we choose a suitable basis for Tfto then the matrix of Hrelative to this is a diagonal matrix with integral entries. Hencethe characteristic roots of H in lltp are integers. These are alsothe characteristic roots of. H in Dt.'We

can now prove the following

THsonpM 12. Let fr be the split three-dimensional simfle Lie alge-bra oaer a fi,eld of characteristic 0. Then for each integer m:0,1,2,..- there exists one and, in the sense of isomorphi,sm, only

86 LIE ALGEBRAS

one irreducible fi-module \Il of dimensi,on m * l- TIt: has a basis

(xr, xr, ..., r,o) such that the representing transformatibns (E, F, H)

corresponding to the canonical basis (e,f,h) are giuen 4y (36).

proofi Let llt be a finite-dimensional irreducible rriodule for tr.

Then the characteristic roots of H are integers. Hence we can

find an integer a and. a vector x * 0 in Sl such that rH : dtc- As

before we may suppose xE:o. Then we obtain that a: m and

that 9lt has a basis (tcr, Nr, . . .,1c*) such that (36) holdl. These for-

mulas are completely determined by the dimensionalityt m * | of !)t'

Hence any two (rn * l)-dimensional irreducible modules for $ are

isomorphic. It remains to show that there is an irredupible (m + 1)-

dimensional module for S for every rn :0,1, ' ' '. ' To see this

y/e let lJt be a space with the basis (xo, trr,' ' ', r-) and we define

the linear transformations E, F,If by (36). Then we have

x{EH-HE) : (- mi * i(i - I))(m - 2(i' - l))ri-r

- (m - 2i,)(- mi * i(i - t))x+,

- 2(- rni * i,(i - 1))x*'

:2x rE ,

x;(FH- HF) : (m - 2(i * 1))16,', - (m - Zi)x*' l- - Zxt+t

: - Z x ; F ,

*{EF-FE) : (- mi. * i(i - r))n + @Q + 1) - $ + r)i)xt

: (m - 2i)r;: X ; H .

Hence E, F, and H satisfy the required commutation rblations and so

they define a representation of S. As befor€, ffi is, S-irreducible'

The theorem of complete reducibility applies herp also and to-

gether with the foregoing result gives the structure of any finite-

dimensional S-module.

g. The theorema of Levi and Maleev'Hariehlchand'ra

The .,radical splitting" theorem of Levi asserts that if E is a

finite-dimensional Lie algebra of characteristic 0 rwith solvable

radical Gi, then E contains a semi-simple subalgebfa 13 such that

E:8 + 6 . I t w i l l fo l low tha t 8 n 6 :0 so tha t S :8OG and


g =El@. Thus the subalgebra 8 is isomorphic to the difference

algebra of E modulo its radical. Conversely, if E contains a sub'

algebra 13 isomorphic to E/9, then 5J is semi'simple. Hence I n I :0

and since dim8: dimG + dimE/€: dimG * dim8, E - I + 6.

We note next that it suffices to prove the theorem for the case

9' :0, that is, g is abelian. Thus suppose 62 + 0. Then if

E : E/6t, dim E < dim E. . Hence if we use induction on the dimen'

sionality we may assume the result for E. Now -@

:61Q2 is the

radical of E and Etd =E/9. Hence E contains a subalgebra

E : E/@. As subalgebra of E, E has the form Erl6' where E, is

a subalgebra of E containing @'. Now 6t is the radical of E' and

ErlQ'=r E/g so that dim Er < dim E. The induction hypothesis can

therefore be used to conclude that Er contains a subalgebra I =

E/6, and this completes the proof for E.

We now assume that @2 :0 and for the moment we drop the

assumption that E : Ei6 is semi-simple. Now 6 is a submodule

of E for E (adjoint representation). Since @' :0, 6 is in the kernel

of the representation of E determined by_the module 6. Hence

we have an induced representation for E : E/@. For the cor-

responding module we have sD : [s, r]' s 9 6, D e E.

We can find a 1:1 l inear mapping o:b-+b' of. A into E such

thatF - b. Such a mapping is obtained by writing E : 6 CI CI

where 6 is a subspace. Then we have a projection of E onto S

defined by this decomposition. Since I is the kernel, we have an

induced linear isomorphism a of E onto 6; hence into E. It b --

s * g , s € 6 , g e 6 , t h e n b y d e f i n i t i o n b " : g a n d D - g s o t h a t

F : A: 5 as required. Conversely, le-t D -' b' be any 1 : I' linear

mapping of E into E such that b' :6. Then 6 : Eo is a com-

p t e m e n i o f 6 i n E . I f s e @ a n d b e E , t h e n s b : s d : [ s 9 ] - [ s & " ]

holds for the module multiplications in 6.

Let br,5r e E and consider the element

(37) lbi, b{l - tb,b,1" € E .

If we apply the algebra homomorphism b:! ot'

use of the property 6" : b we obtain lb9-bil -

Ibr6rl" :'l6Er1: lb'bzl. Hence we see that

E onto E and make

tE,El _ lb,b,) and

g(br, br) = 1b'r, -blzl

- fbrbrl" € g .

one verifies immediately that (D,, br) - g(br, br) is a bilinear mapping

o f E x E i n t o g .

88 LIE ALGEBRAS

Now suppose CI : E' is a subalgebra of E. T!"q tai, Sfl e 6;

hence g$i,6r)e 6 n 6 : 0 so that we must have-g(D,,bu) :0 for all

b, ,br . -The

converse is also c lear s ince g(br,br) :0r impl ies that

l6'r,5{t - [brbr)" € 6. Hence 6 is a subalgebra if and i only if the

bilinear mapping I is 0.If E' is not a subalgebra, then -we seek to modify o to obtain a

second mapping r of E so that E' is a subalgebra. Suppose this

is possible. Then we have a 1:1 linear mapping r of E into E

such that F? - b and, t|i|tl - lb,br7' = 0 for all b,, br. Now let p :

o - r. Then p is a linear mapping of E into E such that

F:F -T :5 -5 : Q.

Hence bo e g, and we can consider p aS a linear mapping of E

into 9. Also we have

e(b;, br) - lb{,El - lb,b,l": [ti + bi,6g +

-ull - [b,b,]o - [b,6'l': L6i5il - lbg,til - lb,b,l'

If sb is defined as before, we have t6-: [sD'I. Thr1s, if we can

somehow choose a comp]eryent of I which is a subalgebra then

the bilinear mapping 9(5,, &r) of E x E into 6 can be expressed in

terms of the linear mapping p of E into 6 by the formula

(38) s(6,,b,) - blb, - bg6' - lb,6,lo

Conversely, suppose we have a linear mapping p of E into 6

satisfying this iondition. Then r:6 -p is another 1:1 linear

,.nupping- of 6 into € such that =br

: b- and one can re-trace the

steps to show that [tl, Uil : lbrbrl' so that E" is subalgebra.

Our results can be stated in the following way:

Criterion. Let E be a Lie algebra, 6 an ideat il n such that

6, : 0 and set E: E/6. Then 6 is a E'module relatiae to the

composition sb - [sD]. Also there exist 1:1 linear rnappings 6 oJ

E into E such that F: D, b e E. If o is such a mapping then

s(br,br) -lbibll-lb,b,)" e q '

Moreoaer, 6 has a complementary space which is a subalgebra if

and only if there erists a linear mabbing p of E into @ such that

g(5r,6r) : blb, - 596, - fb,brlo .


We observe next that the bilinear mapping 9, which we shall

call a E factor set in 6, satisfies certain conditions which are con-

sequences of the speciat properties of the multiplication in a Lie

algebra. Thus it is clear that

(39) g(b,E) - o

which implies g(5r,8) - - g(br,br). We next write

Ib{,6{l : Ib,brl" a sQ,, b,)

and calculate

Itft , 6{lb{l : [[6,&,]" u",l + Ig(b,, F.,)oi"l- llb,brl6rl" * s(lb,br),F l + Ig(b,,6),t{l .

If we permute 6r,6r,Dg cyctically, add, and make use of the Jacobiidentities in E and E, we obtain

(40) s{b,6,1,a11 + [s(b,, 6j, 5j1 + g(16_,b_"1, b_,,)+ ls(br,b),5i11 e([5'b'), 6r) + [g(D', b,),bil - 0 .

Our proof of Levi's theorem will be completed by proving the

following lemma, which is due to Whitehead.

Lsuur 6. Iat 8, be a finite-dimensional semi-simple Lie algebra

of characteristic 0, Ilt a finite-dimensional S,-module and (lylr)--+g(lulz)

a bilinear mapping of 8, x 8, into TJt such that

( i ) s ( l , l ) : 0 ,(ii) g(fl,tzl,lr) + g(lr,lz)ls * g(Urlrl,l,)

i g(lr, lr)l' * g(llsl'), lr) + g(ls, l)12 - 0 .

Then there erists a linear mafling I --'f of 8 into Tlt such that

(iii) g(lr, lr) : lllz - lgl' - l|'lrlo .

Proof: Let S, gr,2i, t4i, T be as in the proof of Whitehead's first

lemma: 0 is the kernel of the representation, 8r is an ideal such

that 8 : R O,8r, (ur), and (ai), l : 1, .. ., ffi, are dual bases of It

relative to the trace form of the given representation, and I'is the

Casimir operator determined by the u; and. ud. We recall that l"

is the mapping r - I,l!,( xnr)u' in fi. Set /s : ui in (ii) and take

the module product with z'. Add for e. This gives

90 LIE ALGEBRAS

0 - Lg(ULlzf, ui)ui * g(lr l)T,,

+ xg(tlzu,f,lr)uo * 2@(lr, u;)I,)ydt , t

+ :.g([zrlrf, l2)ui + Z@@;,lr)l)uit L

: 9(1,, l)r + I,;q{lJ,l, u;)urt

+ \t(lzuil, lr)u' * \g(tr, u\[lruilt t

* T,(g(/', u;)ui)l, + >ig((tui,lrf ,lr)uit t -

* \t(ur lr)ltrui) + 2(g@tlr)ui)1, .t t

If we make use of luulj - Eauilt, luill _ Zgnfii,F;t : - ajt (cf. (28)) we can verify that(41) zg?r, ut)upil : \g(tr, [u;l,])ui(42) Zg(ur, tr)Urui) : Zg(u;lr),l,)ui .

These and the skew symmetry of g permitfour terms in the foregoing equations. Hence

and recall that

r

the cancellation ofwe obtain

(M)

(43) - 9(lr,l)r : l,,g(Ur,lrl, u;)u"

* \@(1,, ui)u')h t Z@@r,l)ui)I" .

If I' is non-singular we define

,L

lo - Zg(1, ui)u'T-'{ : r

Then (43) gives the required relation (iii). If /. is nilpotent, then,as in the proof of Whitehead's first lemm?, ffi -0, S-8, so thatthe representation is a zero representation. Then (ii) reduces to

(i i ') s(UJzl,lr) + g(tlrlrl,l,) * g(UslJ,lr) :0 .

Now let r denote the vector space of linear mappifrgs of g intotjt. we make this into an 8-module by defining f.or A€x., r,/eg,x(Al): -[xl]A, that is, Al: -(adl)A. It is east to see thatthis satisfies the module conditions (cf. S 1.0). For each / e g wedefine an element Ar e I as the mapping r + 9(x,/) e gn. Thenl - Ar is a linear mapping of I into I and

III. CARTAN'S CRTTERION AND ITS CONSEQUENCES 91

xApgrt: g(x,UJ"l) ,xA4lz: - g(f i l r l , l r) ,

xA4lr - - g(fxlr),lr) .

Hence the skew symmetry of I and (ii/) imply that

(45) Agp2t: Arrl, - Atrlt

Thus the hypothesis of Whitehead's first lemma holds. The con-

clusion states that there exists a p e X" such that A1: pl. Thismeans that we have a linear mapping p of I into IJt such that

(46) g ( x , I ) - - l r , l 7 o .

By definition of f; as module, this gives (iii). This proves the result

for the case /- nilpotent. If f is neither non-singular nor nilpotent'

then we have the decompositiorr of llt as Ut. @ Ilt' where the llti

are the Fitting components of SJt relative to f and these are + 0.

These spaces are submodules and we can rilrite 9(lt,lr) : 9o(lu lr) +gr(lr,lr), gt c TIt6. Then the g; satisfy the conditions imposed oD g,

so we can represent these in the form (iii), by virtue of an induc'

tion hypothesis on the dimensionality of !lt. This gives the result

for fi by adding the linear transformations for the lltr.

As we have noted before, the lemma completes our proof of

Leui's theorem. If A is a finite-dimensional Lie algebra of charac'

teristic zero with radical6 then there exists a semi'sirnple subalgebra

8, o f E such tha t E :809.A subalgebra I satisfying these conditions is called, a Leai factor

of E. A first consequence of Levi's theorem is the following result:

Conoru,nv 1. Let E, 6, and I be as in the theorem. Then

t 8 e l : E ' n 6 .Proof: We have E : 8@9 so that E' : t881 1 t8gl. Since

t!$el tr 6 have E' n 6 : ([88] n 6) + t8gl - [Eg].We have seen that tg8l S It the nil radical of E (Theorem 2.13),

so we can now state that E' n g g yt. We know also that the

radical of an ideal is the intersection of the ideal with the radical

of the containing algebra. Hence E' n 6 is the radical of E'. We

therefore have the following

Conorl,.onv 2. The radical of the deriaed algebra of a finite'dimensional Lie algebra of characteristic 0 is ni'lbotent.

92 LIE ALGEBRAS

We take up next the question of uniqueness of the I Levi factors.

It will turn out that these are not usually unique; hpwever, they

are conjugate in a rather strong sense which we shall now define.

We recalt that if. zelt, the nil radical of E, then ad a is nilpotent.

Since ad e is a derivation we know also that A : exP (ad a) is an

automorphism. Let lI denote the group of automorphisms generat'

ed by the elements exp (ad e), z e !t. Then we have the following

conJugacy

Theorem of Malcec'-Harish-Chandra. Let E: 6 @ I where 6 isa solaable ideal and 8, is a semi'simple subalgebra and let 8' be asemi-simple subalgebra of E. Assume E finite-dilnensional and ofcharacteristic 0. Then there exists an automorphism A e 2l suchthat 8f q 8.

Proof: Any /r € 8r can be written in one and onl$ one way asl, : ll + l{, where /l e S and /i e 6 so that we have the linearmappings I and d of ,8, into 8 and 6, respectively. Since 8r issemi-simple, 8, o 6 : 0; hence ,l is 1 : 1. lf- lz e 8r lhen

llrlrl-ll,Irl^ +IIJ.it": t/i/|l + uitn + ltitil + ltitil . '

It,t,)x - ullll ,lt,t,)" - vitn - lflil + ll",l{1 .

The second of these equations shows that UrlrT' € [EF] I It, the nil

radical of E. Since 8{ : 8, this implies that l{ etn fdr every /' €,8r

and so 13, g 8 O tt. We shall prove bi induction thdt there exists

an'automorphism Ai € ?I (A, - 1) such that 8fi g I + !t'i) whereltit is the ith derived algebra of [t. Since It is solvable this will

prove the result. Since we have proved that 8r g I + Tt it suffices

to prove the inductive step and we may simplify the notation and

assume that 8r g I + Tl'&'. Then we shall show that there existsA e lI such that 8f g I + 1{rc+t). If we use the notation introducedbefore, 5J, g U + tt(&) implies that fi € tt'&), /, € 8,. The first equa-

tion in (a8) implies that if we set z/r --lz,t l l , z € nf&), h€ 8,, thenthis rnakes Il'o' into an 8r-module. Now 1{ft+t) is a, submodule so

that 9t(h/!|t&+r1 is an 8r-module relative to Zlr:litij where e e It(*)and t :z*91{tc+r) . We now take thecosetsrelat ivei tot t (&+r) of theterms in the second equation of (a8). Since tl{lil € 9t

(e+'), we have

(u)

Hence

(48)


Jt,t,l" : vrtil - wTl -Tih -Et, .

Now set /(/,) - ti then I, -,f(t,) is a linear mapping of 8r into the

lr-module 91t'tr191t&+t' and the foregoing equation can be re'written as

(4e) f(U,t,D - fQ)lz - f(I,)l' .

Hence by Whitehead's first lemma there exists a 2 C g1ttryltrc+t)

such that f(l'):21t, which means that

(50) n : V7ll ot l{ = lz,/ll (mod ltt+rr; '

Let A - exp (ad e). Then

tf : t, + Ual + Uzllll,zlzl + . . .= h * Urzl (mod !t(&+r))

= /f + l{ + ltizl + lt{zl(mod 9t(&+r))(51)

= /l (mod tl 'u*") .

Now 8f = 8, and (51) shows that 8l g I + Tt'&*". We can therefore

prove the result bY induction on &.

Conor,urny 1. Any semi'simpte subalgebra of a finite'dimensionalLie algebra of characteristic zero can be imbedded in a l*ai' factor.

Proof: If. A is as in the theorem, then 8r is contained in the

Levi factor 8'-t.

Conor,uny 2. If E- 8,06:,lJz @6 where 8r and 8z are serni' '

simple subalgebras then there exists an automorphi'sm A e 2I such

that 8f : 8r.This is an immediate consequence of the theorem.

70. Cohomology groups of a Lie algebra

The two lemmas of \Mhiteheail can be formulated as theorems in

the cohomology theory of Lie algebras. Historically, these con'

stituted one of the clues which led to the discovery of this theory.

Another impetus to the theory came from the study of the topology

of Lie groups which was initiated by Cartan. In this section we

give the definition of the cohomology groups which is concrete and

we indicate an extension of the "l- non-singular" case of White'

head's lemmas to a general cohomology theorem. Later (Chapter

LIE ALGEBRAS

V) we shall give the definition of the cohomology groups whichfollows the general pattern of derived functors of Cartan'Eilenberg.

Let 8 be a Lie algebra, fi an 8-module. lt i 2 t, an'd-dimensionalNt-cochain for 8 is a skew symmetric f-linear mapping of 18 x8 x'' . x8(i t imes) into !]t. Such a mapping/sends an i ' tuple (lr, lr, " ' , l i),/o e 8, into f(lu - - -,li) e !]t in such a rvay that for fixed values oflr, . . . , lo-r, lq+rr ..., /; the mapping ln-+f(lr, ' ' ' ,1;) is a l irrear mappingof 8 into St. The skew symmetry means that f is changed to -/

if any two of the /i are interchanged (the remaining ones un-changed) . lt i: 0 one defines a $-dimensional \I\-cochain for 8 as a"constant" function from I to llt, that is, a mappinE I n u, u afixed element of lft. If / is an i-dimensional cochain (or simply "ana-cochain"), i )- 0, / determines an (f + l)-dimensiondl cochain /d,called the coboundary of f, defined by the formula

t+ r(52) fd(l',' ' ', /t*,) - X(- l)'*'-of(lr, ' ' ' ,

q = r

t + l A

+ E t- L) '+qf( l ' , . . . , I .n, -. .4 1 r = l

Here the ^ over an argument means that this argument is omitted(e.g., f(11,fr,7r1 -f(lr,t)). For r:0 this is to be interpreted as(/d'Xr) - il|, if / is the mapping r -> u c $!t.

The set Co(8, fi) of f-cochains for llt is a vector space relativeto the usual definitions of addition and scalar multiplication offunctions. Moreover /-'ld is a linear mapping, the coboundaryoperator, of Ci(8,IIt) into Ct*'(8, tn), i > 0. Besides the case

(53)

we have

fil(l) - sa1 , if. f: x -+ tr. ,

(54) fd(lb l,) : - .f(1,)1, + f(1,)1" - f(llJi) ,

(55) (.fd)(l ' , lr, lr) : f( lr, l")l '- f( l.r, lr)1, a f(11 ,l) lt- f(lr,ll'lrD + f(lr, U'l'l) - f(l',[/r/'l) .

An a-cochain / is called a cocycle it f6: 0 and a'coboundary if

f - g8 for some (f - l)-cochain g. The set Zi(8,0n) df i-cocycles isthe kernel of the homomorphism d of Ci into Ci+', sb Zi is a sub-

space of Ci. Similarly, the set B{(8, Dt) of e'coboundaries is a sub-space of. Ci since it is the image under d of Ci-t. It can be proved

fairly directly that Bi g Zi, that is, coboundaries lare cocycles.

tn, ' ' ' , l r*r) ln

, tr, " 'r l i*rrUolrl)


This amounts to the fundamental property: d2:0 of the coboundary

operator. we shall not give the verification in the general case at

this point since it will follow from the abstract point 9f viery later

on. At this poirit we shall be content to verify fd' :0 f.or f a

0- or a l-cochain. Thus if. f - u, that is, ,f is the mapping x + r,t

then fy(t) - yg and f62Qt,lz) - - ulzlt * ulrlz - utlrlzl'1 by the

definition of a module' If / is a l-cochain' /d(/" /z) is given by

(54). Hence, bY (55)'

.f B' (I r, l r, l r) :' f (t )I rt, * f (t')l'l' - f (Ll'l "1)l'

* f(t)t,tz - f(1,)1,t, + f(l,l,l)l, - f(l,)l'l'

1 fQ1)lzh - fU.l'zl)I, '1 f(ll'lzl)l' - f(l')UJ'l

+ f(ltslt,t,11) - f$J,l)l, * f(l')ll'ts7- f(l,tl,l,ll) + /(Jt,t'l)/' - f(l,)url'J+ JW'lUsI) '

one checks that this sum is 0; hence f6' :0 for any L'cochain f'

Once the verification D2 :0 has been made, one can define the

i-dimensi.onal cohomologt group (sbace) of P olyti'ae to the module wt

as the factor space ff1t, lft) = Zi(g, tJt)/Bd(8, tJt). lf i :0 we agree

;; ;;" B,:0 since there are no (i - l)'cochains. Hence in this

case it is understood that I/o(8,lJt) : Zo(g,Wt)- This can be identifi-

ed with the subspace l(ilt) of elements z e !)t such tl1p;t ul: 0 for

all t. such elements are called intariants of the module lIt'

H'(g,fi) : 0 means that Zi(g,Ift) : Bd(8, TJt), that is, every l-cocycle

is a coboundary. For i:1 this states that if l->f(l) is a linear

mapping of 8 into !ft such that - f(tr)t, + f(t,)I, - f(ll|tzl): 0, then

there exists a u in lft such that f(t):ul' This is just the typeof

statement which appears in Whitehead's first lemma' Similarly'

Whitehead's second lemma is a statement about the second coho'

mology groups. In fact, these two results can now be stated in

the following waY.

Tnponnu 13. If g is f.nite-dimensional semi-simple of characteristic

Q, then I/l(8, sJ|) : 0 and, H,(g,fi) : 0 for eaery finite-dimensional

mod,ule Ult of 8,.It is easy to see that if tX - $Jt, O S, where the l]ti are sub-

modules of Dt, then F/t(8, fi) : F/i(8, tJt,) o Frd(8' rJt')' This and

the theorem of complete reducibility permits the reduction of the

F/r(8, tlt) for finite-dimensional fi to the case 1lt irreducible and

96 LIE ALGEBRAS

here one distinguishes two cases: (1) In8 + 0 and (2) Ut8 :0. Inthe second case irreducibility implies dim IIt : 1, so; Ilt can beidentified with the field @. Then an r'cochain is.a skew symmetrica-linear function of (/t, ...,1t) with values in0, and sirtce the rep'resentation is a zero representation, the coboundary fofmula redu-ces to

(56) fd ( l r , - . . , l i * r ) : H t - l ) ' ro . f ( l r , . . . , fo , . . . , t , , . - . 1 , ,1 ;a1 , l lo l , l )

It turns out that "r'r-*r-simple

Lie algebras the cohomologygroups with values in l}t -- O are the really interesting ones, sincethese correspond to cohomology groups of Lie groups. On the otherhand, the case lltS + 0 is not very interesting (for sertri'simple 5J,

finite-dimensional irreducible llt) except for its applications to thetheorem of complete reducibility and the Levi theoredl, since onehas the following general result.

Tnnonru 14 (Whitehead). Izt I be a finite-dimensional serni'siruple Lie algebra ouer a fi.eld of characteri,sti,c 0 and let TIt be a

finite-dimensional irreducible module such that SJIS * 0. Then

/ / ' (8 , ! ) t ) :0 fo r a l l iZ0 .lf. i:0 the irreducibility and IJIS + 0 imply that u8:0 holds

only for u :0. This means that F/o(8, fi) : 0. The proof for

i > 0 is similar to the proof of the case: I- non-singular, in the

two Whitehead lemmas. 'We

leave the details to the r€ader.

1.L. More on eomplete reducibilitA l

For our further study of this question we require a notion of a

type of closure for Lie algebras of linear transformations and an

imbedding theorem for nilpotent elements in three'dimensional split

simple algebras. The first of these is based on a spedial case of

a property of associative algebras (the so-called Wedderburn principal

theorem), which is the analogue of Levi's theorem on L[e algebras.The result is the following

THnonpu 15. I-et \ - Olrl be a finite-dimensional algebra (associa'

tiae with identity l) generated by a single elernent x ouer 0 of charac'

teristi,c zero and let ft be the radical of 11. Then \l contains a serni'

simPle subalgebra \\ such that \I: ?I' O ft.Proof: Let /(i) be the minimum polynomial of r andi let

I I I . C A R T A N ' S C R I T E R I o N A N D I T S C o N S E Q U E N C E S 9 T

(57) fQ) : rr(J)"tnz(),)"'"'n,(f,)"

be the factorization of f( ) into irreducible polynomials with

the leading coefficients one such that nQ) + r.t[) if i + i and

deg zi(,i) > 0" We note first that if all the ei = 1: ?I -has

nS lon-

zero nilpotent elements (cf.. p. 47), so lI is semi-simple and there

is nothing to Prove. In anY case, set

(58) f'(l) : r'Q)rc2(l)"'n,(l)

and z : f r ( r ) . Then i f e:max(ei) , z" : ( f r (x))" : rcr(x)e" ' r , (x)c:$ '

so that z is nilpotent. Since lI is commutative, the ideal (e) gene-

rated by z is nilpotent; hence (e) g s. on theotherhand, ft@)=o

(mod (e)). Hencs the minimum polynomial of the coset i : tr + (z)

in ?I/(e) is a product of distinct prime factors. Since .f generates

a/Ui), this means that ll(e) is semi-simple. Hence (z) - ft. It

follows also easily that the minimum polynomial of r : r * fr is

fr(l). Hence it suffices to prove that lt contains an element y whose

minimum polynomial is fr(A). \4Ie shall obtain such an element by

a method of "successive approximations" beginning with Jr': I'

To begin with we have f{r):0 (mod S) and r= nr(modS)' Suppose

we have already determin ed, xr such that fr@r) = 0 (mod gry&) and

r = rr(mod S). Set rr+, : I* * eu where r,o is to be determined in

fte so that fr(rt +r): 0 (mod no*'). we have, by Taylor's theorem

for polynomials,l t l r - t '

f,(xr+r) - f,(xr* w) -f{x) + f!(x*)w *t#w' + "' '

Since the base field is of characteristic 0, ,f,(i) has distinct roots

in the algebraic closure oI O. Hence /'(i) is prime to the deriva-

five f trQ). It follows that V -ffi : f

'r(r*) * S has an inverse D

in ?I/fi. Set a,o - - f,(x)a. Then w = 0 (mod S&), so that

f,(xx+) = ft@) + f ''(x)w (mod Yte*')

= fr@i - f'r(r*)afr(rr) (mod no*')

= fr(xi - f{tck) (mod S**')

= 0 (mod nt*').

Thus we have determin€d .f,r+r such that f{n*r) = 0 (mod s&*t) and

I z frrc+r (mod fr). Since S is nilpotent this process leads to a y

s u c h t h a t f { y ) : 0 a n d y : r ( m o d f r ) . H e n c e U ' - o l y ) s a t i s f i e s2[ : lI, o a. since the minimum polynomial of .7 is .f'(A), it follows

98 LIE ALGEBRAS

that the minimum polynomial of y is /,(i) also.We can now prove

Tnsoneu 16. Izt X be a linear transformation in a ,fi,nite-di,men-sional aector space oaer a rteH of characteristic zero. Then we canwrite X: Y * Z where Y and Z are polynornials in $ such thatY is semi-simple and Zis nilpotent. Moreoaer, if X - 11 * Zr whereYr is semi-simple and Zt i.s nilpotent and Yt and Zt cdmmute wi,thX, then Yr : Y, Zt: Z.

Proof: The existence of the decomposition X: Y *,Z is obtain-ed by applying Theorem 15 to the algebra @fXl. Now supposeX : Yt a Z', where I.r and Z, have the properties stated in thetheorem. Since IZ and Z are polynomials in X they commute withY, and Zt. We have Y - Yt: Z, - Z. Since Z andl, Zt are nil'potent and commute, Z - Zt is nilpotent. Since Y land Yt a;resemi-simple and the base field is of characteristic zerb, the proofof Theorem 11 shows that Y - Yt is semi-simple. Since the onlytransformation which is both semi-simple and nilpoten{ is 0,

Y - Y t : 0 : Z - Z t .

Hence Y : Yr, Z: ZyWe call the uniquely determined linear transformatidns Y and Z

of Theorem 16 the semi-simple and the nilpotent compodents of X.

Dprrr.rrrrou 3. A Lie algebra I of linear transformptions of afinite-dimensional vector space over a field of characteristic 0 iscalled almost algebraic* if it contains the nilpotent and semi-simplecomponents of every X e 8.

To prove our imbedding theorem we require the following twolemmas.

Lpuui 7 (Morozov). Let 8 be a fi.nite-dimensional Lie algebra ofcharacteristic 0 and suppose 8, contains elements f , h such rthat [fn1 -- 2f and h e tS/1. Then there exists an element e € 8, buch that

* This concept is due to Malcev, who used the term spli,ttablic. We havechanged the term to "almost algebraic" since this is somewhat weaker thanChevalley's notion of an algebraic Lie algebra of linear trarjrsformations.Moreover, we have preferred to use the term "split Lie algebra" in a con-nection which is totally unrelated to Malcev's notion.


(59) lehl :2r, lef l - h ( l fh l : -Zf) .

Proof: There exists a ze9, such that h-[zf]. Set F:adf,H : ad, h, Z: ad z so that we have

( 6 0 ) l F H l : - 2 F , H : l Z F i .

The first of these relations implies that F is nilpotent (Lemma2.4). Also

tlzhl - 22, fl - llzf lhl + Lzlhfll - Z[zf I: 0 * 2 h - 2 h : 0 .

Hence [zhl - 2z * h where fr € 0, the subalgebra of elementsr such that lxfl - Q. Since tFHl -- - 2F, if. b e S, then

bHF -- b(FH + 2F): Q .

Hence bH e S and so SH s S. Also we have

[ZF' ] : lzF lFr- '+ FIZFIF;-s + . . . + Ft - ' IzFl: f {P i - r + FHF|- '+ . . - + F i - tH

and since HFk : FkH +%kFk, we have

(61) [zF'l : Fi-'(H + 2(i- 1) + H + 2(i - 2) + "' + H)- 'rt-t(H

+ (t - 1)) .

Let b e A n SF'i-t. Then b - aFi-' and bF : aFn : 0. Hence

iaPi-'(H + (t - 1)): a(zFt - Fnz): (az)Fi e 8Fi .

Hence b(H + (t - 1)) e R n !Fi. It fotlows from this relation andthe nilpotency of. F that if D is any element of S then

(62) bH(H + L)(H + 2) ... (H * m) - sfor some positive integer m. Thus the characteristic roots of therestriction of H to R are non-positive integers. Hence H - 2induces a non-singular linear transformation in S and consequentlythere exists a yr e S such that yt(H - 2) : rr where rr is the ele-ment such that lzh):22 * rr. Then lyrh):Zlt * xr. Hence if weset z : z - Jr we have lehl - 2u. Also lefl: lzfl - h. Hence (59)

holds.

Lpuu^n 8. Let t, be a Li,e algebra of linear transformations in a

fi.nite-dimensional aector space oaer a field of characteristi,c 0. Sufifoseeuery nilbotent elem.ent F + 0 of 8, can be imbedded in a subalgebra

lOO LIE ALGEBRAS

with basi.s (E, F, H) such that tEHl - 28, l"FHl - - 2F, tEFl - H.

Let ft be any subalgebra of ,J which has a complementary space Il

in g inaariant under multiplication Dy s: I : 'Q G) Yt' ltts]i g tt' Then

fi has the ProPerty stated for 8.Proof: Let F be a non -zero nilpotent of R. Then we can choose

E and // in 13 so that the indicated relations hold. Write f/:

Hr* Hr, H, € A, I /2 en, E- Et* Ez, Er€. S, E e Tl . Then we

have -2F - IFH) -- [FH') + [FH,] and' IFH,I e fi, [FH2]:e ft. Hence-zF - [FH,l. Also 17 -- lEFl: lE,F1 + lE,Fl. This lmplies that

Ht : IE,FI e tAFl. Thus f/, satisfies for F, S the conditions on 'I{

in Lemma 7. Hence there exists E', H' in n such that IFH'I -

- 2F, IE'H'l -- 2E', [E'F] -- H' . The subalgebra generated by

F, 8,, H, is a homomorphic image of the split three'dimensional

simple algebra. Since F + 0 we have an isomorphism, so that

F, E' , H' arelinearly independent and satisfy the required conditions

We can now establish our second criterion for complete reducibility-

Tssonpru 17. Let g be a l.ie algebra of linear transformations in

a fi.nite-dimensional aector space Yt ouer a field of chayacteristic 0'

(1) Assume g completely reducible. Then euery non-zbro nilpotent

elernent of g can be imbedd,ed in a three'dimensional sflit sirnfle

subalgebra of g and 8, is almost algebraic. (2) Assume that eaery

non-zero nilfotent elernent of t can be imbedded in a three'dimensional

simpte subalgebra of g and that the center 6. of 8 is almbst algebraic.

Then 5J is comPletelY reduci'ble.

Proof: (1) Assume ll is completely reducible and lbt 6 denote

the complete algebra of linear transformations in 9Jt- Let F be a

nilpoteni linear transformation and let st - tjtr e ilftf O ' ' ' G) 9Jt'

be a decomposition of IJt into cyclic invariant subspaces relative

to F. Thus in 9lt; we have a basis (tr', Ji,,., ' ", r^;) such that r1F:

ri+r, fi^rF : O. We define f/ and E to be the linear transformations

leaving'"u.ty !Jl; invariant and satisfying xiH : (mt - ZJ)xi, xnE:0'

r iE:q-m,i +iU -1))r ; - , , j>0 (cf . (36). Then as in $8, [E/{ ] :

2 E , | F H } _ - 2 F , t E r ] : H . T h i s s h o w s t h a t F c a n b e i m b e d d e din a subalgebra AE + OF + OH of. the type indicated. We shall

show next that we can write @t :8 O tt where S is a sub'

space such that [!t8] g tt. It will then follow from Lemma 8 that

u.r"ry nilpotent element + 0 of I can be imbedded in a split three'

dimensional simple subalgebra of [3. We recall that d^u as module

relative to Il (adioint representation) is equivalent to Dt I tll*, tn*

III. CARTAN'S CRITERION AND ITS CONSEQUENCES TO1

the contragredient module. It is also easy to see that rlt* is com-pletely reducible. Hence, by Theorem 11, tn S rlt*, and consequent-ly oz, is completely reducible relative to 8. Since I is a submoduleof @z relative to 13, there exists a complement !t such that Gz:I O tt, [9t8] s tl. This completes the proof of the first assertionin (1). Now let x be any element of I and let y and z be thesemi-simple and nilpotent components of X. Then

ad6X: adGY * adsZ ,

[ad6lzad6Zl:0 and ad.gZ is nilpotent (Z^:0 implies (adgZ)2*-':0).Also the identification of G with !x I Dl* and the proof of Theorem11 show that ad6lz is semi-simple. Hence ad6Y and ad,6z are thesemi-simple and nilpotent components of ad6X and so these arepolynomials in adgX. Since 8ad6X:c.8 and aci6lr, ad6Z are poly-nomials in ad6X, 8ad6Y s 8, I ad6Z g t. Thus L-+ILyl, L-\LZ)are derivations in Il. we can write ,8 : .8'O 6 where lJ' is semi-simple and o is the center. Since the derivations of t' are allinner it follows that any derivation of I which maps G into 0 is aninner derivation determined by an element of 8'. since z is apolvnomial in x, IXCI - 0 implies [zc] -e. This impries that thederivation L-rVZl maps 6 into 0. Hence there exists a Z, e g,such that lLZl - [LZrl, L e 8. Since Z is nilpotent, ads,Zr:adg,z is nilpotent. since ll' is semi-simple, the result just proved(applied to ad8') implies that there exists an element u e fJ' suchthat lad,g,Z', adg, U] : 2 adg,Zr. Then lzrul : 2Zr, which impliesthat Zr is nilpotent. Since lXZl: g, lxzrl: 0 and since Z is apolynomial in X, lZZ,l: Q. It now follows that Z - Z, is nilpotent.Since [L, Z - Zr]:0, L € 8, and Z - Z, is in the enveloping as-sociative algebra 8*, z - z, is in the center of [J*. since

-g* i.

completely reducible and z - z, is nilpotent, this implies Z - 21: es o Z : Z r e 8 , . H e n c e a l s o Y : X - Z e g . T h i s c o m p l e t e s t h eproof that I is almost algebraic. (z) Assume g has an almostalgebraic center and has the property stated for nilpotent elements.Let € be the radical of I and let F e tsgl. Then we know thatF is nilpotent (corollary 2 to Theorem z.g). rf. F is not zero itcan be imbedded in a three-dimensional simple subalgebra.R. Sinces n @ + 0 and s is simple,,R s 6 which is impossible because ofthe solvability of. @. Hence F: 0 and [86] * Q. This impliesthat 6 : O the center. By Levi's theorem ,8 : O @,8, where gr isa semi-simple subalgebra. Since 6 is the center this implies that

LOz LIE ALGEBRAS

8, is an ideal. We can now invoke Theorem 10 to pfove that 8

is completely reducible, provided that we can show 1 that every

C e 6 issemi-s imple. Nowweareassumingthat C - D+ E where

D is semi-simple, E nilpotent, and D and E are in 6. It' E + 0 we

can imbed this in a three'dimensional simple subalgebra. Clearly

this is impossible since E is in the center. This coinpletes the

proof of (2).It is immediate that if 8 is almost algebraic, then the center 6

of I is almost algebraic. Hence we can replace the as$umption in

(2) that 6 is almost algebraic by the assumption that I is almost

algebraic. We recall that the centralizer 6s(S) of a subset S is the

set of elements y e I such that [sy] :0 for alt s e S'j This is a

subalgebra of 8. We shalt now use the foregoing criterion to prove

Tsnonpn 18. Let g be a comptetely redacible Lie algebta of linear

transformations in a finite-dimensional aector space Tlt', of charac'

teristic zero and, let & be a completely reducible subalgebra of 8'

Then the centralizer 8r: 6e(8') is completely reducible. iProof: Let X € 82. Then since I is almost algebraid, the semi-

simple and nilpotent parts Y, Z of X are in 8. Since these are

polynomials in X, tCX):0 for C €,8r implies lCYl F 0: tCZl'

Hence Y, Z e.8, and 8, is almost algebraic. We shalll show next

that I : ,8, O 8a where 8, is a subspace of I such that [8a8r] tr '8r'

It will then foltow from Theorem L7 and, Lemma 8 that every nil-

potent element of 8, can be imbedded in a three'dimensional split

iimpte algebra. Then 8z will be completely reducible by Theorem

L7. Now we know that adgSr is completely reducible (proof of

Theorem 17). Since 8 is a submodule of 6 relative to 8r1, I is com-

pletely reducible relative to adg8r. Thus we may writd

8 : f i , O l l t , O " ' G ) D t *

where [!]ti[,l g TJI;, i : !, . . ., k, and tJJti is irreducible relative to

adg8,. we assume the !)ti are ordered so that Plt;8,J :0, i -

1, 1..,h, and [f ir8,] +0 if j > h. Since the subset & of elements

zi such that [zt8r] :0 is a submodule of g]t,i, it is immediate that

8z : lltr + "' * !]tr -

Set,8s : IJlp+r + ... * I ltr. fhen I : 8rO8'. If. i > h, the\[!]tr '8' l +0

and [1]t;8,1 + [[!]tr$.l8,l + ... is an lr-submodule + 0 of [ti. Henceglti : pltr.8'l a ttlltr8rl,8'l * ' ' '. This implies that


g, - [.8r8,] + [[g3g'Jg'] + ... s [gg,] .

On the other hand, I : ,8, @,Iai hence, [88,] - [8r8,J s ga. Hence8, : [88,J and

[9,8,] - [[gg,lgz] tr [g[9,9,]l + tlsg,lg,l tr [gg,] : g, .

This shows that 8c is a complement of .8, in 8 such that [fJs.8e] s 8a,which is what we needed to prove.

Exereiees

In all these exercises the characteristic of the base field will be zero, andunless the number is indicated with an asterisk the dimensionalities of thespaces will be finite.

1. Show that if S is a cartan subalgebra of 8 then 0 is a maximal nil.potent subalgebra of 8. show that the converse is false for onz@ > 2).

2- Let 6 be a nilpotent Lie algebra of linear transformations in ![l andlet sl - !n0 @ il?t be the Fitting decomposition relative to 0. Show that if@ is infinite, then there exists an A e 0 such that fio = Wto;., xlh : fllta,ffic* the Fitting components relative to A.

3. Show that the diagonal matrices of trace 0 form a Cartan subalgebrain the Lie algebra 8 of triangular matrices of trace 0. Show that g iscomplete

1. Let I be the subalgebra of e2s of matrices A satisfying s-rAts = - |where

s: (; ,J)

(This is isomorphic to an orthogonal Lie algebra.) Show that the diagonalmatrices

diag { l t , . . . , f , r , - ) r ,

form a Cartan subalgebra of B.5. Same as Exercise 4 but with S reptaced by

e : ( 1 : ) .r - 1 1 0 ) '

6. Generalize Exercise 2.9 to the following: Let B be a Lie algebra, 0a nilpotent subalgebra of the derivation algebra of g. Suppose the onlye lemen t l €8 such t ha t lD=0 fo r a l l D€S i s l : 0 . Then p rove tha tg i snilpotent.

7. Show that if 8r is a semi-simple ideal in 8 then 8=8r@gz where gris a second ideal.

- i r l

104 LIE ALGEBRAS

8. Let S be an ideal in 8 such that 8/S is semi-simple. Show that there

exists a subalgebra 8r of I such that 8: SO8r.9. Let 8 be simple over an algebraically closed field @ and let /(4, b) be

an invariant symmetric bilinear form on 8. Show that / is a rhultiple of the

Kitling form. Generalize this to semi-simple 8.

10. Let Slr be the n-dimensional irreducible module for d split three-

dimensional simple algebra S. Obtain a decomposition of. llllns lll' into

irreducible submodules.1l*. Let e, h be elements of an associative algebra such thdt llehlhl: 0.

Show that if h is algebra,in, that is, there exists a non-zero polynomial d(f)such that 6(h) :0, then lelal is nilpotent.

l2*. Let ?I be an associative algebra with an identity element I and suppose

?I contains elements e, f, h such that lehl -- %, Ifhl - - 2f , lelJ : h. Show

that i f d(h) e?I is a polynomial in h then €id(h):6Qt,+2i)ei, i :0,t ,2,

6(lift : fi6(ll + 2i). Also prove that if r and n are posifive integers,

r 5 n, then I

lrlzj r-zJ' f c" , i f I @+n-t)s i - t+ r

J = 0 t = l

r

I ' ' ' Ie"flf l ' ' '"f l =

. / n \ t n - r \ .wnere cmr| =

\ I l\ r _ zj)r .

l3*. 21, h, e, f as in Exercise 12. Show that if e- : 0, then

zm- l

g ( h * m - i ) : o

14. Prove that if e is an element of a semi'simple Lie algehra I of char-

acteristic zero such that ad e is nilpotent, then eR is nilpotGnt for every

representation ,B of 8.15. Prove that if I is semi-simple over an algebraically closod field then

8 contains an e * 0 with ad e nilpotent.16. Prove that every finite-dimensional Lie algebra * 0 over tan algebraic'

ally closed field has indecomposable modules of arbitrarily highlfinite dimen'

sionalities. (Hint: Show that there exists an e e I and a reprbsentation .B

such that eE is nilpotent * 0. If $t is the corresponding module, then the

dimensionalities of the indecomposable components of ff|, D?8tIt, $8m8![t, '"

are not bounded.)17. Prove that any semi-simple algebra has irreducible modules of arbi-

trarily high dimensionalities.18. Show that the derivation algebra of any Lie algebra [s algebraic.

(Hint: Use Exercise 2.8.)19. A Lie algebra I is called reiluctiae if ad I is completely reducible.

Show that I is reductive if and only if I has a 1 :1 completOly reducible

representation.


20. A subalgebra s of I is called reductiae in I if. adgs is completelyreducible. Prove that if 8 is a completely reducible Lie algebra of lineartransformations and s is reductive in I then s is completely reducible.

21. Show that any reductive commutative subalgebra of a semi.simple Liealgebra can be imbedded in a Cartan subalgebra.

22. Show that any semi-simple Lie algebra contains commutative Cartansubalgebras.

23. Let A be an automorphism of a semi-simple Lie algebra 8. show thatthe subalgebra of elements gr such that U(A - L)^ = 0 for some ?c is a reduc-tive subalgebra. (Hint: Use Exercise 2.5.)

24. (Mostow-Taft). Let G be a finite group of automorphisms in a Liealgebra. show that I has a Levi factor which is invariant under G.

25. Let /.(i) = det (tl - ad a), the characteristic polynomial of ad a in aLie algebra 8, and let D be a derivation of U. Show that if t is an indeter-minate, then

/"+nn()t) = fo(A) qmod t2) .

(Hint: use the f.act f,,1()') : .fo?) if A is an automorphism and the fact thatexptD: l * tD+( tzDzpD+. . . is a wel l -def ined automorphism in gp, pthe field of power series in t with coefdcients in @.)

26 . Wr i t e f " ( l ) = l " - q (a ) l o - t +n (a ) l o -? + . . . - r - 1 - l ) , r r ( a )p - t . . . andlet rt(ar, . . . , ar) be the linearized form of rt defined by

c t { a r , . : . , a t ) * " . * o r )

- ! r 1 ( a r * . . . * d 1 * . . . * a d * ) 1 6 ( o r * . . . * , 6 1 * . . . * , 6 r * . . . * a i

) r a ( o r * . . . . + 6 t * . . . * 6 r * . . . * 6 * . . . . * o r ) * . . . l .J < K < L

show that ri(or, ' ' ' , ar) is a symmetric i-l inear function and that

r t (atD, az, . . . , a) * f i (er , ezD, a l t . . . , a) * . . . I t ,1(ar , . . . , ( t r t -1, atD) = e

for any derivation D.

:f ["(o'

CHAPTER IV

Split Semi-Simple Lie Algebras

In this chapter we shall obtain the classification of simple Liealgebras over an algebraically closed field of characteristc 0. Thiswas given first by Killing, modulo some errors which were cor-rected by cartan. Later simplifications are due to weyl, van derwaerden, coxeter, witt, and Dynkin. In our discussion we shallfollow Dynkin's method which is fairly close to Cartan's originalmethod. However, w€ shall formulate everything in terms of"split" semi-simple Lie algebras over an arbitrary base field ofcharacteristic 0. It is easy to see that the assumption of algebraicclosure in the classical treatments is used only to ensure the exist-ence o f a decompos i t ion o f the a lgebra as 8 :6@[email protected]@.. .Otowhere $ is a Cartan subalgebra and the 8o are the root spacesrelative to €. This can be achieved by assuming the existence ofa "splitting cartan subalgebra" (cf. g l). It appears to be clearerand more natural to ernploy this hypothesis in place of the strongerone of algebraic closure of the base field. For the benefit of areader who has some familiarity with the associative theory itmight be remarked that the split simple Lie algebras which aresingled out in the classical theory are the analogues of the simplematrix algebras A" of the associative theory.

A part of the results of this chapter (the isomorphism and ex-istence theorems) will be derived again in Chapter VII in a moresophisticated way. In Chapter X we shall take up the problem ofextending the classification from algebraically closed base fields-or from split Lie algebras-to simple Lie algebras over any fieldof characteristic 0. It should be noted that the classification weshall give is valid also for characteristic p ;e 0 unde r fairly simplehypotheses which are strongershown by Seligman and, in anSeligman.

than simplicity. This has beenimproved form, by Mills and

[107]

108 LIE ALGEBRAS

l. Propertiee of roo|'s and raot s4xtceg

We shall call a Cartan subalgebra b af. a finite-dimensional Lie

algebra 8, a splitting Cartan subalgebra (abbreviated s.c.s.) if the

characteristic roots of every adgh, h e &, are in the base field. We

shall say that g is split if it has a splitting Cartan subalgebra. If

the base field @ is algebraically closed, any finite-dimensional I is

split and any Cartan subalgebra is a s.c.s.Example. Let I : Ant and let $ denote the subalgebra of diago-

nal matrices. We have seen ($ 3.3) that 6 is a s.c.s. in Anr,, so A*t

is sptit. Next let O be the field of real numbers and; let A be a

matrix whose characteristic roots fr (in the complex field) are dis-

tinct but not every Ei e o. Then the polynomial algdbra @t.41 is

a Cartan subalgebra of 8. The characteristic polynomiall of ad A is

IlT.t-rQ - G, - Fi)) and some of its roots are not in O. 'Hence OtAl

is not a s.c.s.In the remainder of this chapter 8 will denote a ,split finite-

dimensional semi-simple Lie atgebra over a field A of characteristic

0, O will be a splitting Cartan subalgebra of I and wer shall write(a,b) f.or f (a,b): tradaadb, the Kil l ing form on 8. We know

that (a, D) is non-degenerate. Our assumption on O is that adg$ is

a split algebra of linear transformations. Hence we knpw that we

can decompose 8 as

( 1 ) 8 : b O 8 , O t l e O . . . O 8 cwhere d, 9, .. ., d are the non-ze11o roots. These are llinear func-

tions on 6 and ,8, is the set of elements ro € 8r such that

r , ( a d h - a ( h ) ) ' : 0 f o r s o m e r : r ( h ) , / z e $ ( c f . $ 2 . 4 a n d S 3 . 2 ) .In the same way 0: 80, the Fitting null component of I relative

to ad $. We have [t]"8p1 I 8a+p if. a * I is a root while [8"8p1 - g

if. a * I is not a root. Our first task will be to obtairl additional

information on 6, on the rcots a and on the corresppnding root

spaces 8". We shall number these results by Roman numerals.

I. If a and p are any two roots (including 0) and P + - a, then

8,a and 8,p are orthogonal relatiue to the KiUing forrn.Proof: We show first that 6 : 8o -l- '8" if d + 0' Let h e 0'

0d € ga and choose h' so that a(h') + 0. Then the re$triction of

ad h' to 8, is a non-zero scalar plus a nilpotent and so this is non-

singular. It follows that for any *k

: I,2, "' we can find an

et, egc such tbat ea- [.. . tnp|ff i- lr '1. Since the Kil l ing form

Iv. SPLIT SEMI.SIMPLE LIE ALGEBRAS 109

is invariant we li,,"or: ([. . . ret'h') . .. h'], h)

- 1et\,tff in'hl " ' l) .

Since O is nilpotent, & can be chosen so that lh' " ' lh'hl " 'J : 0'

Then the above relation implies that (ed' h):0' Thus 6 r 13" for

a * 0. Now tet g * - a and let ep e 8a. As before, write eo :

lett h'1. Then (en, eo) - (lel'h'\, €F) : - (h' ,lett epD. I{ a * I is a

root it is non-zero and lettepl€ 8o+p. Hence (h',lettepJ):0. If

a * F is not a root, Iet\ep): 0 and again (h',let\epJ) : 0. Hence

(ea, ep): 0 and fJ" I 8p.

II. 6 i,s a non-i,sotropic subspace of 8, (relatiue, to (a, b)). If a is

a root, then -a i.s a root and 8,, and 8,-r are dual spaces relatiae

to (a, b).Proof: If. ze$ and z L6, then e I8 since, bY I, e IScfor all

a*0. Then z-*0 by the non-degeneracy of (a,b). If c is a root

and -a is not a root, then 8, Ilp for every root F. Then 8" 18

contrary to the non-degeneracy of the Kifiing form. Also the

argument used for a :0 shows that if. z + 0 is in 8a, then there

exists a w G 8-o such that (z,w)+0. Simi lar ly, i f w +0is in8- ' ,

then there exists a z Q.8o such that (2, w) + 0. This shows that

8" and ,8-o ar€ dual spaces relative to (a, b).We recall that the matrices of the restrictions of ad h, h e O, to

8o c?n be taken simultaneously in the form

($2.6). If dim[o: 1a ?1d h,h e 6, then this gives the formula

l.'^;̂l( 3 )

We recall also that a(ht) -- g for every h' e 6'. These results

imply our next two results.

III. Tlrcre are I linearly independent roots where l: rto: dim6.

Proof: The roots are linear functions and so belong to the conju-gate space 6* of O. We have dim 6* - /. Hence if the assertionis false, then the subspace of O* spanned by the roots has dimen'

110 LIE ALGEBRAS

sionality l' < l. This implies that there exists a non-zero vectorh e b such that a(h): 0 for every root a. Then (3) implies that(h, k): 0 for every k e 6, contrary to II.

ry. g is abelian.Proof: If. h' € O' : [00], then a(ht) : 0 f.or all a. Then (h,, h) :

0 for all k and ht :0. Hence b, :0 and 6 is abelian.The fact that the restriction of the Killing form to b is non-

degenerate implies that if. p(h) is any element of O*, that is, anylinear function, then there exists a unique vector h, e 6 such that(4) (h, h) - p(h) .The mapping p -, he of 0* into 0 is surjective andwe define

( 5 )

Then

1 : 1 . I f . p , o e 0 *

(p, o) : (hp, hn)

( 6 ) @, o) - p(h,) : 6(hp) ,and it is immedia.te that (p, o) is a non-degeneratb symmetricbilinear form on O*.

V. Let- e, be an element of g, such that [erhl: d\h)er, h e 6,and let 0-o be any element of g_,. Then

(7 ) te-oeof : (€-r, eo)ho .

Proof: We have

(le-"coJ, h) : (e_d, [e*h)) : (€_r, a(h)er): a(h)(e_r, e*)

((e-r, ea)ho, fu) - (e-u, eo)(ho, h) : (e-*, er)a(h) .Hence (7) follows from the non-degeneracy of. (h, k) in b.VI. Euery non'zero root a is non-isotropic relatiue to the bilinear

form (p, o) in b*.Proof: We note first that ad e, is nilpotent if a is a non_zero

root and €a € 8,n. For this it suffices to show that if ',"f , gp, then

there exists a posi t ive integer f r such that [ . . .n*f f i ] ln l :0.Consider the sequence

fxpe"f, [[xpe,]e,|, il[ree,fer]e,1,The vectors in this sequence are either 0 or belong respectively

IV. SPLIT SEMI.SIMPLE LIE ALGEBRAS 111

to the roots p * a, F + 2a, I + 3a, "'. sinCe there are, onlv a

finite number of distinct roots for O it follows that [ ' ' 'lrBe")' ' ' €a):

0 for a suitable &. we can choose eo * 0 in 8" so that lerh) = a(hY'

(cf. (2)) afid e-o€ 8-o so that (e,,L-r) : L' The latter choice can

be made since 8",8-" are dual spaces relative to the Killing form'

If p is any element of O* *e define 8p :0 if p is not a root and

8e is the root space corresponding to p if p is a root' Set

R:Oer+Oh ,+ ! , 8 - * "t : 1

It is clear from the rule on the product of root spaces that

x i = ' 8 - * ' i s a n i l p o t e n t s u b a l g e b r a o f S . A l s o a h , + x L ' 8 - * " i s asubalgebra containing Xt'8-r" as ideal' Hence Oh' * Xt'5J-*" is

solvable. If. r-"e 8-"' then le,x-") is a multiple of h"' by (7)'

Also if. k > /, then fe"g-*rlE 8-*-,a ?Ird lerh,l: a(h,Y': (d, a)e'r '

This shows that s is a subalgebra of I and that if (a, a) :0,

then Ohr+ XL,8-ro is an ideal in S' Since this ideal is solvable

and has one-dimensionat difference algebra, (a, a) - 0 implies that

.R is solvable. Then adg$ is a solvable algebra of linear transfor'

mations acting in 8. sitt"u ad e, is 'nilpotent,

this element is in

the radical of the enveloping associative algebra of adgn (Corollary

2 to Theorem 2.8). Hence the same is true of (ade") (ade-*),

which implies that (er,e-r) : tr ad e* ?dT-a :0, contrary to (ea, e-o) :

1. Hence (a, a) + 0.VII. If a is a non'zero root, then /to: dimS':1' Moreouer'

the only integral multiptes ka of a which are roots Qre d, 0 and - a'

proif: Let e,, e-o, ft be defined as in the proof of VI (cf . (8))'

Then S is an invariairt subspace of I relative to ad h, h e 6' since

le,hl : a(h)e,, lh,hl: 0 and lg-r,hl E g-rr. The restriction of ad h

io g_"" has the single characteristic root -ka(h). Hence we have

trg ad h : a(h) (I - n-" -

and, in particular,

( 8 )

(e)Sincead e,Hence

trs ad (hr) - (a, a) (l - n-, - Zn-r, -

S is a subalgebra contain\ng e, and e-, it is invariant under

and ad, e-, and since le-rerl -- hr, [adge-", adner] : adfth*

tr (adp/?,) : Q. Since (a, a) + 0 this and (9) imply that

| - n-, - Zn-r, - '" - 0. This occurs only if rt-, -* l ' /t-z':

? t - s a - - . . . - 0 . T h u s -3a, .. - are not roots and nl-u: t-

T12 LIE ALGEBRAS

Since we can replace d by -a in the argument, we have also thatn6 - - I and 2a ,3a , . . . a re no t roo ts .

If a is a non-zero root, 8o : Oe, and [erh) - a(h)el Moreover,(7) shows that if we choose e, and €-a, so that (er, e-r) : - 1 then

[e,e-, ] : h, . Then [e,h,7: (d,a)€r, fe-rhr) : - (d,a)er. I f we set

(10) €L : € , ,

then

(11) leLh'"1: ?.eL ,

, 2e-'Y-d, -

i7 t\ d , a )

hL : ,2h ' . ,\e, a)

le ' - "hLJ- -k ' - , , leLe ' - " | =hL.

Thus (eL, e'-,, h',) is the type of normalized basis we havle consideredbefore for a split three-dimensional simple algebra, This proves

VIII. If a is a non-zero root then Ah" + 8, * 8-" is a split three-dimensional simple subalgebra of 8,.

At this point we have all the information we need qn the multi-plication in 6, the products of elements of O bv elefnents of the8, and the products fe,e-*\, €o € 8r, e-, € 8-,. It remains to in-vestigate products of the form [e"ee) where a. and p iare non-zeroroots and € + - a. We shall obtain the results required after wehave established in the next section a basic result oh representa-tions of semi-simple Lie algebras.

2. A baeie theorem on representations and itsconsequences for the structure tkeora

We shall need only a part of the following theorem ht this point.Later the full result will play an important role in the representa-tion theory.

THponpru 1. Let t, be a rtnitu-dimensional split serni-simple Liealgebra ouer a field of characteristic 0, ) a sblitting Cartan subal'gebra, 5J : D ty \ Oeo the decomposition of 8, into root spaces rela'tiue to b. Let h, and (p, o) for p, o in the conjugatp space 0* bedefined as before (cf. (4), (5)) and let e, and e-, be chosen so that

Ie,e-,) - h,- Let Dl be a finite-dimensional module for 8, R therepresentation. Then glt is a s\lit module for 6 and t,if IIl,t is theueight module of $ corresponding ta a weight l, thBn the lineartransformation induced by h* in [lt r is the scalar A(h)t. Let 8,''' :

0 + Ae, + Ae-,. Then 8'"' ls a subalgebra and Dl is a completelyreducible g'n'-module. Any irreducible 8.''' -submodule It of fll has

Iv. SPLIT SEMI.SIMPLE LIE ALGEBRAS

a bas is (Yo, Yr , ' ' ' , ! * ) such that

l ih - - (M - ia)(h) l ; , i : 0, 1, " ' , rn

( I 2 ) ! ; o - o : ! ; + r , ! ^ € - o : 0 , i : 0 , 1 , " ' , t u - l

l o€ r : 0 , ! ; €a : r y (a , a )Y r - r , i : 1 ,2 , " ' , l r t

u)here M is a weight and 2(M, a)l(a, a) : m. Moreoaer, if y i.s any

non-zero uector such that yh : M(h)y and yeu: 0, then ! generates

&n irred.ucible gto -submodule of Tft. Let A be a weight of b in

YJt and, let X be the collection of weights of the form I * ia, i an

integer, a a fi.xed non'zero root. Then 2 is an arithmetic progres'

sioi with first term t! - r&, difference d, and last term I * qa and

we haae

W-e-nrIf x is a non-zero uector such that xh - (l + qa)(h)x, then r gener'

ates an irreducible g''t -submodule and eaery weight belonging to 2

occurs as weight of 6 i,n this submodule. If A is a weight, then

., 2(A. a)1 : t l - - a

\d, e)

dim TJtz : dim IJt,r,

113

(13)

(14)

is also a weight and

(15)

Proof: By III we can find a basis for S* consisting of / roots

&r, &2, - - . , &t. The corresponding elements h"r, h"r, "'r hr, form a

basis for 6. Also gi: Ahq * Oe'o ! Oe-o1 is a split three-dimen-

sional simple Lie algebra and, by (10), e!"r, e'-rr: Ze-'J(e;, di),

hLt: Zh"ol(a;, ao) is a canonical basis for 8r. If we recall the form

of the irieducible modules for such an algebra given in $ 3'8 and

use complete reducibility of finite-dimensional modules' we see that

there exists a basis for tn such that h'A, hence also h|'r:

(Ll1)(a;, o)h'fl, has a diagonal matrix relative to this basis. This is

equivalent to two statements about h|r: the characteristic roots of

h*o are in A and, hI, is a semi-simple linear transformation. Since

the hf, commute, a standard argument shows that there exists a

basis (c1r,r.tr,...,ur) for IJI such that every hf , has a diagonal

matrix relative to this basis (cf. the proof of Theorem 3.10). Since

the h,n form a basis for 6 it follows that hn hasadiagonalmatrix

TL4 LIE ALGEBRAS

relat ive to (u), that is , we have: u1h: 1/h)u1, h e 6, j + 1,2, . . . , N.Thus it is clear that O8 is a split abelian Lie algebra of lineartransformations and that the Ai: ,li(h) are the wdights of 6.Moreover, ft8 is the scalar A(h)L in the weight spacb fiz (whosebasis is the set of a; such that Ai: i). This prdves the firststatement. Now let a be a non -zero root and consider the subspace8'"' : b + Oe" -y Oe-". Since S is an abelian subalgebr4 and Le,hl -

a(h)e', [e-"h]: - a(h)€-,, 8(d) is a subalgebra. Let 6o be thesubspace of 0 defined by a1h7 - g. Then 6 : 6o + Oh, and 8t'\ :

Oo O S, where S is the split three-dimensional simplq Lie algebrawith basis (e,, e-,, h,). We have [0'F] : 0, so Oo is the center of8'''. Now we have seen that ftn is semi-simple for pvery he6.Hence the main criterion for complete reducibility (Theorem 3.10)shows that !)t is completely reducible as an 8'"'-module. Let Itbe an irreducible 8'''-submodule of I)?. Then It conthins a vector

t + 0 such that yh - M(h)y where M is a weight. If we replacey by one of the vectors in the sequence y, yeY, y@!)'; ... we maysuppose also that ye' : $. Let E: etf , F: e'!r, H): hB, where(eL, e'-,, hL) is the basis for S given in (10). Then I

yH : yhL - z-:w(hi) y - zlM' 1),

ld, d) \d, d)

and yE:0. The argument of $ 3.8 shows that 2(M, a)l(s' a) is a non-negative integer m (strictly speakingm.L) and that (y,yF, "',yF^)is a basis for an irreducible S-module. Also we have yF-+' :0and induction on f shows that (yF\n:(M-ia)(h)yFo. Hence}ayF'is an 8(''-submodule of Tt and since Tt is ilrreducible as8'''-module, !t :l,AyFi. We now modify the basisl for It by re-placing yFi by

yr: y(e! , ) ' : ( (ara)\ 'nr ' .

I\ 2

Then we have !;h: (M - i,a)(h)yr Jt€-o : !r+r f.or i: 0, 1, . . ., t l t * Land y^e-, - 0. Also the last formula of (3.36):

UF')E : ( -mi + i ( i - l ) )yFt- ' , i : 1, . . - , / / t t

l i€a: Gmi + ;(t - D)Yy*, ,

becomes

ry. SPLIT SEMT.SIMPLE LIE ALGEBRAS

which is the same as the last equation in (12). Thus we have

established (12). The argument shows also that any non-zero y in

un satisfying lh : M(h)y, !€o:0 generates an irreducible 8{'}-

submodule of IJt. This proves our assertions about 8'''. Now let

/l be any weight and let J be the set of weights of the form

tl*ia, a a fixed non-zero root, f an integer. The weight I is a weight

for O in one of the irreducible 8'''-modules It into which fi can

be decomposed and we may suppose that !t is generated by y such

that yh - M(h)t , !ea:O. Then 1:M-ha where 2(M,a) l (a,&):

m a n d 0 < k 3 m . L e t q b e t h e l a r g e s t i n t e g e r s u c h t h a t A + q d

is a weight. Then if, x + 0 is chosen so that xh * (A + qa)(h)r,

we have xe':O, since A + (q * L)a is not a weight. Hence x

generates an irreducible 8''''submoctule of dimensionality s * 1

where

, - 2(A * qa, aL _2(A, a) t 2Q .( a , a )

- ( a , a )

' -

On the other hand, fu[ - A i ka is a weight, so k S q, and we

have m :2(M, a)l(a, a) : 2(/1 + ka, a) l(a, a) :2(1, a) l(a, a) + 2h.

Then

115

(16)

(17)

h-m:-H#-kq-s:-u8,,8*Q,

so & - m 2 q - s. Now the weights in the module !t generated

by y and those in the module generated by x are, respectively,

A * h a , A + ( k - t ) a , . . . , 1 * ( k - m ) a

A * q a , 1 + ( q - I ) a , " ' , A + ( q - s ) a .

S ince h=qand q-sSk-m, .a IL those in the f i rs t p rogress ion

are contained in the second and since A was arbitrary in J, it

follows that J coincides with the second sequence. Since I is con-

tained in th is sequence we have g-sS0. I f we set / : - (q-s) ,

then the last term becomes A - ra where r ?:0. Also 2(A, a)l(a, d):

s-Zq-r-q, which proves (13). We have At :A-12(A,a) l (a,a)Ja:

A+(?q-s )a and - r<Zq-s<q by our inequa l i t ies . Hence

A' e X and so 1' \s a weight. It remains to prove (15). We ob-

serve that the argument just used shows that 1' also occurs in

the first sequence in (17), that is, At is a weight in every irreducible

116

(18)

(1e)

LIE ALGEBRAS

8'")-submodule It which has I as weight. We have dim lt,r : 1 :

dim ltz,. Hence if In - J O ltr is a direct decomposition of Dt into

irreducible 8'''-submodules, then dim llt,r is the numben of !t, which

have I as weight for S. Hence dim Dtz : dim IJlz'.This result applies in particular to the adjoint reprlesentation of

8. Our result states that if I and d ate any two lroots, a*0,

then the roots of the form I + ia' a an integer, form An arithmeticprogression with difference d. We call this the a'string of roots

c o n t a i n i n g B . I f t h e s t r i n g i s g - r a . , P - ( r - l ) a , " ' , ' F * q a , t h e n

A number of consequences can now be drawn frorri Theorem 1.

We continue the numbering of $ 1.IX. If a, B and, a * I are non-zero roots, then fe'ep] + 0 for any

e, * 0 in 8, and any ep * A in 8,p.Proof: The a-string containing P is I - ra, "', 0 * qa and

q = 1. None of these roots is 0 since no integral multiple of a is

a root except 0, *a. Since the root spaces corresmnding to non-

zero roots are one-dimensional this holds for our strifrg. Let r be

a non-zero element of 8B*n,. Then lxhl - (9 + qa)(h)x. If we

choose e-" so that lere-r): hr, then the theorem $hows that r,

tr ad e-r, x(ad, e-r)'r . . . t r(ad, e-t)'*o are non-zero and lhese span the

spaces $p*or, BB+tq-l 6, "', &p-rr. In particular, ep is a non'zero

multiple of r(ad e-,)q. Formula (12) implies that

W:e-nL

which is not zero since r > 0 and q > 0. Hence leBeul + 0.

We see also that lfeBe,le-,} - - (q(r + l)12)(a, a)ep. This formula

is valid also if F + a is not a root since in this c?s€, [epe"): 0 andq -- 0. Also, if. p - - a, then the a'string containlng P is a, 0,-d, so r :0, Q :2. The right'hand side of the formula is -(a, d)8-,

and the left-hand side is lle-,e,le-,] : - lh,e-,] - le-rh'pl: -(d, d)€-,,

so again the formula holds. We have therefore prorled

X. Let a and F be non-zero roots and eB € 8p, €, Q 8., €-, e 8-a

satisfy le"e-,f : h,. Suppose the a-string containing B is B - fd, ''',

P, " ' , 9 * qa. Then

(20)

W. SPLIT SEMI.SIMPLE LIE ALGEBRAS IL7

We remark also that if a and -d are interchanged, r and q are

interchanged and we have le-,er) - - hr: h'r, (-a, -a) : (a' a)'

Hence another form of (20) is

(20') ltepe -,1e,1 : #

(a,,a)ea .

xI. No muttipte of a non-zero root a is a root except 0, d, -&.

proof: Let I : ka be a root. Then 2(a, 0l@, a) = 2k is an inte-

ger and we may assume this is odd and positive. Then it is

immediate that the a-string containing 0 contains r : Qlz) a' This

contradicts the fact that a:ZT cannot be a root'

XII. Tlu a-string containi,ng g (a, P + 0) contains at most fourroots. Hence 2@, ill@, d):0, fl, *2, *'3'

Proof: We may assume that I * d, -a since the a-string con-

taining a consists of the three roots a, 0, -d. Assume we have

at least five roots. By re-labelting these we may suppose that

9 - 2 e , 9 - a , 9 , 9 + d . , P + 2 r r a t e r o o t s . T h e n 2 a : ( 9 + Z a ) - g

and 2(9 + a) : (9 + 2a) + F are not roots. Hence the p-string

containin g F + Zahas just the one term I + 2a. Hence (9 + 2a, p1 -

0. Similarly p -2a- I and F -2a * g are not roots, so that

(9 - za,F) : 0. Adding we obtain (F, 0) : 0, contradicting the

fact that the non-zero roots are not isotropic. We now have

2 ( a , i l 1 @ , & ) : ( r - q ) w h i l e r + q + 1 < 4 . H e n c e r 3 3 , q < 3 a n d

2(a, 9)l@, d) :0, tl, +2, +3.

From now on we shall identify the prime field of @ with the

field e of rational numbers. As before, let 0* be the conjugate

space of 6 and now let 6f denote the Q-space spanned by the

roots. We have seen that the roots span the space O*' We now

proveXIII. dim Of : I : dim O.Proof: It suffices to show that if

6* consisting of roots, then every

of the a; with rational coefficients.

Hence (9, s) : ) i ; (c1, a;) , i :1,2,

(21) ffi:p,ffi^,,This system of equations has integer coefficients L2(ai, a)l(a1, a)),

IZ(F, u)l(di, a)J and the determinant

(ar, ar, ' ' ' , a,t) is a basis for

root P is a linear combinationWe have I - 2 f,s* Ai €, O.

. . . , / a n d

118 LIE ALGEBRAS

(22) u"(W):#det((a i , a1)) +0,J

since (p, o) is non-degenerate and the d; ?tr-, a basis for O*. Hence(21) has a unique solution which is rational. Thus the )'; arerational numbers.

If I is a weight of 6 for a representation of 8, then We have seenthat 2(A, a)l(a, a) is an integer for every non-zero rbot d. Theargument just given for roots shows that A is a rational linearcombination of the roots di, i:L,2, " ',1. Thus the weightsle6 t r .

XIV. (p, o) is a rati.onal number for p, o e $t and (p, o) fs a

Positiae defi.nite symmetric bilinear form on 6tr.Proaf; We recall the formula (3) for (h, k) h, k e 6.' If we take

into account the fact that the 8" are one-dimensiona[ we can re'write this as

(23)

Let p, d e 6*. Then

(p, o) : (hp, ho) : \a(he)a(h')

Hence

(24)d

Now let F be a non-zero root. Then (p, p) : 2,(9,W)'. Let the

€-string containing d be

e - fuFF, a - (r"p - I)p, " ' , d * Q"pp .

Then, by (18), 2(a, 9)l@, B) : rrp - QoF and, (a, 9) : l(rrp * a"e)l2l(P, P).Hence

/ r . o - n - , o \ 2(P, P) : +( :2* )@' i l " .

Since (p, p) * 0, X,(roF - Q,p)" * 0 and

(25) (8,0) :2?,P - Q,P)'

is a rational number and (a, 9) : l(r,p - q"e)l?l(F, F) is rational for

(h, k) :. d a rloot

Iv. SPLIT SEMI.SIMPLE LIE ALGEBRAS I19

any roo ts d ,P . I f p ,o e6 f , Q:Z lp ror , o - l ' l v ;a i t l i r v ;e Q, d ;

roots. Then (p, o) : Z t-tivi(s,r, a) E Q. Also (p, p) : I (p, 4)' = 0

and (p, p) : 0 implies that every (p, a) :0. Then P :0 since the

roots d span S*.We recall that in a space with a non-degenerate bilinear form

(p, o), if a is a non-isotropic vector, then the mapping Sr:

2b. a)P - , P - - a '

l&' 4)

leaves fixed every vector in thesends a into -d,. We call thisThis belongs to the orthogonal

(26)

is a linear transformation whichhyperplane orthogonal to a andthe reflection determi.ned bY d.group of the form (p, o).

we have seen (Theorem 1) that if A is a weight of o in a rep'

resentation of 8, then At : AS,: A - l2(A, a)l(a, a))(a) is also a

weight. The reflections S*, d a root, generate a group of linear

transformations in Oi called the Weyl group W of 8 (relative to

O). This group plays an important role in the representation

theory for 8. The result we have just noted is that the weights

of a particular representation ate a set of vectors which is invari-

ant under the Weyl group. In particular, this holds for the roots.

If two elements of. W produce the same permutation of the roots

they are identical since the roots span Of,. Since there are only

a finite number of roots it follows that W is a finite group.

3. Simple systerns of roots

We now introduce an ordering in the rational vector space 6f.For this purpose we choose a basis of roots dt, &zr ''', &I" and we

call p : Zl,lrar, h € Q, positiue if the first non-zero ),i is positive.

The set of positive vectors is closed under addition and under

muttiplication by positive rationals. lf o, p € 6; we write o > p if.

6 * p > O. Then OJ is totally ordered in this way and if 6 > p

t h e n o * t l p * t a n d A o > ) , p o r A o < ) ' p a c c o r d i n g a s , l > 0 o r

t < 0. we shall refer to the ordering of sf just defined as the

leri.cographic ordering determined by the ordered set of roots

(ar, ar; . .., a,t) (which form a basis for 6i).

Lprrru.r, 1. Let pr, pz, "' , p,, c $t . Suppose the pr > A and (pu, pr') s 0

if i + j. Then the p's are li,nearly inde\endent oaer Q.

120 LIE ALGEBRAS

Proof: Suppose pr: Ef-t, l,pr: 21'opn + L l l 'pr wherp 1S4, s 5h - l ^ , t i > 0 , t l ' S 0 . S e t X i i p q : ( 1 , 2 1 ! r ' p r : " . S i n c e d r t ) 0 , o * 0 .We have (o, r) : >, ),'q^'r'(pq, o.) 2 0. Hence

(p*, o) : (o, o) * (o,r) ) 0 .

On the other hand, (pr, o) - 2l[@*, p) 3 0 which is a contradiction.Hence the p's are Q-independent.

DprrNrnoN 1. Relative to the ordering defined in 6i we call aroot a simple if. a> 0 and a cannot be written in the form F + rwhere I and r are positive roots.

XV. Iat tr be the collection of simple roots relatiad to a firedlexicographic orderins of bt. Then:

( i ) I f o , 0 e r , a + P , t h e n a - 0 i s n o t a r o o t .( i i ) I f a , F e n a n d a + g , t h e n ( a , 9 ) < 0 .(iii) The set tr is a basis for bi oaer Q. If B isany hositiue root,

then 0 -- Z"enkra where the k" are non-negatiue integers.(iv) If A is a positiae root and B 6 o, then there erists a,n d, e rE

such that B - a is a positiae root.Proof: ( i ) I f . a, letr and a-P is a posi t ive root, then a:

9*@-9) contrarytothe def in i t ion of r . l I a- F i+ a negat iveroot we again obtain a contradiction on writing B : (9 - a) + a.(i i) Let I - /&, I -(r -I)a, " ', F + qa be the a-strinFcontaining

P. Then 2(a, ill(a, d) : r - q. Since r :0 by (i) and (4, 4) > 0'(a, F) < 0. (iii) The linear independence of the roots tontained inTc is clear from (ii) and the lemma. Let F be a positive root and

suppose we already know that every root T such that F > r > 0is of the form Zrenkr&, k' a non-negative integeri We mays u p p o s e a l s o t h a t 9 d t r , s o t h a t 9 : 0 ' . * 9 2 , f u > 0 . T h e n 9 > 9 +and pr:lh',a, pr:.\htiu, h'", k'r: non'negative intisers; hence

9 : ZtkL + p'J)a which is the required form for p., If p is anegative root then -F is a positive root. Hence F : X kra wherethe ho ?te integers 5 0. The first statement of (iii) fpllows from

this and the linear independence of the elements of r. (iv) Let F bea positive root not in z, The lemma and (iii) imply that there is

a n a € z r s u c h t h a t ( 0 , 4 ) > 0 . T h e n Z ( F , a ) l @ , & ) : r * q > 0 ( r , q

a s b e f o r e ) . H e n c e r ) 0 a n d 9 - a i s a r o o t . I f p - a < 0 , t h e na , - F > 0 a n d a : B + ( a - F ) c o n t r a r y t o : a € r . H e r l c e 9 - a > Oand F - (P - a) * a where a € r.

We now write 7t : (&t, &zt ..., cs) and we call this the simble

ry. SPLIT SEMI.SIMPLE LIE ALGEBRAS I2T

slsteln of roots for I relative to O and the given ordering in 6J.

We have seen that every root F - Zk;at where the k; are integers

and either all, ki ) 0 or all &r < 0. This property is characteristic

of simple systems. Thus let fr : (dr, dzt ''', er') where / : dim O

and the dt Ata roots such that every root B:}k;aa where the kt

are integers of like sign (rationals would do too). Evidently our

hypothesis implies that ft is a basis for 6f. We introduce the

lexicographic ordering of Oi based or the a;:

! , l 6 a i > 0 i f i r : " ' - / . h : 0 , i e a l ) 0 , h < l '

Then the positive roots g : X k&; ?te those such that the k, > 0

and some h, > 0. It is clear that fio d; is a sum of positive roots.

Hence the a; are simple. Since any simple system consists of /

roots the set E : (&r, dzt . .., dt) is the simple system defined by

the ordering.lf n : (&r, dr, . , ,, dt) is a simple system of roots' the matrix

(Aii), Aii:2(a;,,a)l(e;,d;) is called aCartanmatrix for 8 (relative

to o). The diagonal entries of this matrix are Ai+:2 and the

off diagonal entries are A;t:0, -1, -2 or -3. (XII and XV (ii).

If i + i the d+ zfid, e5 1tE linearly independent so that if dir is

the angle between ar and d; then 0 3 cos' 0;5 11. This gives

0 S4(a;,a)' l(at,a;)(ai,a) < 4; hence 0 5 A;tAii <4- This implies

that either both A;t and A1a are 0 or one is -1 while the other is-L, -2, or -3. The determinant of the Cartan matrix (Arr') is a

non-zero multiple of that of ((a;, a)). Hence det (A;) + 0.

We now choose 0a1€ g6r, €-o5 Q B-o6 so that ler6e-r1\ : hor arrd

we now write

(27) 2;: €at , fr: fu-rJ(d;, a,i) , hi:2h611(ar a;) -

These elements have the canonical multiplication table for a split

three-dimensional simple Lie algebra: le;h;1:Zer, [frhi - - 2f;,

lef;7- h;. Also we have Lefi l:0 if i + i since dt- di is not a

root, le;hi:Z(au a)l(ai, a,i)ea: Ane; and lfthil -- - Anfr. The

last relations include le;hil: ?,ei, lf;hl - - Zfo. Thus we have the

following relations for the ea, fi, hai

(28)

lh;h) - sleJi : d.;th; (d,i : I if. i : j, dri : 0 if. i + i)

le;hi : Ai$;

l f th i : z Ai ; f ;

t22 LIE ALGEBRAS

i , i : L , 2 , " ' , 1 .We wish to show that the 3/ elements et,fi,h; (or the 2/ elements

er, fr since h - lefil) generate 8. Moreover, we shall show thatwe can obtain a basis for 8 whose multiplication tdble is com-pletely determined by the Ait. This will prove that 8 iS determinedby the Cartan matrix.

If p - \k;a; is a root then we define the leuel lF | : 2 le; l.The level is a positive integer and the positive roots 1of level oneare just the ai e n.

XVI. The set of roots is determiwd by the simple lpystem n and.the Cartan matrix. In other words, the sequences (kr, kz, ' . . , kr)such that >kiai are roots can be determined from the matrix (Ad.

Proof: It suffices to determine the positive roots. rThe positiveroots of level one are just the ai e r. Now suppose we alreadyknow the positive roots of level S fl, n a positive integer. Wegive a method for determining the positive roots of the next level.By XV these are of the form 9:a*oj , a)0 of le ivel 14, d ierE,Hence the problem is to determine for a given a ) 0 bf level n thea1 e r such that a* ai is a root. lf. a:&j, a* dt is not a root.Hence we may assume that a:Lkier, and some h > A f.or i + j.

Then the linear forms o - oi, a - 2ai, . .. which 4re roots arepositive of level less than z. Hence one knows which of these areroots. Thus the number r such that the a1-string coptaining a isa , - r L i , , . . s d t . . . , a * q a t i s k n o w n . w e h a v e

4 : r - 2(a, a)l(at, di) :, - i k,;,Ai; ;

hence q can be determined by the Cartat ,|atri*. Since a. + di isa root if and only if 4 > 0 this gives a method of ascertainingwhether or not a * a1 is a root.

Example. Suppose the Cartan matrix is

(2s) (_3 -),

that is,Z(at a)l(at, a,t) - -L , Z(dr, az)l(ar, as) - +3 .

Since dr - &z is not a root these relations imply that the ar'stringcontaininE a.z and. the az-string containing dl are, respectively,

& z t d z * d r

4 t t & t t d z t

(30)d r * 2 d 2 , a t * 3 a z

(31)

IV. SPLIT SEMI.SIMPLE LIE ALGEBRAS 123

The only positive root of level two is dt * &z' Since az * Zar is

not a root the only positive root of level three is a, * 2a* Since

2a, * Zaz is not a root the only positive root of level four is

e , r *3a2. We have Z(ar+3ar ,a , r ) l (a rdr ) -2 - 3 : -1 , wh ich im '

plies that (ar + 3az) * &r:2a, + 3az is a root. Since ar * 4az is

not a root Zar * 3az is the only positive root of level five. Since

(2at *Sar ) + d r :3 (ar t , a r ) and (2ar* 342) + d2 :Z(a t+Zaz) a re

not roots there are no roots of level six or higher. Hence the

roots are

td1 , !.a2 , t(a, * a) , t(a, * 2a2) ,

-r(a, * 3a,) , t(Zar + 3az) .

A simple induction on levels shows that any positive root I can

be written as

( 3 2 ) F : a o , * a t r + " ' * d \ c ,

arre z in such a way that every partial sum

a q * " ' + d t m , r n < k

is a root. The number ft in (32) is the level of 0. We shall now

abbreviate ['' 'fxrxrl " ' x,] as lxfiz "' x,1, x; Q' 8'' Then the result

IX implies that

(34) l e q e 4 . . . e q , ) + 0 ;

hence this element spans 88. It follows also that

lfnrfrr "' forl + 0

(33)

(35)

and this element spans 8-p. Since the a; € z form a basis for 0*,

the h"n and hence the h; form a basis for b. since I -

6 + > igu + g_p) summed over the positive roots, this proves

XVII. Let r : (dr, dzt ''', a,t) be a simpte system of roots for 8'

relatiae to b and. let €;, f;, hi be as in (27). Thpn tlrc 3l elements

ei, f;, hi gennr&te g. For each positiue root I we can select a rep'

reientati in of P:nq* dtz+ "' + dik s' that a6r+ "' * a;^ is

a root for euery m 3 h. Then the elements

(36) ht , lei " ' o;1,7 , lftr " 'fn*l

determined by the positiae roots B forrn a basis for 8''

We shall be interested in the multiplication table of the basis

(36). For this we require

L?I LIE ALGEBRAS

XVIII. Int P be a positiue root and let the sequencq ir, ir, '' ', i*be determined by B as in XVII. I-et !',2' , .. ., k' be a'permutationof L,2, ' . ., k. Then l0\,€;2, " ' e;rr,f ts a rational t multi'ple of

feheir - - . €irol, the multiplier being determined by the A;i, A simi'larstatement holds for the f's.

Proof: This is clear for k:1, so we use inductionrand assume

the result for positive roots of level k - L. lf irc:'i,v, :7, thenwe may assume that the sequence chosen for the rqot 6 - ai is

i r , i r , . . . , i * - r . Then the resul t for k - 1 impl ies that fe;r , " '€;o,- t , f :

tle4.. . g;x_rf where / is a rational number determinedr by the A;1.

Then l€ i r , , . . nt , r , ) : lzh, " ' € i (n-r ,e i ) : t le\ " ' eh-Pi) = t le\ " ' e i* \ .

Next suppose in: i * in' and write

lenr, " ' e ik, f : leh, " ' o i r ,€ i " ' e ik, \

where the displayed e; is the last one occurring in the expression-

If any of the partial sums &\, * "' + a;-, is not d root, then

fetr, ... 0;rr,l:0. Since this f'act can be ascertained from the

knowledge of the Cartan matrix the result holds ipr this case.

Now suppose every d;t, * " ' + d4*, is a root' Then lett," 'e;r 'J*0'

Since 1f ri: 0 if i + j, ladf i, adeil: 0 and

lenr, " ' €;r,0i " ' eik,f i l : l€\, " ' eir"eif i " ' e;v') '

By (19), fe i r , ' " e;r ,e i f i l : -q(r * L) lenr," 'e i , , ) (s ince i r r , * ' " * d;r '

i s a r o o t + 0 ) w h e r e t h e q a n d r a r e i n t e g e r s , q > 0 , r > 0 a n d 4

andr are determined by the a, str ingcontaining 9=d\r* . . ' *d;r , .This string is known from (A;i). A similar argument shows that

lorr, "' oi1r,f i€if : slerr, "' eih,) :

where s is a non'zero integer which can be determined from the

Ari. It follows that

s[etr , ' " e iy, l : le i r , " ' e i* , f $17

- -q(r * 1) [e;r , " ' €4,eis+2) ' ' ' ' 0+n'€i1 '

Thus le;r , " 'e4, l : t l^ i r , " ' €;r ,e i r r+2), " 'e ik 'e i \ where /1 is a rat ional

number which can be determined from the Aii. Thi$ reduces the

discussion to the first case. The /'s can be treated Similarly.

We can now prove the following basic theorem' ,

Tnponnu 2. Let r : (dr, dzt "', az) be a si'mhle sYstern of roots

for a split semi-simple Lie algebra 8, relatiae to a splltting Cartan

W. SPLIT SEMI.SIMPLE LIE ALGEBRAS 125

subalgebra Q. I*t the et, f+, h'i,. i :L,2, ' l l , -be

generators of 8'

aefinea in Q7) and let the basis i,r, lerr'' ' 0;*1, lf'r'' ' {'-1 {:f I

be

ai specirt,ed, in XVII . Then the multi\lication table for this basis

has rational coefficients which are determined by the Cartan matrir

(Ai.' Proof: We have [hthi: 0 and le;r' ' ' e+*hil: 2h'='Aii*le;r"' €irc7,

bv (28i. SimilarlY , lfr, "' fr*h51to consider products of e terms and f terms. Since ferr''' -eitrl:

1 . . . le ; re r r l " ' e i * \ , ad{e6 , " e , ;n l : [ " ' l adearadear l " ' adea* l .and

i",lior" ' i eo*ll -'xl" ' lad eaad irr! " ' ad' ei*l' This can be obtained

6y op"tutin! on r with a certain (non-commutative) polynomial in

adeir, ...,adei*. It follows from this and a similar argument for

the i's that ii suffices to show that [[e;' "' e;*leil, llf1r^"' f'*lei'

fle;r " ' eiklf il, llfrr''' fr*\f i are rational combinations of oUr base

ilements where the "oeih.i"r,ts

can be determined by the Atl. The

argument is the same for the last two as for the first two so we

consider the first two only. To evaluate lfeir"'eircleA we ascertain

first whether or not 6 * ei, F: ai'r+ "' + aib is a root' If not'

then the product is 0. on the other hand, if T : P + 4' is a root

then, by XVIII, l!err... eiftleil is a rational multiple of the e'base

element associated with r, the multiplier determined by the Cartan

matrix. Next consider llfrr"'fr*1ei. If k : ! the product is 0

unless i l: j , in which case lfareil: -ht. lf ' k>:2 we shall show

by induction that the product ir "

rational linear combination of

,fjbu." elements. Thus, if k :2 the product is 0 unless / - i' of

j -- i,. For llfr,f ileiJ we have llf;rf ileil: lfirlf&il + llf'reilf il:-- lf trh5l: Airrfry since i, + i and so If;reil :0' The relation just

derivld impliel atso that Llfftrleil- - Ai;rf;r' Now assume k>2'

If no i, : j, then the product is 0. otherwise, let e'+r be the last

index in [,f,, "'fo*j which equals 1. Then

l l fo r ' ' ' fo , f i ' ' ' f4 le i l : l l f ; r ' ' ' f , , f , ie i f ' , * r ' ' ' f t * l

: - [ ' ' ' 1 . f , , ' ' ' f i , lh i ] ' ' ' f , * l + l l [ f ; r ' ' ' f i , )e ] fo ' ' ' f ' * f '

The first term is a rational multipte determined by the A;i of an

/-base element. The induction hypothesis establishes the same

claim for the second element. This completes the proof.

Suuulnv. Before continuing our analysis, it will be well to

summarize the results which we have obtained. For any split

semi-simple Lie algebra I with splitting Cartan subalgebra O we

tz6 LIE ALGEBRAS

have obtained a canonical set of generators e;,fr hr, i F t,2, "',1,satisfying the defining relations'(28). These were obtained bychoosing a simple *y.ie. z of roots. The characteristic property

of z is that every root a: \lkna;, ai e, ft, where tfie &r are alleither non-negative or non-positive integers. Such systems areobtained by introducing lexicographic orderings in the lspace 6J ofrational linear combinations of the roots and selecting lthe positiveroots which are not of the form a I 9, d, P > 0, in such an order-ing. A canonical set of generators associated with rT is hi:Zh*nl(ai, a1), €i: €o,t, ft: 2e-r6l(ai, a) where zai is 4nY non'zetoelement of 9rn and e*r6 is chosen in ,8-"a so that lepne-r) - hat,.We observe ihat hr is uniquely determined by da while er c?D bereplaced by any $tei, $r * A, in O. Then /r will betreplaced bypt,'fr. The elements er, fn, ho constitute a canonical basis for a splitthree-dimensional simple Lie algebra 8;. Moreover, [t is easy tocheck that if (eL,fl,h') is any canonical basis for $o such thath't e 8a n O, then h't: h;, e'r: pren, f l - pttfr.

lf e;,frhi zre canonical generators we obtain a cahonical basisfor I in the following manner: the basis /l; for 6 ahd for eachnon-zero root P a base element zy: fz\" 'e;* l ot [ f . r " ' f t * l ac-cording as p > 0or F < 0 where (ir, " ',ir) is a sequenpe such thatd;r* " '+ dirr : tF and every part ia l SUIr I 4r , + "1 ' * d+n is aroot.

The multiplication table for the canonical basis is rational and

is determined by the Cartan matrix (Au), Ari :2(ai, a)l(di, ai).

The 4;i are integers, A;r: 2 and if. i + i, then either 4r, -- 0 : Aii

or one of the numbers Ait, Aio is -1 and the other iF -L, -2 ot-3. We observe also that the group of orthogonal linpar transfor-mations generated by the reflections

S; = S'r : F * t - ?9' q:.'l r,- (ai, a;)

which is a subgroup of the Weyl group W is finiter (Later we

shall see that this subgroup coincides with I7.) TheLreflection S,i

can be described by the Cartan matrix. Thus if we t4ke the basis(dr, dr, ''', dr) for Oi, then S; is completely describedl by

, , js t : a i - f f i dt : d j - A; iat '

The conditions we have noted on the A6i "te

redundanlt. However,

ry. SPLIT SEMI.SIMPLE LIE ALGEBRAS 127

sometimes one subset of these conditions will be used while at

other times another will be used.

4. The isomorphism theorem. SimplicitA

Theorem 2 of the last section makes the following isomorphism

theorem almost obvious.

Tnsonnu 3. I-et 8,, 8' be sflit semi-simple L|e algebras oaer a.

fietd O of characteristi,c 0 with sbtitti,ng Cartan subalgebra.s O, S'

of the sarne dimensionatity l. I*t (ar, dzt ''', dr), (al, aL, "', a!) be

simble systems of roots for I and 8' respecti,uely. suffose the

Cartan matrices (2(a;, a)l(a;, a6)), (Z(a!t, d1)l@!t, a!)) are identical'

Let e;,f;,h;, e't,f l ,hla, i -1,2, " ',1, be canonical generators for I

and g, as in (27). Then there exists a unique isomorphism of 8

onto g' mab\ing 0; on eL,f; on fl.,ht on hL.Proof: By XVI we know that x h&; is a root for 6 if and only

if >kp,t is a root f.or b'. If F is a positive root for s we write

B : a \ + . . - * d i r s o t h a t d \ * " ' + a ; - i s . a r o o t , m 3 h . T h e n

P' - ilr+ "' + ailr* and everY d!, + "' + d!i* is a root' We can

choose as base eliments in the tanonical basis (36) for I and 8'

the elements lenr ' " €h), l f r r ' " f r*7, lu|r ' " e 'n) , t f l ' " ' f {*) ' Then

Theorem 2 shows that the coefficients in the multiplication table

for the basis for I and 8/ are identical. Hence the linear mapping

which matches these base elements is the required isomorphism.

The uniqueness is clear since the el, f;, hi are generators.

The result we have just proved is basic for the problem of

determining the simple Lie algebras. It is useful also for the

study of automorphisms (which we shall consider later) of a single

Lie algebra. We note here that the result implies that there exists

an automorphism of I mapping e;-+f;, f;+ €i, hi--+ -h;. This

follows by observing that a!, : -ai, i :1, ' '', /, is a simple system

and e't: fr, fl : e,t, h't - *h,;. is a corresponding set of generators.

We shall need this in $ 7.

DsrrxrtloN 2. A simple system of roots E : (dv &zt "', dt) is

called i.ndecomposable if it is impossibte to partition zr into non-

vacuous non-overlapping sets z',2r" such that Ati-- 0 for every

d t € n t , a 1 € n t t .

Tsponpu 4. I e's simfle if and only if the associated simple

L28 LIE ALGEBRAS

systern r of roots is tndecomposableProof; suppose first that rc : (dt,. ' ., dt) U (ar*r, " ', a,r), L = k < l,

so that A;i:0, i < k, i > h. Choose canonical gener4tors e;,fi,h;,and let 8r denote the subalgebra generated by the €t,ft, h, i I k.It is readily seen that 8r:0, * X'8r where Ot is the subspace of

6 spanned by the h and the summation is taken overt the roots rwhich are linearly dependent on the ai, Hence 0 c 8r ct8. If. r) k,j < k, then Ar'" :0 and since di - dr is not a root, this impliesthat a5* d, is not a root. Hence le7e,1:0 as well as lft,erl:0.Also lhie,l - g and consequently ar is in the normblizer of 8r.Similarly, /" is in the normalizer of 8r. Since 8, is contained in

its own normalizer it follows that I is the normalizer df 8r. Hence,8, is an ideal and 8 is not simple. Conversely, supfose 8 is notsimple. Then 8 : 8r O 8z where the ,8; are non-zero ideals. Let abe a non'zero root and let er€$r. Then 0r:0lr\ l,et', eS\ e8;and, le,hl - a(h)e, for h e O implies that lettnl - o(h)el'. Since 8,is one dimensional this implies that either 8" g 8r or 8o S 82. Since

[8r8r] - 0 and [8"8-,] + 0 we have either 8, * 8-' S I' or

8"*,8-" I ,8s. In particular, we may order the canonical generators

za,f i so that eyfrL " ' , er , f t € 8r, €t+t , f*+u ?"t l t t f r€8r. Since the8i are non-zero ideals, L<k < / and g-[e{e,f,11 -left, l- A,Pi if.j < k and, r > k. Hence Ai,: A,i:0 and the simple system ofroots is decomposable.

5. The determination of the Cartan mqtrlees

The results of the last section reduce the problem of determiningthe simple Lie algebras to the following two problems: (1) determi-nation of the Cartan matrices (Ai corresponding to indecomposablesimple systems of roots and (2) determination of simple algebrasassociated with the Cartan matrices. We consider (1)l here and (2)

in the next section. We observe first that the indefomposabilitycondition amounts to saying that it is impossible to order theindices (or the at) so that the matrix has the block fbrm

(B 0 \\0 c )

where B, C are not vacuous.\il'e now associate a diagram-the Dynkin diagratn-with the

Cartan matrix A;;. We choose I points drt &zt '", dr. a4d we connect

IV. SPLIT SEMI.SIMPLE LIE ALGEBRAS

a; to et, i + j, by AuAt; lines. Also we attach to each point ai

the weight (a;, a;). If. Ati: 0 : Air, a, and' ai are not connected

and if A;i *0, An* 0, then AtrlAu : (d* ai)l(at, a)- Hence AnlAtt

and AuAr.; c€lr be determined from the diagram. Since Arr is non'

positive this information determin€s ,4ii and An. Thus the matrix

can be reconstructed from the diagram of points, lines and the

weights. We consider two examples:

129

3 1 -0-6 ' Gz

r^n\ d'1 4z( . t / /

1 1 1o - o - o .

d1 d,2 &g

1 1 A. O-o , .c t l .

& t - t d t

For G, we have Arr lArr : (dt ,a) l (az,az):3, ArrArr :3 which im'

plies that Arz - -1, A,-: -3. Hence the Cartan matrix is the

matrix in (29). For /r we have Au: 2, Ap: Azt : Aza : Asz:. i . _ Ar-r , , , : Ar, , l - t - -1 and al l the other A;t :0. Hence the

matrix is- 2 - 1- 1 2 - 1

- 1 2(38)

. - 1- 1 2

In determining the Dynkin diagrams we at first drop the weights(ai, ai) on the points and consider only the collection of points andthe lines joining these.

'We have / points dt, &2, '", dt., ?nd ai

?1d ax', i.+j, are not connected if A;tAi;:0 and are connectedby AuAi; :1,2, or 3 l ines i f . .AuA,r+0. The ar are l inear ly in-dependent vectors in a Euclidian space Eo over the rationals. Thiscan be imbedded in the Euclidian space E: Eon over the field Rof real numbers. If. 0;t denotes the angle between ai and c; thenAriAn: 4 cost 0i; and cos dir ( 0.

Any finite S€t 41, dz, . . ., &t of linearly independent vectors ina Euclidian space (over the reals) will be called an allowableconfi,guration (a.c.) if 4 cosz 0;i : 4(a;, a)z l(a;, a)(ai, a;) : 0,1,2 or

3 and coS d;i < 0 for every i, j, i + j . Ihus cos d;; :0, -L, -tv/T

or -]2/X and accordingly 0u:90o, 120",135o, or 150o. We may

130 LIE ALGEBRAS

positive mrlltiple of. aa.replace a; by the unit vector z; which is aThen the conditions are

( g g ) ( u t u ; ) : 1 , 4 ( u i , u ) 2 : 0 , 1 , 2 o r 3 , ( u ; , u ) 1 5 0 ,

i + j , i , j : ! , 2 , . . - , 1 .

The Dynkin diagram (without weights) of an a.c. z isla collectionof points %;, i :1, . - . ,1, and lines connecting these dccording tothe rule given before: il,; and ui aty not connected if (u* u) :0

andu;andu5 are connected by 4(ui,ut)' :1,2 or 3 l ines otherwise.An a.c. is indecomposable if. it is impossible to partition z intonon-overlapping non-vacuous subsets z/, z" such that (t't;, u) : 0 if.ttr.; Q zrt , u.i e rc't . The corresponding condition on the Dynkin dia-gram is connectedness: If tr, u € n, then there exists a sequencegair:04, %i2, "',cl irc - zr such thatutraad tttr*, are connbcted in thediagram. If the Dynkin diagram is known, then all,(urar) willbe known. We shall determine the Dynkin diagram$ for all theconnected a.c. after a few simple observations as folloWs.

1. If S is a Dynkin diagram, the diagram obtained by suppressinga nurnber of Points and the lines incident with these is the Dynkindiagram of the a.c. obtained by dropping the uectors correspondingto the points.

2. If I is the nurnber of uertices (boints) of a Dynkin diagrarn,then the number of pairs of connected points (u, u, (u, a) + 0) is lessthan l.

Proof: Let u : |!u.i. Then l

0 < (u , u ) : I * ,Z ( t t t , u ) l

If. (uru) + 0, then Z(ut,u) < -1. Hence the inequality shows thatthe number of pairs ilrr ui with (ur u) + 0 is less than /.

3. A Dynkin diagram of an a.c, contains no cycles.,(A cycle isa sequence of points ur, . . - , xrk such that ur, is connected to u;+ri = k - L and ur is connected to ur)

Proofz The subset forming a cycle is a diagrami of an a.c.violating 2.

4. The number of lines (counting multiblicities) issiling from auerter does not exceed three.

Proof: Let u be a vertexl u1,'uzr , , ., ur the vertices connected tou' No two ur are connected since there are no cYcles' Hence


(a;, u) -- 0, i + i. In the space spanned by u and the ?/t w€ c?r

choose a vector 2o such that (ro, uo) :1 and uo, ur, ' ' ' , orc are mutually

orthogonal. Since u and, the u;, i > \ are linearly independent, u

is not orthogonal to ?/o and so (u, u) * 0. Since u : 2t@, r)t)ui,

(u, u) : (tr, ai' * (u, ur)' + "' + (u, ur)' : L .

Hence 2f @, rr)' 11 and 2! 4(u, a;)2 I 4. Since 4(u, ui)z is the

number of lines connecting u and I); wc have our result.

5. The only connected a.c. containing a triple line i's

A

l tz i o:o

This is clear from 4.6. Let r be an a.c. and let ar l)zt ' '', ttr be uectors Of n such that

the corresponding points of the diagram form a simple chain in the

sense that each one is connected to the next by a single line' Let t'

be tke collection of aectors of t whi,ch are not i,n the simple chain

urt . - -, uy together with the uector u : 2!ur Then n' i's an a'c'

Proof: We have 2(ai, ai+) - -1, i : I, "', k - L' Hence (u, a) -

h + Zin./a;, a). Since there are no cycles (ut, a) : 0 if i < i

u n l e s s j - i + L ' H e n c e ( u , a ) - k - ( f t - t ) : t a n d u i s a u n i t

vector. Now let u € n, tc * ar Then z is connected with at most

one of the uit say ui, since there are no cycles' Then (u, u) :

(u,2!o) : (u,at) and 4(u, a)z : 4(u, a)' -- 0,L,2 or 3 as required'

The diagram of z' is obtained from that of ft by shrinking the

simple chain to a point: Thus we replace all the vertices ,i by

the single vertex a and we join this to any ue n, tt + ?/d by the

total number of lines connecting u to any one of the a1 in the

original diagram. Application of this to the following graphs

131

(40)

reduces these respectively to

(41) o:o--o , o:o

O - O - O - - O . . . O . - - O : : O

oEo_o_o . . o o_ "< ' \_o

o \ -Zo> o - o - - o c . . o - s /

o ' / \ o

o o__^ /o

o ' o / - \ o

Since the center vertex in each of these has four lines from this

t32 LIE ALGEBRAS

vertex, these cannot be diagrams of a.c., by 4. Hende we have7. No Dynhin diagram contains a subgraph of the fdrm @U.8. The only possible connected Dynki.n diagrams haee one of the

following forms

o . . . o _ o A l

O . . . O - - O - O - O . . . O - O

th llz ilp-r tlp Uq Uq-r Uz l)1

(42) *' iI*, ?.

11lr-r 1I

o - o . . . o - - f - o . . . o - oI,fi l, l.z Up-r Z Dq-r U2 At

Proof: If a connected Dynkin diagram S contains I triple linethen it must by Gz by 5. If S contains a double line it containsonly one such line and it contains no node, that is, graph of the

form o-o/o This is clear from ?. Also it is clear that\--o

S cannot contain two nodes. This reduces the possibilities to thoseof (42).

We now investigate the possibilities for 0, Q, / in the secondand third types in (42). For the second type set u 1 >f iu;, u :

Lljui. Since Z(ut, ui+r) - -1 and Z(ui, ui+t) : -1 we tlave

(43)p ^ p - l

(u,u) - i i ' - t i ( i + L): P" - P(P - L)12

: oro + r)12 ,(44) (u,u): q(q + l)12 .

Also (u, a) : fue(uo, on) : @qlT)Z(ur, an) and

(45) (u, u)' : P'q'12By Schwarz's inequality :

(46) ry'ryry'

(47)

ry. SPLIT SEMI-SIMPLE LIE ALGEBRAS 133

Since fq > 0 this gives (P + L)(q + L) > 20g, which is equivalent

t o ( P - r l c q _ L ) < 2 . H e n c e t h e o n l y p o s s i b i l i t i e s f o r t h e p o s i t i v eintegers p, q are

f :1, q arbitrarY; 4 :1, fr atbiltary;

f : 2 - q .

The first two cases differ only in notation. Hence we have

9. The onty connected Dynki,n di,agrams of the second tyfe in

(42) areo r r Q - O - o B t : c t

(48)

Finalry *. .:;;;;:",. , l '*o. set u:2t-t iui, 0:

>i-tjai: q1 : 2l-rhwr,. The vectors u., u, u) are mutually orthogonal

and e is not in the space spanned by these vectors. Hence if 0b 0r,0t,

respectively, are the angles between z and t4, a and u then

.or; d, * cos2 dz * cos2 0r 1 | (cf. the proof paragraph 4 above). Now

cos' a, : (Lt, z)'l(u, u)

: *(p - I)'l(p(p - L)12) (bv (4:l))

: (f - L)lzp

+('-i *, -+ + ' -+) . 'or

(4e)

We may suppose f ZqZr (>2) . Then P- '=q- '5_r- 'and (49)

implies that 3r-1 > 1. Since r 2 2, this gives r :2. Then (49)

g i v e s P - ' + q - ' > r . H e n c e 2 q - ' > | a n d q < 4 . H e n c e z € q < 4 '

if. q -- Z, then the condition is that P-' > 0 which holds f.ot all p.

lf. q:3, then the condition is f-' > U6 and f < 6. Hence in this

case p : 3, 4, 5. Thus the solutions for p, q, r are

(50) r :Q :2 ' f a rb i t ra ry ;

r : 2 , 4 : 3 , f : 3 1 4 , 5 .

1*1+J>r .p q r

:t(-+)Simitarly, cos' 0z - L(I - Llq), cos'd, : *(1 - Llr) so that we have

tu LIE ALGEBRAS

This proves10. The only connected Dynhin diagrams of the third,,type in (42)

are

(51) 0 _ _ o . . .

Dt: l -o1 . .d1 &2

o

II

o - o _ o

o

III

D,

(52)

We have now completed the proof of the following

Tsponpu 5. The only connected Dynkin diagrarns are At, I >- L,Bt : Cr, l2 2, D7, I > 4 and the fiue "erceptional" diag/ams Gr, Fr,Eu, Er, Ea giuen in (42), (48), (51), and (52).

We now re-introduce the weights on the diagrams. I This willgive all the possible Cartan rnatrices: We recall that in the Dynkindiagram obtained from the simple system T : (dr, dzt - . ., &t), AriAi;,i+ j, is the number of l ines connecting ar and ai. ,If. A.i i*O,Aii + 0, then Ai;lAu : (ai, a;)l(a1, a) and Au or Ai; + -I whilethe other of these is -1, -2 or -3. Since nothing is changed inmultiplying all the d4 by a fixed non-zero real numbdr, we maytake one of the at to be a unit vector. If the diagrarn has onlysingle lines, then all the (a;, ar) :1 since the diagram is connected.Hence the weighted diagrams for At, Dt, Er, E7, Es ?te,

At t l -3 . . .1 - ] />r ,d1 d ,2 & t - t 4g

o - o - o _ o - _ o

oII

II

o + o - o - o - o - o

oII

I

Io - o - o - o - o - o - _ o

Es

E,

S'

Lo,o'I

1 11 1' ' * i i * i - , ' t>4'

E a :

W. SPLIT SEMI.SIMPLE LIE ALGEBRAS

t lou

I 1 11 I 1o - o - o - o - o t

d.1 dz da, dl d's

d4 dz 4s &a d'r 45

t io'

Ea: l-t-l-l-Jt-L-ld.1 dz ds d.a ds d.6 d7

For Gr we have chosen the notation so that

Gz:3- l .d4 Ol2

For Fr we may take the weights as follows:

F+ i l_:_2"-? .d..1 d.z da &q

For Br and Cr we take the following diagrams:

2 2 2 2 1 , _o - o - o . . . o - o , 1 2 241 dz d.0, &t-t &t

1 1 1 2 , \o - O . ' . O - O : O , l > 3

d1 4z dt-r d7

These diagrams give the possible Cartan matrices.

6. Construction of the algebras

The time has now come to reveal the identity of the principal

characters of our story-the split simple Lie algebras. With everyconnected Dynkin diagram of an a.c. which we determined in $ 5we obtain a corresponding Cartan matrix (Arr) and there exists forthis matrix a split simple Lie algebra with canonical generatorsei , f ih i , i : t ,2, . . . ,1, such that lerhl j : At&i . We shal l g ive thesimplest (linear) representation of the algebras corresponding todiagrams Ar, Br, Cr, Dr, Gr, F, and Eu. Later (Chapter VII) anotherapproach will be used to prove the existence of split simple Lie

135

136 LIE ALGEBRAS

algebras corresponding to the diagrams E, and Er.We recall that if I is an irreducible Lie algebra of linear trans-

formations in a finite-dimensional vector space over a field ofcharacteristic 0, then 8 : I, O O where I, is semi-simple and 0 isthe center. Hence such an algebra is semi-simple if r and only ifG :0. We note also that if ,8 is semi-simple and 8 contains anabel ian subalgebra 6 such that 8:6@Oer@OepO.. . where&, 9,. . . are non-zero mappings of 6 into O such that Ieuhf : a(h)en,h e 8, then O is a splitting Cartan subalgebra for 8 and 8 is split.We shall use these facts in our constructions.

Let O be the algebra bf linear transformations in an (, + t)-dimensional vector space llt over O, I > L. It is well known (andeasy to prove) that 0 acts irreducibly on llt. We have 0z : EL$AIwhere 8 = Gl is the derived algebra and is the set of linear trans-formations of trace 0. Evidently, aoy Gi-invariant subspace is @z-invariant. Hence 8 : Gi is irreducible. Also the decompositionCIz : Si O O1 shows that the center of Ol is 0. Hence @i is semi-simple.

We now identify G with the algebra O*r of. (l * l) x (/ + 1) ma-trices with entries in O,8 with O'r*r, the set of matrices of trace0. We introduce the usual matrix basis (e;), i, j : l , ...,1* 1, inOr+r so that

(53) €;i€*n: 6i*Qi^

A basis for I is

( 5 4 ) h x : € r n - e r + r , t + r , h 3 l ; € ; i , i + j - 1 , , " , 1 + t .

Set h - }l,rorh*. Then the set of h's is an abelian subalgebra 6of / dimensions and

(55)lert, hl: (or, - @r)on ,

fer*r . , ,h l - ( r * a,)or+r, , , T -

fe, , r*r ,h l - -Q * o4)e, ,s1 , r + " ' r l

The l2 * / l inear funct ions h-1o)r - o) r t h- -+T * @r, h- t+- ( r + co, )

are distinct and are non-zero weights of ads0. We have E: O + >8'where a runs over these weights. It follows that 6 isl a splittingCartan subalgebra and the a are the non-zero roots. Set

Ix,,I

S : 1 ,

(56) &r : 0)r - 0)z t

dl-t : 0)1,-t - (t)l ,

& z : ( D z - ( D g s " ' ,

d r , : T * a t ,

Then

(57)

ry. SPLIT SEMI.SIMPLE LIE ALGEBRAS

a r r * c t x + r * " ' + a 1 : ( t o v - c D n + r )

* (orsr", - ox+z) + "' * Q * ttt)

: y * o * t h : l r 2 r " ' r l - L ,

a ; * d ;+ t+ " ' * a l : (a t - o ;+ r )

( 5 8 ) * ( a r ; + , - t o ; + z ) + " ' * ( a t - @ i + t )

- <Di - 0)i+t , r= is j< t - r .

This shows that every root has the form }h;a; where the k; are

integers and ka > 0 for all i ot hi 3 0 for all i. Hence the at form

a simple system of roots for 8 relative to b. Equations (57) and( 5 8 ) s h o w t h a t a i * e * t i s a r o o t , 1 S i < I - I , d t * Z a ; a 1 i s n o t

a root, 3Bd ar I at is not a root if i > i + 1. This means that the

Cartan integers Au have the following values:

(5e) r< i< lA ; r : 0 i f j > i + t o r j < i - 1 .

Hence the Dynkin diagram is the connected simple chain

1 1 1A t : o - _ o . . . o - o

d1 d,2 dt-t (I1

It follows that I is simple of type /r (with Dynkin diagram Ar).

Hence we have

Tnponpu 6. Let 8, be the Lie algebra of linear transformations of

trace zero in an (l * l)-dimensional aector sPace oaer A. Then 8' is

a shlit simhle Lie algebra of type At.Assume next that Dt is z-dimensional over 0 and is equipped

with a non-degenerate bilinear form (x, y) which is either symmetric

or skew. Let I be the Lie algebra of linear transformations A

which are skew relative to (x,y), that is, (rA,y) - -(r,yA) (cf'.

$ 1.2). We require the following

Lpuru.r, 2. 8, is irreduci,ble if n > 3.Proof: Let Tt be a non-zero invariant subspace relative to 8 and

let z be a non -zero vector in It. Let u be any vector in the or'

thogonal complement Azr. Choose a d, OzL and consider the linear

transformation A: x - (x, u)a - (u, x)u. One checks that A e 8.

Moreover, zA: -(o,z)u+ 0 is in !t. Hence !t contains Oer for

138 LIE ALGEBRAS

e v e r y e € I t . I t f o l l o w s t h a t d i m T t ) : / t - 1 a n d d i m [ : z u n l e s sTt : AzL for every z e It. Then It is totally isotropic {nd dim Il <

lnlz). (cf. the author's Lectures in Abstract Algebra'll, p. 170).

This implies that n:2, contrary to our assumption.We shall now distinguish the following three cases: B. (r, y) is

symmetric and n : 2l + 1, , = 1. C. (x, y) is skew so necessarily

z is even dimensional, say, n :2i, I > L. D. (x, y) is symmetric,

n - 2l4, t > I. Moreover, we shall assume that in the symmetric

cases (B and D) the bilinear form has maximal Witt index. This

means that Dt contains a totally isotropic subspace tt of / dimen-

sions.B. Let (ut, uz,

(oi. A linear-(u;, utA) for i, i : l , ' ' ' , f l . If uoA : Zrcdixl 'tr, then these con'

ditions are that Zrd*6ri: -Xrdtxdix, or in matrix fofm,

(60) as: -sat , a - (au) .

Since the form is of maximal Witt index the basis cdn be chosen

so that00

6 ' L t

columns and avoll II, p. 168).

by a non-zero

where 1r denotes the identity matrix of / rows and

is a non-zero element in CI (cf. the author l2l,Since nothing is changed by replacing the form

multiple of the form we may assume that

If we partition a in the same way as s:

: ',,:)'

where a,it € Or, ur, Irz vtl I x L matrices and ur, uz ate | *

then a simple computation shows that (60) holds if and/ matrices,only if

a z z : * a ' r , ,

. . . , i l n ) be a bas is fo r l l t and le t (a ; ,4 ) : o i i , s :transformation A e 8, if and only if (unA, u) -

":,,) ,' : ( [

':(i l, :,)(61)

,:(n'llt :

a l z :(62) - 0 .

Iv. SPLIT SEMI.SIMPLE LIE ALGEBRAS 139

These conditions imply that the following set of elements is a

basis for 8, identified with the algebra of matrices satisfying (60)

(63)

l l t : € i + t , i + L - € i + t + t , i + t + t

€ o r o J : € i + t , i + t - e i + l + l , i + l + L i

€ o t r * r o 1 : € i + r + r , r + t - 0 5 + t + t , i + r i

€ - a t r - , o 1 : € 7 + r , i + z + r - e i + r , i + I + r i

g r t : € r , i + t - ? i + t + r , t

t - . t : € i + t , t - e t , i + t + t

where i, j : L, -. .,1. The set O : tX a;h;j is an abelian subalgebra

of 8. The linear forms which are subscripts can be identified

with the linear mappings h -- a(h), where h : Zorh6 ?trd we have

le,hf - de6,1 d - 0)i @j, oi * ari, etc. It follows that 0 is a split'

ting Cartan subalgebra and the d ate the non-zero roots. IJ acts

irreducibly on Dt and if z - ho * 2 p,gn is a center element, then

lzhl - 0 gives Lpra(h)eo :0. Since the e, are linearly independent

this implies that p,a(h): 0 for all' h. Since a * 0 we have 0o : 0'

Hence z:ho. But then lzer) :0 gives a(h):0 for a l l a- Since

there are / linearly independent a (e.g. the o)it i : l, ' ' ',1) we

have ho:O. Hence the center is 0 and I is semi'simple and split.

We now assume I > 2 and we set

(64) &r: (Dr - 0)2 t d z : 0 z - ( D g t " ' ,

dl , - t : 0)7-1 - (D7 , d l : 0) t .

One checks that this is a simple system of roots with Dynkin

diagram Br:

2 2 2 2 1

;=;,";"u-, ;-&,'

Hence 8 is sirnple of type Br. We therefore have the following

Tnsonnn T. Let 8 be the Lie algebra of linear transformations in

a (21 * I)-dirnensional space, I > 2, which are skew relatiae to a non'

d.egenerate syrnmetric bilinear form of maxi'mal Witt index. Theng is a s\tit si.mhle Lie algebra of tyfe Bu

C. Let lft be of dimension 21, I > L, (x, y) non-degenerate skew

bilinear form in llt. Let I be the Lie algebra of linear transfor-

mations which are skew relative to (x, y). We can choose a basis

+ j< j< j

140 LIE ALGEBRAS

(trr, ur, .. ., uzt) so that the matrix q - ((u;, u)) is

(65) o: (-1, l')As in B, one sees that I can be identified with the Lie algebra ofmatrices a e O27 such that aq: -qat. This implies ttlat

(oo) a: (a" o") , aii € or ,

\4zr azz/ '

where

(67) a z z : - a l r , a l z : a p , a l t : a z t .

Hence I has the basis

h; : € i ; - €t+; . t+ i t

0oyo6 : e i i - €g+z , i+ t r i + i

( 6 8 ) € - o t t r - t o 1 : € i . i t t . * e i , r * r , i < j

0 o 4 r t t 1 : € i + t , 1 * € Y t , t , i < j

g * r . r : € ; , ; + t , , o z - Q : o i + r , i , t

where i , i :L ,2, . . . ,1 . As in B, one proves that O=t>ar ; /z ; ) isa Cartan subalgebra and the subscripts in (68) define the rootsrelative to 6. Also 13 has center 0 and So, by Lerrima 2, 8 issemi-simple. The roots

(69) oh : ar - o)zr " ' t dt - t : o)r- r - o) t t dt :2@t,

form a simple system with Dynkin diagram Ct. if , > 3. Thisproves

Tnnonpu 8. Let 8 be the Lie algebra of li.near transforntations ina Zl-dimensional space, I > 3, which a're skew relatiua to a non'

degenerqte skew bilinear form (the syruplectic Lie algebm.) Then 8,

is a sflit simfle Lie algebra of tyhe CuD. Dt, 2/-dimensional, I > 2, (x, y) symmetric of maximal Witt

index. Here we can choose the basis (zi) so that t - ((ur, zr)) has

the form

(70) t -- (0, 1')- U , 0 /

and 8 can be identified with the Lie algebra of matrices a satisfy'ing at : -ta'. These are the matrices

ry. SPLIT SEMI.SIMPLE LIE ALGEBRAS 141

(?1) o : (1" 1"\ , a;i € or, ,\4zr azz/

such that

(72) azz: -a!r, , alrz: -aP t a!n: -azr '

Then 8 has the basis

h i : A a - A r , + i , t + i

( 7 3 ) Q o Y o l : e i i - € + + t ' i + t ' i + i

e o 4 * o t 1 : € i + t , i - 7 i + t , ; , i < i

0 -o4 -o1 :01 , r+ t , - ^ i , i + r , , i < i

where i, j :1,2, " ',1. 6 : {X ouh;} is a splitting Cartan sub'algebra and the subscripts in (73) define the roots. The center of

I is 0; hence I is semi'simple. Set

( 7 4 ) & t : 0 ) t - Q t b " ' t & t - r : &) t - t - 0 ) r , r dL : 1ar ' l - r * to t '

Then these ai form a simple system of roots which has the Dynkin

diagram D, if. I Z 4. We therefore have

Tnnonnu 9. Int g be the L|e algebra of linear transformations

in a Zhdin ensional space, I > 2, which are shew symmetric relatiue

to a non-degenerate symmetric bilinear form of marimal Witt index'

Then if t > 4, I fs a sfli't simple Lie algebra of tyhe Dt.

The four classes of Lie algebras A, B, C and D are called the,,great, classes of simple Lie algebras. These correspond to the

linear groups which Weyl has called the classical groups in his

book with this title. It is easy to see directly, or from the bases,

that we have the following table of dimensionalities

tybeAt

(75) Br,CtDr,

dimensi,onalityt(t + 2)t?t + t)t(zt + L)t(zt - L) .

The determination of the simple systems and the general iso-

morphism theorem (Theorem 3) and the criterion for simplicity

(Theorem 4) yield a numhr of isomorphisms for the low dimensional

orthogonal and symplectic Lie algebras. The verifications are left

to the reader. These are1. The orthogonal Lie algebra in 3'space (three-dimensional space

I42 LIE ALGEBRAS

llt) defined by a form of maximal Witt index and thb symplecticLie algebra in 2-space are split three-dimensional sirirple and soare isomorphic to the algebra of matrices of trace 0 itr Z-space.

2. The orthogonal Lie algebra of a form of maximal Witt indexin 4-space is a direct sum of two ideals isomorphic to split three-dimensional simple Lie algebras.

3. The symplectic Lie algebra in 4-space is isomotphic to theorthogonal Lie algebra in S-space defined by a symmgtric form ofmaximal Witt index.

4. The orthogonal Lie algebra in 6-space of a formt of maximalWitt index is isomorphic to the Lie algebra of lindar transfor-mations of trace 0 in 4-space.

The remaining split simple Lie algebras: types Gr,4r, Ea, Et znd.Es zre called excepti,onal. We shall give irreducible representationsfor Gr, Fr, Eu but we shall be content to state the resirlts withoutproofs, even though some of these are not trivial. A completediscussion can be found in a forthcoming article by the 4uthor (t111).

Our realizations of G" and F. will be as the derivatfion algebrasof certain non-associative algebras, namely, an algeb4a of Cayleynumbers 6 and an exceptional simple Jordan algebra i4f.

Following Zorn, the definition of the split Cayley algehra or uector-matri.x algebra 6 is as follows. Let V be the threeidimensionalvector algebra over O. Thus I/ has basis i, j,k over @ and hasbilinear scalar multiplication and skew symmetric vector multipli'cation x satisfyingi i, j, h are orthogonal unit vectors and

(76) : j x j : k x h - 0 , i x j : h , ,

: i , k x i - j

Let 6 be the set of 2 x 2 matrices of the form

(Tz) (a a\ A

\ ; - g )

, a , P e o , a , b e V .

Addition and multiplication by elements of O are as u$ual, so thatO is eight-dimpnsional. We define an algebra product in 6 by

;) :

The split Cayley algebra is defined to be 6 together \iith the vec-tor space operations defined before and the multiplicafion of (78).6 is not associative but satisfies a weakening of thd associative

i x i

i x k

(?8) (; 'r)

(, ( a y - @ , d ) m * B a * D x d \\ r D + P d + a x c P s - ( b , c ) /

W. SPLIT SEMI.SIMPLE LIE ALGEBRAS

law called the alternatiue law:

143

(7e) tc'! : x(xy) , !x2 : (Yr)x .

Let 8 be the Lie algebra of derivations in 6. The unit matrix

1 is the ident i ty inGand since 1' :1, ID:0 for every der ivat ion

D. The space 6o of elements of trace 0 (a * I : 0) coincides with

the space spanned by the commutators [ry] : fri - !x, r, y e E.

Hence GoD g 6o for a derivation. Thus 6o is a' seven-dimensionalsubspace of 6 which is invariant under 8. The representation in

Go is faithful and irreducible.

lf. T is a linear transformation of trace 0 in V, and T* is its

adjoint relative to the scalar multiplication, then it can be verified

that

(;

is a derivation in 6. The set of these derivations is a subalgebra

8o isomorphic to Olz. In any alternative algebra any mapping of

the form Da,o : fafir] + [arbR] + lanbn), where a, b are in the alge'

bra and az,en denote the leftand the right multiplications (x-+ar,

x -, xa) determined by'a, is a derivation. In 6 any derivation has

the form D"r,or, * Drz,t n a Do, where

"u) -'(-our."[)(80)

(81)e r :

A n :

€ z :

bzr :

(31)(: 3)

(l3)(3 6)

and Do € 80. If 6 is a Cartan subalgebra of 80, 6 is a Cartan sub-algebra of 8. If we identify 8o with AL", we can take S to be theset of matrices of the form orht * ozhz, ltr: et - e$, hz: €zz - €ss,Then 6 is a splitting Cartan subalgebra of 8 and the roots of O in

8 are: t@r, *o)zt t(<o, - lrl,z), t(ar, * arz), +(Zar * arz), t(c,rr + Z,c|.z).The center of 8 is 0 and since I acts irreducibly in 60, I is semi'simple. The roots dr: o)r - o)zt dz: o)z form a simple systemwith Dynkin diagram Gr. Hence I is split simple of type Gz.

The Cayley algebra 6 has an anti'automorphism x-+i of period

two such that 1 : !, I : -)c if r € 60. Let Mi denote the space of

3 x 3 hermitian Cayley matrices defined by this anti-automorphism.Thus Mrt is the set of matrices of the form

r44

(Bz) .:(1;

If x, Y e M{, then

LIE ALGEBRAS

)Cs tt\

E , r r l , t t € 6 , E r e O .xr Etl

(86)

(83) X.Y : L(XY + YX) e M: ,

where the product XY is the usual matrix product. iThen Mf isa non-associative algebra relative to the usual vector;space opera-tions and the multiplication (83). Moreover, the mulltiplicatibn inMs8 satisfies

(84) X..Y: Y'X, (x" .Y). x - x2 .(Y. x) .These identities are the defining properties of a clas$ of algebrascalled tordan algebras. A consequence of these ide4tities is thatif R/ denotes the mapping X-' X- A - A. X, then

(85) Dt,n: [R1Ral

is a derivation in the algebra.Let 8 denote the Lie algebra of derivations in Mf and let 8o

denote the subalgebra of those derivations which map the elements

e t : d iag {1 ,0 ,0 } ,es - diag{O,0, 1}

ez: d iag {0, 1,0} ,

into 0. Then 8o is isomorphic to an orthogonal Lie ialgebra of asymmetric form of maximal Witt index in 8'space. Let O be aCartan subalgebra of 8o corr€sponding to the Cartan subalgebraselected above for the orthogonal Lie algebra (type D.). Then Ois a splitting Cartan subalgebra of 8.

The derivation algebra maps the 26'dimensional space Io of. matri-ces of M! of. trace 0 into itself. The representation in /e is faithfuland irreducible. The center of 8 is 0, so 8 is semi-simple. Theroots of 6 in 8 are

l c o i t c o t , i 1 i : L r 2 r 3 r 4 . l

t @ ; t

t 1;, wherl Aa: *(rt + @z * arg * orr) - ori ,+ M ; , M r : * k o r * u z * a . g * c o t ) ,

M z : L @ r * o z - @ s - a , t ) ,

Ms: L@, - arz * ara - @t) ,

M.: Lko, - 0)2 - @s * o\) .


The roots

a t : * r h - * t r - L t r - L t r ,

ds : (t)g - 0)a 1

form a simple system with the diagru* l-!-2"-3 of typ"

Fr. Hence I is split simple of type Fr.We now use the notation D for the derivation algebra of. Mt

and we let I denote the set of linear transformations in Ms8 which

have the form '

( 8 8 ) L : R t + D , t r A : 0 , D e 0 .

This is clearly a subspace of the space of linear transformations

in Ml. Moreover, if tr B :0 and .E e E, then

(8e) lRrR"l € S , [R/E] : R.a.E

Since [O, EJ E D these relations impty that I is a Lie algebra oflinear transformations. Now I acts irreducibly in MsE and has 0center. Hence I is semi-simple. Let S denote the Cartan sub-algebra of S defined above and let 6 : AR"r-es * OR"r-r, + A. Then

O is a Cartan subalgebra of 8 with roots

t a ; ! a i , i < j : 1 , 2 , 3 , 4

! (ttti *. *(toa - arr)) ,

! (4 + l2@, - ars)) ,-r (M,t + l@t - &rJ)

where 1; and Mi arre as in (87). The roots

+ *(aru - alr) ,

145

d Z : ( D + t

d t : ( D z - O s

d r - - - @ r

& t : @ s * 0 ) t t & s :

d z : 0 ) t - ( l D z , d g ,

+ *(a, - eoa) , do

- 0 )z - 0 )s t

0)t

form a simple system of type Es. Hence 8 is simple of type 86.

An enumeration of the roots gives the following table of dimen-

sionalities:

(e1)

tybeG2

F,Ea

'We state also (without proof) that Et

mensionalities 133 and 248 respectively.

dimensionalityL4527 8 .

and Ea exist and have di-

146 LIE ALGEBRAS

Conclusion. Every split simple Lie algebra over Fn arbitraryfield of characteristic 0 is isomorphic to one of the ll-ie algebrasA r , l 2 L , B t , l > 2 , C t , 1 2 3 , D 1 , l > 4 c o n s t r u c t e d a b o v e o r t o a nexceptional Lie algebra Gr, Fr, Eu, Er, or EB, We shall see later(Chapter IX) that the algebras listed here are not isomorphic.Hence the results give a complete classification of split simple Liealgebras over any field of characteristic 0. In particullar, we havea complete classification of simple Lie algebras ov$r any alge-braically closed field of characteristic 0.

7. Compact forms

One of the most fruitful and profound ideas in the theory ofLie groups is the method introduced by Weyl for transferringproblems on representations of Lie groups and algebrab to the caseof compact Lie groups. Weyl gave this method the $triking titleof the "unitary trick" and he used it to give the first proof of

complete reducibility of the representations of serrii'simple Lie

algebras over algebraically closed fields of charactefistic 0. Heused it also for studying the irreducible representdtions, deter-mining the characters and dimensionalities of these representations.We shall consider this later on. Weyl's method permits the appli-

cation of analysis-via the theory of compact groups*to Lie alge'bras. The essence of the method has been formalized lby Chevalleyand Eilenberg in the following form.

A property P of Lie algebras is called a linear property if (1) P

holds for 5J implies that P holds for 8o, I any exteilsion field of

the base field of L and (2) P holds for [3o for some eftension fieldg implies that P holds for 8. A trivial example is the statement

that dim g: n. Cartan's criterion implies that the property of

semi-simplicity is a linear property for Lie algebras of qharacteristic

0. Also it can be proved that the property that the finite-dimen'sional representations of such algebras are completely reducible isa linear property.

We now introduce a certain class of semi-simple ll.ie algebrasover the field R of real numbers and we shall see that the validity

of a linear property P forall of these algebras implies the validity

of P f.or all semi-simple Lie algebras of characteristib zero. The

class we require is given in the following

IV. SPLIT SEMI-SIMPLE LIE ALGEBRAS

Dprwrrron 3. A Lie algebra 8 over the field R of real numbersis called compact if its Killing form is negative definite. (This

implies semi-simplicity.)The importance of compact Lie algebras is that the groups as-

sociated with these algebras in the Lie theory are compact groups.The following result is an immediate consequence of the classifica-tion-due to Cartan;of simple Lie algebras over the field of realnumbers. The proof we shall give, which does not make use ofthis structure theory, is due to Weyl.

Tnponnru 10. Let g be a semi,-simple Lie algebra ouer the fietd,C of complex numbers. Then there exists a compact Lie algebra8u ouzl the field of real numbers such that the "complexification"(8")c =,8.

Proof: Since C is algebraically closed our structure theoryapplies. Let er,,.fi, h; be canonical generators for 8 and let thecanonical basis be chosen as in $4. We have seen that there existsan automorphism a of I such that ei : f;., fi : er. Then eT' : et,

.ff' - fi and since the e;, f; generate 8, oz : L. If we recall theform of the canonical basis determined by lhe e;,fr fti it is clearthat 8I :8-, for every non-zero root a. Hence if we choose anyeo + 0 in 8", then e"" + 0 is in 8-, and (e", e'") : 4, * 0. If wereplace e, by eL : ]re, we obtain (eL, e'f) : f,?al, and, since we areworking in an algebraically closed field, we may choose ,1" so that]Ltlr: -L. Hence we may suppose that (er,ei) : -1 for every ,u.If we choose €-a: €X for the positive roots a (relative to someordering in 6i), then we shall have this relation for all non-zeroroots, since oz : L. We now choose as basis for 8: (hy hz, - . -,

hr, €', 0-or .. .) where 0-a : eX and these elements satisfy

(e2)

for every non-zeroF + + a , t h e n

(e3)

(er, e-a) - _1

root a. If a and F are non-zero roots and

i f a* F is a root andrelation we obtain

lerepl: Nrgor+p * 0

[e,epf : 0 otherwise. If we apply o to this

(94) le-re-pl : N,pa- w+pt or le-re*p)

We know also that (e,,0-,) - -1 implies that

t47

lene-"1 - ltn and

148 LIE ALGEBRAS

this element is a rational linear combination of thE h;,. Now if

a * B is also a root, then i

(95) lNrp;r+p, Nrpe-r-pf : Nl,phr+p: Nlp(h, * hq) .

On the other hand,

LN rpL, +p, N,pe - r-pf - lle oe plLe -,e -p11::'!:;::';,i:::i:i')"ff!j,,i)-,i)-,'_ q(r ! t) (a,a)leBe_el +

q'(r'-+ l) (p, p)le6e_dl ,z

- - Y t ' 2

by (20). This can be continued to

- q(r ! l) (a, a)ou * W G, B)h, .z

' 2

\ F t r

Since a and p are linearly independent, h, and, hB are linearly

independent. Hence the tast relation can be comp4red with (95)

to see that Nje : lq(r * I)l2l(a, a) is a positive ratlionat number.

It follows that alt the Nrp, a, p roots, are real nUmbers. Since

lerh;l - a(h;)e, and a(h) is rational it is now clear that the multi-

plication table for the basis we have chosen has repl coefficients.

Hence the set of real linear combinations of this bdsis is an alge-

bra 8r over the real field R such that 8rc : 8. Since €X,: €-o,

hf, : -hi, 6 induces an automorphism of period two in 8t. We

now modify this automorphism by intertwining it with the standard

automorphism p --+V of the complex field. Thus if we denote the

chosen basis by (ur), we let r be the mapping: E pru, --+ ) Vgt -

This is a semi-automorphism of 8 in the sense thpt r is a semi-

linear transformation and [ry]" - lf y"l. In fact, we have (x + y)' :

tr ' * y", (p-r)' - ir ' , peC. The fact that r satisfies [ry]" : lr 'y"f

is clear from the reality of the multiplication.tabl$ of the a. It

is clear also that r is an automorphism of I considefed as an alge'

bra over R; hence the set ,8, of fixed points under r is an R'

subalgebra of 8. We shall show that 8* is the re{uired compact

fo rm o f g . I t i s c lear tha t * : ! . Hence 8 , , : {x * x " lx e8} . I t

follows that every element of 8, is a real linear Combination of

the elements zr + ul, y'A@t - u[). If we use the form of the

il.; ?nd the relations /ai : hi - -h;, eL: eZ: Q-' wel see that every

element of 8,, is a real linear combination of the elpments

Iv. SPLIT SEMI.SIMPLE LIE ALGEBRAS

( 9 6 ) { i 1 x n ' i - l ' " ' ' l '

€a * e-a , y'4qer - o-r) ,

where 4 ranges over the positive roots. These elements form abasis for 8 over C and this implies that (8")c :,8. Also we seethat any element of 8" has'the form

* : f1i{qhi + E (p,e, * i,e-,)

where the Fr "r" ,"'"r, p' compl

"1',0 U' the conjugate. Since I -(,8,)c, the Killing form for 8, is the restriction to 8,, of the Killingform in 8. Hence we may use the orthogonality properties andthe relations (er, e-r): -1 to calculate

(e7) (x, x) : -(h, h) - 2 Zp,i,

where h : ZE;ht. Since (h, h) > 0 untess /l : 0, it is clear that(r, r) ( 0 unless .tr : 0. Hence the Killing form is negative definite.

Now tet P be a linear property of Lie algebras which is validfor every compact Lie algebra over the field R of real numbers.Let.8 be a semi-simple Lie algebra over a field O of. characteristic0 and let Q be the algebraic closure of O. Then 8o is split andso this has the form 8oo where 8o is a Lie algebra over the field

Q of rational numbers. Shen 80" is a semi-simple Lie algebraover the complex field C; hence, by Theorem 10, there exists acompact Lie algebra 13, over the reals such that 8,,c :80c. SinceP is satisfied by ,8*, it holds for 8uc: 8oc, hence for ,80 and for8oo : 8o. It follows that P holds for 8.

Exereises

In these exercises all base fields are of characteristic 0 and all algebrasand modules are finite-dimensional. Unexplained notations are as in the text.

1. Show that 0 is a Cartan subalgebra of a semi-simple Lie algebra 8 ifand only if: (1) S is maximal commutative and (2) 0 is reductive in 8, thatis, adgS is completely reducible.

2. Let 0 be a Cartan subalgebra of the semi-simple Lie algebra 8. Showthat if .B is any finite-dimensional representation of 8, then ltn is semi-simple for every h e 8.

3. Let 8r be a semi-simple subalgebra of the semi-simple Lie algebra Iand let Sr be a Cartan subalgebra of 8r. Show that Sr can be imbedded ina Cartan subalgebra of 8.

149

150 LIE ALGEBRAS

4. Use the method of $ 3 to obtain a canonical basis and rnultiplicationtable for /2 and Gz.

5. Use the method of $ 3 to obtain the roots for the split Irie algebra bftype Et (assuming it exists) and show in this way that the dimensionalityof this Lie algebra is 133.

6. Without using the determination of the connected Dyrtkin diagramsR l l

show that ;-6-6 cannot be the Dynkin diagram of a Cartan matrix.(Hint: Show that the set of roots determined by this contains a positive

root p such that 29 is a root.)7. Show that the constants of multiptication of a canonical basis ($3) of

a split semi-simple Lie algebra are rational numbers with deirominators ofthe form 2k3r, k,l integers.

8. Prove that any split semi-simple Lie algebra can be genbrated by twoelements.

9. Let I : @'L, @, the algebra of linear transformations ln an (t + t)-dimensional vector space and let (A, B) be the Killing form in 8. Show that

(A, B) :2( I I 1) t r AB '

(Hint: Use Exercise 3.9)10. Express the Killing forms of the Lie algebras of types Br, Cr. and Dr,

in terms of the trace form tr AB where the representation is the one given

i n $ 6 .11. Show that if I is split, 6 a sptitting Cartan subalgebra, th0n 0 * I">08,

is a maximal solvable subalgebra and lo>o8a is a maximal nllpotent subal'gebra of 8.

12. Determine the Weyl group for the split Lie algebra o{ type Gz and'

the split Lie algebra of typ, Az.13. Let at be a simple root and let a be a positive. roof * a6. Show

that every root of the form a * kae, /c an integer, is positive. Show that

2'*@ * Ica,1, ai = 0 where the summation is taken over the' k such that

a * kat is a root and a is any positive root. Hence prove that lc>o(a,at):(ar, a6), or lc>oa(h) :2.

CHAPTER V

Universal Enveloping Algebras

In this chapter we shall define the concept of the universal en-veloping (associative) algebra 1l of a Lie algebra 8. The principal

function of 1l is to reduce the theory of representations of I tothat of representations of the associative algebra ll. An importantproperty of 1l is that I is isomorphic to a subalgebra of llz. Inthis way one can obtain a faithful representation of every Liealgebra. For finite-dimensional Lie algebras we shall obtain in thenext chapter a sharpening of this result, namely, that every suchalgebra has a faithful finite-dimensional representation. In thischapter we obtain the basic properties of ll. Some of these willbe used to prove an important formula due to Campbell andHausdorff on the product of exponentials in an associative algebra.We shall give also the Cartan-Eilenberg definition of the cohomologygroups of a Lie algebra.

The discussion here will not be confined to finite dimensionalalgebras nor to algebras of characteristic zero. In fact, a part ofthe chapter will be devoted to some notions which are peculiar tothe characteristic P + 0 case. The main concept here is that of arestricted Lie algebra of characteristic 2, which arises in consider-ing subspaces of an associative algebra which are closed relativeto the mapping a -, ao as well as the Lie product labl - ab - ba.Restricted Lie algebras have restricted representations, restrictedderivations, etc., and we can define a "restricted" universal en-veloping algebra called the a-algebra. We discuss these notionsbriefly and consider the theory of abelian restricted Lie algebras.

7. Definition and baaic properties

The central notion of this chapter-the universal envelopingalgebra of a Lie algebra-is a basic tool for the study of represen-tations and more generally for the study of homomorphisms of aLie algebra I into,a Lie algebra 2Iz where lI is associative with

151

I52 LIE ALGEBRAS

an identity element. Any such homomorphism can bb "extended"to a homomorphism of the (associative) universal envbloping alge'bra ll into !I. Throughout this chapter we shall be cotrcerned withLie algebras and with associative algebras containirlg identities.

We recall our conventions (of Chapter I) on terminology: "algebra"will mean associative algebra containing an identity element 1,

"subalgebra" means subalgebra in the usual sense containing 1,and "homomorphism" for algebras means homomorphism in theusual sense mapping 1 into 1. ]

DnnNtnoN 1. Let I be a Lie algebra (arbitrary dfmensionalityand characteristic.) A pair (U, i) where 1l is an algebra and a is ahomomorphism of I into llz is called a uniuersal enuelbping algebra

of I if the following holds: If lI is any algebra and d is a homo-morphism of I into ?Iz, then there exists a unique homomorphism0t of. ll into lI such that 0 : i?t. Diagramatically, we are given

tl : llz I

i l

l__I e ? [ : l l z

where i and 0 are homomorphisms of I and we can complete this

diagram to the commutative diagram_ 1 l- ++L

\ a f

\:l

o l t *Wt

where d' is a homomorPhism of 11.

A number of important properties of

definition, or are easy consequences of

tations. We state these in the following

Trnonnu 11. Let (V,i), (S, i) be uniuersal enueloping algebrad for

there exists a unique isomorphism i' of lI onto E sucht that

2. lI is generated by the image 8'i.3. Let gy g, be Lie algebras with (11,, f'), (Ur, ir) respdctiae

ut

i lIIa

(lI, f) are builtbasic facts on

into thisrepresen-

8. Then. . . t

J : U .

uniaersal

V. UNIVERSAL ENVELOPING ALGEBRAS 153

enaeloping algebras and let a be a homomorphism of & into tz. Then

there erists a uniqae homomorphi,sm a' of l\ into llz such that ai2 :

ita', that is, we haae a commutatiue diagram;

8r ----i---t 82

ln,IJ

trlllt ,f+ llr

4. Let E be an i,deal in 8 and let R be the ideal in 7I generated

by Ei.. If I eE then i: t +E-rti *,F es a homomorphism of 8,N

into En where E - ll/F and (8, j) is a uniaersal enueloping algebra

for 8,18.5. 17 has a unique anti-autornorphism n such that h - -i- More'

ol)er, t' -- L.6. There is a unique homomorphism d' called the di,agonal map'

f r ing of 71 into U8U such that ai6 ' :a i '&L + 18 ai , ae8' .7. If D is a deriuation in g there exists a unique deriaation D' in

17 such that Di: iD'.Proof:1. If we use the defining property of (U, i) and the homomorphism

0 : j of I into E.r w€ obtain a unique homomorphism i' of ll into

S such that I - ii' . Simitarly, we have a homomorphism i' of S

into t l such that i : j i ' . Hence j : j i ' j 'andi : i j ' i ' . On the otherhand, we have i : il" where 1o is the identity mapping in E. If

we apply the uniqueness part of the defining property of (t, r) to0: j we see that ' i t j t :1o. Simi lar ly, j ' i ' - I* the ident i ty in U.

It follows that i' is an isomorphism of 1l onto E.

2. Let S be the subalgebra generated by I3r. The mapping I

can be considered as a mapping of .8 into Sr. Hence there is aunique homomorphism it of l7 into E such that i -- ii' . Since i : iluand i' can be considered as a mapping of 1l into 11, the uniquenesscondition gives i' :1,. Hence ll - lll, : Ui' g E. Hence E : tl.

3. If a is a homomorphism of ,8t into,[Jz, aizis a homomorphismof 8r into tlzz. Hence there exists a unique homomorphism a' ofllr into Uz such that ira' : diz.

4. We note first that the mapping / --+ li *,fr' of 8 into E - 1l/Sis a homomorphism of 8 into Ez. Since Ei = S, E is mapped into0 by this homomorphism. Hence we have an induced

'homo'

154 LIE ALGEBRAS

morphism / + E + li +,fr' of ,8/8 into E. This is the mapping /.Now let d be a homomorphism of 8/E into llz, ?I an algebra. Thentt: I --+ (/ + t)a is a homomorphism of ,8 into llr. Herlce there ex'ists a homomorphism 7' of tl into 9I such that iqt : 11. If D e E,brt - 0 so that bi is in the kernel of the homomorphism T' . Itfollows that 0 is in the kernel of q' and consequently we have theinduced homomorphism 0': u + fi --. un' of E - ll/.f into lI. Now( r + E ) a - l T - l i 4 ' a n d ( t + A ) j | ' : ( t i + R ) 7 t : l i q t . H e n c e 0 : i 0 'as required. It remains to prove that d/ is unique. This willfollow by showing that (8/E)i generates E. Now, by 2., ll isgenerated by 8i, which implies that E is generated by the cosets,i + A. Since (/ + l3y : li * S, it follows that S is generated bythe set of elements (/ + E)i, that is, by (8/E)f. This completesthe proof of. 4.

5. lf p' is an anti-homomorphism of an algebra !l into an alge-bra E, then one verifies directly that -p' is a homornorphism of\IL into Ez. Let I be an algebra anti-isomorphic to tl under amapping u -n ?ttt' of 11. (Such algebras exist trivially.) The mapping0 : -ipt is a homomorphism of 8 into [r. Hence there exists ahomomorphism d' of 1l into I such that 0 - -ip' : i0'. Let r :0'(tt')-'. Then z is an anti-homomorphism of 1l into itself such thatitr : -i. Hence inz : i and since 8f generates ll and zP is a homo-morphisrir, rcz : L. Hence z is an anti-automorphism in U. Theuniqueness of z is also immediate from the fact that l3e generates ll.

6. The reasoning used to define the product of representationsand modu les ($1 .6) shows tha t a - ra i 81+ L&a i , a€8, , i s ahomomorphism of I into (ll8ll)r. Hence there exists a uniquehomomorphism d' of l l into UBU such that ai,B' : ai,81 + 1&ai.

7. Let D be a derivation in 8. We form the algebra. Uz of 2 x 2matrices with entries in the universal enveloping al$ebra tl andwe consider the mapping

( 1 )

of 8 into Uz. This is a linear mapping and

o : , -, (3' "1,')

oil) - (i' o?ou) (3'lai aDi\ /bi

( z ) \o o t ) \o. : ('o'6"t

aD.i\a t /

lai, bDil + laDi, Dtl\ _ (fabli Ia, blDi\l a i , b i l / - \ o t w b l i ) '

( 3 )

and the ai generate tl,

( 4 )

V. UNIVERSAL ENVELOPING ALGEBRAS I55

Hence d is a homomorphism of I into tlsz. It follows that there

is a homomorphism 0' of tl into tlz such that 0 : i0'' Since

we have for any tc e U,

* , : ( ; t ) ,

where y is uniquely determined by ti'. We write ! : rD' and a

calculation tike (2) shows that D/ is a derivation in 8. Then (3)

shows that ai,Dt : aDi. Hence iDt : Di as required. The unique-

ness of D/ follows from the fact that 8l generates ll, and a deri'

vation is determined by its effect on a set of generators.

We now give a construction of a universal enveloping algebra.

Let t denote the tensor algebra based on the vector space 8. By

definition,

T . -aLOS '@8,O" '@8rO" '

a;o' -- (ai "3u)

( 5 )

( 7 )

w h e r e 8 r : 8 a n d 8 i - 8 8 8 " ' 8 8 , i t i m e s . T h e v e c t o r s p a c e

operations in E are as usual and multiplication in t is indicated

bv I and is characterized bY

( \& . . . A r c ) I ( y , 8 . . . 8y r )( 6 )

\ d

: . r r g . . . A x ; & l r 8 . . . 8 y i .

Let S be the ideal in ! which is generated by all the elements of

the form

l a b l - a & b * b & a , a , b e 8 ,

and let U : t/,Q. Let f denote the restriction to I :8' of the

canonical homomorphism of ! onto ll. We have

l a b ) i - a i & b i * b i & a i- ( l ab l - a8b+bBa)+n: S :0 ( i n 11 ) .

Hence i is a homomorphism of I into llr. We shall now prove

Tnnonpu 2. (U, i) is a uniaersal enueloping algebra for 8'.

Proof: We recall first the basic property of the tensor algebra,

that any linear mapping d of I into an algebra lI can be extended

156 LIE ALGEBRAS

to a homomorphism of ! into?[. Thus let {uili e I} be a basisfor 8, then it is well-known and easy to prove that I the distinct

"monomials" u1r&uirA "' Iuin of degree n f.orm a lbasis for 8o.H e r e u i r S u t r & " ' A i l i n : ? / r r @ U r r $ " ' & u r * i f ; a 1 1 d o n l y i fj , : k , , r :1 ,2 , " ' , / t . The e lement 1 and the d i f fe ren t monomi -als of degrees I,2,... form a basis for t. It is easy to checkthat the linear mapping 0" of t into ?I such that 10" :1,

(zr ' ,8 uir$. . . A ui* \0" * (q i r?)(uir?) " ' (u i*0) is a homomorphismof E into !I such that a0" :e0 for a € I (- 8'gt). Now let d

be a homomorphism of I into ?Iz and let 0t' be its e*tension to ahomomorphism of t into ?I. If. a, b e 8,,

[ab)0" - (ao")(bo" ) + 1bo"11ao")- [ab)o - (a0)(b0) + (b0)(a0)- la|,b?l - (a0)(b0) + (b0)(a0)- 0 .

Hence the generators (7) of S belong to the kernell of 0't . We

therefore have an induced homomorphism 0' of" ll intd ?I such that

aiT' : (a + R)0' : a0" : a0. Thus 0 : i0' as required. The tensor

algebra t is generated by I and this implies that l1 ris generated

by 8,i. Since two homomorphisms which coincide on generators

are necessarily identical there is only one homomorphism d' such

that i0' : 0.

2. The Poincarb'Birkhotr'Witt theorem,

We have noted that if {qli e J}, where / is a set, is a basis

for 8, then the monomials z;, & uirA ' ' ' & ui^ of dbgree n form

a basis for 8,., n Z I. We suppose now that the set "/ of indices

is ordered and we proceed to use this ordering tO introduce a

partial order in the set of monomials of any given degree n > L.

We define the inder of a monomial uir& uir$ . .' A uti^ 3s follows.

For i, k, i < ft, set

and define the index

( 8 ) i n d ( a 1 , & u i r @ " ' @ u t n ) :

f 0 i f j , 5 j *

t r i f j , > i *

Note that ind - 0 if and onlY if 3 jol. Monomials


having this property will be called standard. We now suppose

j* ) j*, and we wish to comPare

i n d ( w , E w r 8 " ' 8 ^ u n ) a n d

i n d ( u t r A " ' & u h n r & u t * A " ' 8 u ) ,

where the second monomial is obtained by interchanging ttrrc, tti*+r.

Let Vlrc denote the ?'s for the second monomial. Then we have

n' t t : r l i t i f i , , j + k, k * I ; Tl* - - ' tJ ; , rc+rt 4 ' r ' **r :4 i .* i f i < k; tL ' t -

I r a + t . i r q L + r , t : ' 4 r c . i i f i > k + L a n d T L , * * r : 0 , 4 n r c + r - 1 ' H e n c e

i n d ( z ; r 4 . . . 8 r y o ) : L- r i n d ( u n & . . ' A i l i n + r 8 w r A " ' 8 u ) .

We apply these remarks to the study of the algebra tl : t/S for

which we prove first the following

Lnurrr.o, 1. Eaery element of T i,s congruent mod I to a O'linear

combi,nation of t and standard monomials.Proof: It suffices to prove the statement for monomials. We

order these by degree and for a given degree by the index. To

prove the assertion for a monomial u1r&uhA "'$un it suffices

lo "rru*e

it for monomials of lower degree and for those of the

same degree z which are of lower index than the given monomial'

Assume the monomial is not standard and suppose ir) ir*t. We

have

u t r & " ' A u l n : u 1 & " ' A % i n + r & u t * @ " ' 8 U n

+ u4@ " ' I (a l * & u t *+r - r t i t t+ t8 w)A " ' & u t * '

Since uJkS uin+t- Itirc+t& ut* = lu1rul**rl (mod 9)'

u h } " ' A u r n T u 1 & " ' A i l i * + r 8 u t r A " ' & u o

* u l r8 " ' S lu t *u i * * , ) I ' " I uo (modS) '

The first term on the right'hand side is of lower index than the

given monomial while the second is a linear combination of monomi'

als of lower degree. The resutt fotlows from the induction hy-

pothesis.We wish to show that the cosets of 1 and the standard monomi'

als are linearly independent and so form a basis for ll. For this

purpose we introduce the vector space S" with the basis tt'trl,rtr ' ' ' tttnt

i rs i r= . . . 3 i1, , i ie / , and the vector space $ : @1 OS'OSrCI " ' .

The required independence will follow easily from the following

158 LIE ALGEBRAS

Lprurur 2. There exists a linear mappi.ng o of T. into * such that

( g ) l o : L , ( u n r & u r r A " ' @ m ) o : I t t ( r r z " ' u t n ,

i f i r 3 i r < . . . < i *

(r0) tu"* "8r"| &'r7,r',:,,)3u.:'*;:,3,

I ut^)o

Proof: Set 1o: 1 and let 8o,r be the subspace of 8, spanned bythe monomials of degree n and. index S i. Suppose a linear map-ping a has already been defined for OI e)8, O . . . O 8,_r satisfying(9) and (10) for the monomials in this space. We extend f linearly too I @ 8 ' @ . . ' . O 8 " - ' O 8 " , 0 b y r e q u i r i n g t h a t ( u 1 r 8 u n r A . . . @ u t n ) o :u.trut2 - .. Irrn for the standard monomials of degree n. Next assumea has already been defined f.or AI O 8, O . . . O,8o-, O 8,0,,-r, satisfy-ing (9) and (10) for the monomials belonging to this space and letu t r$ . . .Au1nbe- o f index i>1 . Suppose jx ) j * * , , . Then we se t

(11) (w '& " ' A u to )o : (u \A " ' S t t i *+ r& u t *A " ' @ u)o

* ( u h A . . . b l u * u t * * , 1 8 . . . 8 u o ) o .

This makes sense since the two terms on the ridht are inOI @ 8' O "' O 8,-' O &n,i-t. We show first that (11) is iindependentof the choice of the pair (j*j**r), j* ) jr*r. Let (jrj*,) be a secondpair with jr. ) j*r. There are essentially two cases: I; / > k + I,l l . l : k + 1 .

I. Set z1* : r4, ltib+r: a, ujt, - tD, tclr*r: fr. Then the inductiOnhypothesis permits us to write for the right hand side of (11)

( " ' a 8 2 8 " ' A x 8 w 8 " ' ) a

+ ( . . . 8o &u& . . . A Wx l& . . . ) o

+ ( . . . Luu lA . . . 8 r8 w & . . . ) o

+ ( . . .& l ua lA . . . & Iwx l I . . . ) o .

If we start with (jrjr*r) we obtain

( . . . A u& aA " . 8 r8bo I " ' ) o+ ( . . . 8 u8u8 . . . A Wx l& " ' ) o

: ( . . . 8a8a8 . . . A x@u8 . . . ) o

+ ( . . .& l uu )A . ' . I r 8 w & " ' ) o+ ( . . . &u828 . . . A Wr l@ ' . ' ) o+ ( . . '& Iuu )A " ' & [wx ]8 " ' ) o .


This is the same as the value obtained before.

II. Set z1r : l l , l , l in+r: u : uit, t l i t+t: w. If we use the induc-

tion hypothesis we can change the right hand side of (11) to

( . . . w8 u8 u . . . ) o + ( . . ' l uw lQc^ u " ' ) o+ ( . . . a & luw l " ' ) o * ( " ' t uu l& w " ' ) o

we can

(13)

(14)

(12)

(16)

Similarty, if we start with

( . . . u 8 a 8 u - . . ) o + ( . . . u @ l a w l " ' ) o ,

wind up with

( . - . w 8 u 8 u . . . ) d + ( . ' . w & l u a l " ' ) o

+ ( . - . l u w l S a . . . ) o * ( " ' u 8 [ a a r ] " ' ) q

Hence we have to show that a annihilates the following element

of . AL O I ' O " ' @,8"-r :

( . . . lawl& u . . . ) - ( ' " u I [utu] ' ' ' )

+ ( . . . a I l uw) . " ) - ( " ' l u ,w l8 u " ' )+ ( . . . t uu l@ w " ' ) - ( " ' w S luu l " ' )

Now, it follows easily from the properties of o in Al @' ' ' @ 8"-rthat i f ( . . . a 8D " ' ) e 8"- r , where a,b e 8 ' , then

( 1 5 ) ( . . ' a 8 0 " ' ) o - ( " ' b & a " ' ) o - ( " ' l a b l " ' ) o - 0 .

Hence a applied to (14) gives

Since llawlul + luluw]|1 + lluuTwl - lluwluJ + llwulul + llualwl: 0, (16)has the value 0. Hence in this case, too, the right hand side of

(11) is uniquely determined. We now apply (11) to define a for

the monomials of degree n and index e. The linear extension of this

mapping to the space 8o,i gives a mapping on tLL O ' '' (E 8"-' @,8",isatisfying our conditions. This completes the proof of the lemma.


THponnr,r 3 (Poincar6'Birhhof'Witt). The cosets of L and the

standard monomials form a basis for 17: g/S.

Proof: Lemma 1 shows that every coset is a linear combinationof 1 * S and the cosets of the standard monomials. Lemha 2gives a linear mapping o of. T. into !S satisfying (9) and (10). It iseasy to see that every element of the ideal S is a linear combi'

160 LIE ALGEBRAS

nation of elements of the form

u t r & " ' A i l r n - u t t D " ' A u i n + t @ u t * A " ' @ u r o- , thA ' " Slut*ui*+ ' l I ' ' ' & ut* .

Since o maps these elementg into 0, So :0 and sor o induces a

linear mapping of 11 - !/S into S. Since (9) holds,i the induced

mapping sends the cosets of 1 and the standarfl monomial

ua18.. . &u6 into 1 and 4cnrt t t r . . . ut* respect ively. Since these

images are linearly independent in S, we have the linear inde-pendence in U of the cosets of 1 and the standardl monomials.

This completes the proof.

Conolunv 1. TheProof: If (u) is

the cosets uii : ui *statements.

maf\ing i of 8, into 17 es 1: L and) AL n 8i - 0.

a basis for I over O, then 1=1*9 and

S are linearly independent. This limplies both

We shall now simplify our notations in the followiflg way: Wewrite the product in ll in the usual way for associative algebras:xy. We write 1 for the identity in tl and we identify I with its image8i in 11. This is a subalgebra of tl6 since the identity mapping isan isomorphism of I into tlr. Also I generates I and lhe Poincar6'Birkhoff-Witt theorem states that if {uilj e I}, / ofdered, is abasis for 8, then the elements

(17) L, ?t'rrct,t, ' ' ' cct, t

form a basis for ll. In particular,then the elements

if 8 has the finite

(18) u!,ut, ... ufo*

C t

ht

(ul - 1) form a basis for tt. The defining property Of U can be

re-stated in the following way: If d is a homomorphiSm of I into

llr, U an algebra, then 0 can be extended to a unique hofnomorphism

d (formerly O') of 1l into !I. In particular, a representhtion R of 8

can be extended to a unique representation R of tl. fhis implies

that any module trt for 8 can be considered in one {nd only one

way as a right ll'module in which xl, x €fi, /e8, is as defined

for I)t as !-module. Conversely, the restriction to I of a representa'

tion of ll is a representation of I and any right ll'mddule defines

a right 8-module on restricting the muttiplication to 8. In the

sequel we shall pass freely frorn 8-modules to ll'modgles and con'

V. UNIVERSAL ENVELOPING ALGEBRAS

versely, without comment.The Poincar6-Birkhoff-Witt theorem (hereafter referred to as the

P-B-W theorem) gives a characterization of the universal envelop-ing algebra in the following sense: Let I be a subalgebra of ?I1,?I an algebra having the property that if {utlj e I} is a certainordered basis for [J, then the elements 1 and the standard monomi-als uq24h... tttr, it S iz -< . . . S i,, form a basis for ?I, Then lt (andthe identity mapping) is a universal enveloping algebra for 8.Thus we have a homomorphism of the universal enveloping algebrall into ?I which is the identity on I!. The condition shows thatthis is 1: 1 and surjective. Hence ?I can be taken as a universalenveloping algebra. Now suppose that E is a subalgebra of 8 andlet 1l be the universal enveloping algebra of 8. We may choosean ordered basis {ui l j e I } for I so that I : Ku L, Kf iL: @,h < l i f . k e K , I e L a n d { u y l k e K } i s a n o r d e r e d b a s i s f o r E .Let E be the subalgebra of ll generated by E (or by the u*.) Thenit is clear that 1 and the standard monomials urcfl*2.. . uksth t 3 h zbe the universal enveloping algebra of E.

Next assume E is an ideal in I and let the notations be as be-fore. Let ,ft be the ideal in U generated by E. By Theorem l,par t 4 . , we know tha t E : l l /Sand the mapp ing a*E- -+a*Sdefine a universal enveloping algebra for 8/E. By Corollary 1,a*E--+a* S is 1:1so we may ident i fy 8/E with the subalgebra(8 + S)/S of Ez. This subalgebra is the set of cosets a * ft and ithas the basis {u * S l/ e Z}. Hence by the P-B-W theorem, thec o s e t s 1 * S a n d u , 1 r u l r . . . u 4 * f r , l r ( l r = . . . < l t , f o r m a b a s i sfor E. Hence if 0 is the subspace spanned by the elements 1, andthestandardmonomials u\utz- . . ut t , thenS f l S:0. We note nextthat any standard monomial of the form

(1e)

where s 2 1 and / ) 0, is in s and these monomials together with1 and u,rL . ilrt, l, S l, Sstandard basis for tl. It follows that the elements (lg) form a basisfor S.

The main results on subalgebras and ideals we have noted canbe stated as follows:

conoueny 2. Let be 8, a Lie algebra,Tr its uniuersal enueloping ar-

161

162 LIE ALGEBRAS

gebra. If E is a subalgebra of 8,, then the subalgebra oflll generated6y E can be taken to be the uniuersal enueloping algebral, of E. If Ais an ideal in 8,, then 8,lE can be identified with (8, + n)/n where fizs the ideal in 17 generated by E and E: ll/S is the uniuersal en-ueloping algebra of 8,18.. Moreouer, the set of standard monomials(L9) forms a basis for fi, if (ui) is an ordered basis for',8, such that(u*) is a basis for E.

If we take E: ,8 in this corollary we see that the ideal U0generated by I in U has the basis consisting of the standardmonomials u;(i2,.. r4ir, i, S i, S . . . ( d". Since the$e elementstogether with 1 form a basis for ll we have the following:

Conou,lnv 3. Let 17a be the ideal in 17 generated by g. Then 71:@1 €) u".

Since the imbedding of 11 in U is 1 : 1 and since any algebrahas a faithful representation, we have:

Conor,r,.rnv 4. Any Lie algebra has a faithful representation bylinear transformati,ons.

The main properties of rl given in Theorem 1 which remain tobe re-stated are:

2'. I generates 11. \3'. Any homomorphism of a

,82 can be extended to a uniqueenveloping algebra Ur of I' into1lz of 82.

Lie algebra 8, into a Lie algebrahomomorphism of the universal

the lniversal enveloping algebra

5'. There exists a unique anti-automorphism r of I satisfyinga 7 c : - a f o t a e 8 , .

6'. There exists a homomophism d' (the diagonal rhapping) of1 l i n t o U 8 U s u c h t h a t a d : c 8 1 + 1 @ a , a E 5 J . i

7'. Any derivation D of. t, has a unique extension to a deriva'tion D of tl.

We prove next:

Conorunv 5. The diagonal mabfi,ng d of 17into U8U rs 1:1.Proof: The decomposition 1J: Al @ tlo gives a de0omposition

U B U - o ( 1 8 1 ) @ ( 1 8 1 1 0 ) @ ( 1 1 0 8 1 ) @ ( U o B t t o ) w h e r e 1 8 1 1 0 i sthe subspace of elements I 8 r, 6 e llo, llo I 1 is the Nubspace ofelements b I 1, b e 1lo and 110 81lo is the subspace df elements

> r 8 b ' , b , b ' € t l o . W e h a v e l d : 1 8 1 , a d : a & 1 + 1 1 8 a , a Q 8 .

Since Uo 81lo is an ideal in ll @ U, it is easy to prove bF induction

V. UNIVERSAL ENVELOPING ALGEBRAS r63

on r that

(u4uir " ' uu)d - uliL "' xt4r@ 1 + l 8 u;r " ' ct ' ir (mod 110 8110)

where (zr) is a basis for 13. Is follows that the set of images under

d of the standard basis 1, rrir "' c4ir, ir 3 i, < " ' 3 i,, is linearly

independent. Hence d is 1: 1.Examples. (1) Let I have the basis u, a with lual -- 1t. Then o

has the basis uiuk and we have the commutation relation

(20)

Hence

(21)

U A - a u : u

o t u : ( u * L ) u

The elements of 1l are the polynomials

t)o * t)f l * azuz + "' * u^u* ,

where the ar e Olu). Multiplication for such polynomials is defined

in the usual way except that we have

(22) uof (a) - f (u * k)uk , h : 0 , 7 , 2 , " ' ,

which is a consequence of (21). This is a type of ring of difference

polynomials.(2) Let I be abelian with basis (u). Then ll is commutative

a n d h a s t h e b a s i s 1 a n d t h e m o n o m i a l s U i ( | i z . . . | , l i 1 , i , S i z < . . .This means that the tti 31te algebraically independent and tl is the

polynomial algebra in these 4.

3. Filtration and graded algebra

An algebra lI is said to be graded, if 2I - xLo @ lln where llt is

a subspace and ?Ii?Ir' tr ?I;+r. An example is the tensor algebra

Y : O l @ , 8 , @ 8 r @ " ' w h e r e w e t a k e t o - O l , t n : 8 t i f i > 0 '

Another example is the algebra of polynomials in algebraically

independent elements in which lI; is the space of homogeneous

elements of degree e. If ?I is any graded algebra, then the ele-

ments of lli are called homogeneous of degree i and every a € lI has

a unique representation a-Zfa't, aie \;, and a;:0 for all but a

finite number of l. The at are called the homogeneous parts of a'

A left (right) ideat E of ?I is called homogeneous if E : XLo(A n Ud)'

This is equivalent to saying that D e E if and only if its homo-

geneous Parts bi € E.

164 LIE ALGEBRAS

An algebra lI is said to be fi.ltered if. f.or each non-negative inte-ger i there is defined a subspace ll'd' such that (1) U(d) c ?I(') ifi< j ; (2) u l l (d) :U; (3) l l ( i ) l [ ( i ) c l [ ( i+,) . I f ! I isgradedl and we setrytil - Xrsr?li then this defines a filtration in lI and lX becomes afiltered algebra. Another standard way of obtaininf a filteredalgebra is the following. Let lI be any algebra and let fi be asubspace of lI which generates l[ as an algebra. This lmplies thatwe can wr i te \ l : r0 I+ tn+t l t '+ . . . where Dtd is the subspacespanned by all products of a elements taken out of [Jt. Set ll'i) :Ol + tn +...+ Dl'. Then it isclear that these ?I'd' de6ne a fi l tra-tion in li. This applies in particular to the universal envelopingalgebra 11 of a Lie algebra and the space Ut - 8 of gendrators of ll.

An important notion associated with a filtered algebra ?I is theassociated graded algebra [: G(2t). One obtains thisi algebra byforming the vector space

( 23 ) GoD- [ -

where we take ?I(-t) : 0.ponent-wise by

, [n : !1t i l /?I (d- t t ,

A multiplication in I is defined com-

(24) (at * l I(d-r)) @i * 11ri-t ' ; -- erai + l I( i+i-r) ,

i f . a;a 21rdr , a i e \ I6 ' \ . l f at : D,(md?[( i - r ) ) and, ai : &6(modl l ( j - r ) )then a;a1 = b;b1 (mod lI(i+'-r)). Hence (24) gives_a single'valuedproduct for an element of [i and an element of [i with result inEn*i. This is extended by addition to [. It is easy to see thatthis gives a graded algebra fi with [r as the space of homogeneouselements of degree f.

LetDe?Iand suppose DelI ' " ' , 6,21r"- ' t . Theelement b:b + ! I ( ' ' - t )is a homogeneous element of degree n inln and is called the leadingterm of b. If. b - 0 we take the leading term to be 0. lf. b and chave the same leading term of degree n, then b - c € U('c-r). Anyhomogeneous element of I is a leading term. If D is a leadingterm of D and c is a leading term of c, then either be-0 or Da isthe leading term of. bc. These remarks imply that if E is a leftideal in lI, then the set of sums of the leading term$ of the ele-ments of E is a homogeneous left ideal E in [. We carl now provethe following

Tnponpu 4. Let \I be a fittered algebra with I as the associated

@

I ' @U't : 0


graded algebra. If E has no zero diuisors + 0, then

diaisors + 0. If I is left (ri'ght) Noetherian, then 2I

Noetherian (defi.nition below).

165

no zero(ri,sht)

Proof: Let b,cell, b + 0,c + 0. Suppose b is the leading term of

6,c- that of c. Then, by def in i t ion, b+0, c*0 and so ue +0.

Then Da is the leading term of. bc and this is * 0. Hence bc + 0.

The statement that a ring is left Noetherian means that the as-

cending chain condition hotds for left ideals of the ring. As is

well known, this is equivalent to the property that every left ideal

is finitely generated. Suppose I has this property. Let E be a

left ideal in ?I and let E be the associated homogeneous left ideal

in I consisting of the sums of the leading terms of the elements

D e E. By hypothesis, 6 has generators (0,,5r, " ', D") and we may

assume that bi is the leading term of Dt € E. We assert that the

D; generate E, that is, every 6 e-E has the form Zc;bt ci € ?I. We

assume that the leading term 6 of b is of degree n and we may

suppose that the result holds for elements with leading terms of

a"iiee less than z. Now b: ) e;bi, E;e fr. By dropping some_o_f

the terms c-;b; (equating homogeneous parts) we obtain b : 2e ihtwhere E1 is homogeneous and etbi is homogeneous of degree n.

Then ci is the leading term of an element ci and the leading term

of ) c f t t i s b . Hence d :b-Zc tb i e l l ( '0 - r ) so tha td is in the le f t

ideal generated by the Di. Hence b is in the left ideal generated

by the Di.We prove next the following ring theoretic result which we shall

apply to the universal enveloping algebra of a finite-dimensional

Lie algebra.

Tnponsu 5 (Gotdie-Ore). Let ?I be a ring (associatiue with 1)

without zero-diuisor * 0 satisfying the ascending chain condition

for left id,eals. Then 2I has a left quotient diuision ring (defi.nition

below).

Proof: Goldie's part of this result is that any two non-zero ele-

ments a,b e lI have a non-zero common left multiple m: b'a -- a'b.

This is equivalent to saying that the intersection of the principal

left ideals lla n llb + 0. To prove this we consider (following Le-

sieur and Croisot) the sequence of ideals Lra, ?Ja * \rab, 2Ia * \rab *2fab2,. . . . By the ascending chain condition there exists a & such that

abk* te? Ia * 2 lab * . . .+ 2 Iabk . Then abk* ' : xo&* xpb + " ' t c *abk ,

tc; € II. Since ?I has no zero divisors o6*+r # 0 so not everY rr:0.

2I hasis left

LIE ALGEBRAS

rf. rn is the first one of these which is not zero we have abk*' :rnabh * xn*rabo*t +...* rrabr,x1,*0. Cancellation of DD, gives

0 * xna - o6h+t-h - $n+fib-. , .-rf ibk-h eilan UD .

Hence lI has the (left) common multiple property. Now ore hasshown that any ring without zero divisors + 0 has a left quotientdivision ring if and only if every pair of non-zero elemerlts o, b h^uea non-zero common left multiple m: a'b: b'a. we rdcall that Dis called a left quotient diuision ring for lI if: (l) O iS a divisionring; (2) w is a subring of E; (3) every element of D has the forma-'b, a, D e lI (cf. Jacobson [l], p. 11g.)

we now apply our results to the universal enveloping algebraof a Lie algebra 8. As before, we emproy the ntrition ofdefined by(2s) ll'o) : @1 , u ( i ) - aL+ I J+g , + . . . +g t , p > 1Let ii - G(ll) be the associated graded algebra. It is ieasy to seefrom the definition of G(u) that, since 8 generates ll, E = g'r4l,or -(oL + 8)lOt generates i[. It follows that if. {usli e 1y,l ordered,ls a basis for 8, then the cosets i l i :ui+ AL i; d-gun.r"t. [.We have ilpy: uiu* * ll") and upl -- urt4 * ll(t) and ,ttrr, - ,ptr1 :luiurc\ € U(r). Hencr: frrui- uiilr. Thus the generatofs commuteand consequently I is a commutative algebra. It follows thatevery element of I is a linear combination of the elements f(- 1;,ilrrilir "' il,r*, it 3 i,and the Poincar6-Birkhoff-wftt th.or.m that the different .,standard,,monomials indicated here form a basis for il. This means that theui a,re algebraically independent and I is the ordinary algebra ofpolynomials in these elements. The general results we have derivedand the properties of polynomial rings now give the following theo-rem on 11.

Tnponnu 61. The uniuersal enueloping algebrallof any Lie atgebra g has no

zero dioisors + 0.

uu

2. If 8, is finite-dimensional, then lI satisfi,es thecondition for left or right ideals and \ has a left

ascending chainor right quotient

diuision ring.Proof :1. Since I is a polynomial

elements over a field, I has noring in algebraically independentzero divisors + 0. Consequently tl


has no zero-divisors * 0, by Theorem 4.

2. If 8 has the finite basis ur, uz, . . . t unt then I is the poly'

nomial algebra in frt, iliz, . . . , ilr. This satisfies the ascending chain

condition on ideals by the Hilbert basis theorem. Hence ll is left

and right (Noetherian) by Theorem 4. Hence 11 has a left and right

quotient division ring by the Goldie-Ore theorem.

4, Free Lie algebras

The notion of a free algebra (free Lie algebra) genetated by a set

x: {nl j e l} can be formulated in a manner similar to that of

the definition of a universal enveloping algebra of a Lie algebra.

We define this to consist of a pair (8, D ((88, i)) consisting of an

algebra I (Lie algebra 88) and a mapping i. of. x into 8(88) such

that if d is any mapping of. X into an algebra ?I (Lie algebra E),

then there exists a unique homomorphism 0' of 8(&8) into lI(E)

such that 0 : i|t. It is easy to construct a free algebra generated by

any set X. For this purpose one forms a vector space Dt with basis

X and one forms the tensor algebra 8(: !) : OLO rn O (m 8 tX) @ " '

based on Dt. The mapping f is taken to be the injection of X into

8. Now let 0 be a mapping of X into an algebra lI. since x is

a basis for llt, 0 can be extended to a unique linear mapping of

tlt into ?I and this can be extended to a unique homomorphism d

of I into lI. Hence $ and the injection mapping of X into $ is a

free algebra generated bY X.It is somewhat awkward to give a direct construction for a free

Lie algebra generated by X. Instead, one obtains the desired Lie

algebra by using the free algebra $ generated by X. Let 88 denote

the subalgebra of the Lie algebra 8t, generated by the subset X.

Let 0 be a mapping of x into a Lie algebra E and let 1l be the

universal enveloping algebra of E, which (by the Poincar6'Birkhoff-

Witt theorem) we suppose contains E. Then 0 can be considered

as a mapping of X into O, so this can be extended to a homo'morphism d of S into 11. Moreover, d is a homomorphism of &zinto 1lz and since d maps Xinto a subset of E (g U), the restrictionof.0 to the subalgebra 88 of 8r Senerated by X is a homomorphismof 88 into E. We have therefore shown that 0 can be extendedto a homomorphism of 88 into E. Since X generates S8, d isunique. Hence &8 and the injection mapping of X into &8 is afree Lie algebra generated by X.

168 LIE ALGEBRAS

We note next that g (and the injection mapping) is the universalenveloping algebra of 88. Thus let 0 be a homomorphism of gginto a Lie algebra 216 2[ an algebra. Then there exi]sts a homo-morphism 0 of. 8 into ?I which coincides with the restriction of dto X. Then d is a homomorphism of B, into llz and so the restric-tion 0t of. 0 to 88 is a homomorphism of 88 into ?[2. iSince x0' :x0 f.or tc e x and X generates 88, it is clear that 0t cdincides withthe given homomorphism d of 88 into ?[2. Thus we have extended0 to a homomorphism of I into ?I. since 88 genefates ff it isclear that the extension is unique. Hence & is the rfniversal en-veloping algebra of 88. l

The two results which we have established can be $tated in thefollowing

Tnponpu 7 (witt). Let x be an aTbitrary set and let 8 denote thefree algebra (freely) generated by X. Let 88 denote tlw subalgebraof 8" generated by the elements of x. Then Ss rs a free Lie algebragenerated by X and 8 is the uniuersal enueloping algebra of tg.

For the sake of simplicity we shall now restrict our,attention tothe case of a finite set X : {*r, xzr . .., )c,}. Then

Dt - o r ,@ tL rz@ . . . @ ar ,

a n d t ! - O L e D t O ( t X 8 t X ) O . . . , a n d w e w r i t e 8 : O { x , , . . . , x , } .The algebra I is graded with Tn^- tn8m8.. .AI I (z- t imes)as the space of homogeneous elements of degree /n. A basis forthis space is the set of monomials of the form rc;lc,;.2. .. xt*, ii :L,2, . . . , ri hence dim 1)t- : ym .

An element a e S is called a Lie element if a e 98. iWe proceedto obtain two important criteria that an element of 18 be a Lieelement. we observe first that it is enough to treat the case inwhich the given element a is homogeneous. Thus 0onsider thecollection of linear combinations of the Lie elements of the form

(26) l. . . llxEr4)xn,) ... x.t^\ ,

i i : L , 2 , . . . , r , f f i : L , 2 , . . . . T h e J a c o b i i d e n t i t y s h o w s t h a t t h i ssubspace is a subalgebra of 8r. Since it contains the r it coincideswith $8. we therefore see that every element of []8: is a sum ofhomogeneous Lie elements. Hence an element is a Lie element ifand only if its homogeneous parts are Lie elements. we see alsothat if a is a Lie element which is homogeneous of degree m then


a is a linear combination of elements of the form (26).

Let 8/ denote the ideal !n e (gfn I tlt) e . . . in 8. An element

of I is in 8' if and only if it is a linear combination of monomials

rir . xi,n, m 2l. Since the different monomials of this type form

a basis ior s' we have a linear mapping o of tl' into'?i8 such that

( 2 7 ) x i d : x ; , ( x o r " ' x ; * \ o : [ " ' f x i r x ; r l " ' x ; ^ l , f f i > 1 '

We consider also the adjoint representation of 88. Since $ is the

universal enveloping algebra of t!8, this representation can be ex-

tended to a homomorphism 0 of fi into the algebra 0(&8) of linear

transformations in the space $8. We have

(x " :' ?!::,' ii,,i',,:,:!);i,;' ^o' ;,'r "r r t')

This implies that

(28) @u)o : (uo)(u9) ,

lf a, b € []8, then

u e 8 ' , 0 € t i

lablo - (ab)o - (ba)o : (ao)(b|) - (bo)(a|)

(9) : (ao) ad b - (bo) ad a- lao, bl + la, bo) .

Thus the restriction of 6 to 88 g 8' is a derivation in 88. We

can use this result to obtain the following criterion.

TsBonBu 8 (Dynki.n-Specht-Weuer). , If A is of characteristic 0, then

a homogeneous element a of d'egree rn ) o is a Lie element if and' onl'yif ao : vna where o is the linear ma?bi'ng of 8' into fi8, defined by(27).

Proof: The condition is evidently sufficient (even for characteris-tic p, provided f tr m). Now let a be a Lie element which is homo'geneous of. degree m. We use induction on rn. We have seen that

a is a linear combination of terms of the form (26). Hence it suffices

to prove that ao : ma f.or a as in (26). We have

f" ' fxqx. ; r ] " ' t r " int l l : [ [ " ' fx i rx;r ] " ' tc im-rfd, Nt^ l

- [ [ ' ' ' lnrxul " ' )cr^-r f , x6o\: ( m - l ) t . . . L x \ x h l . . . x t * \ + t . . . f r 4 r 4 l . . . x ; , * l- m l . . . l x4x ;2) . . . r i *7 ,

LTO LIE ALGEBRAS

which is the required result.Our next criterion for Lie elements does

tion to homogeneous elements. This isnot requite the reduc-

Tsuonou 9 (Friedrichs). Let fi : 0{rr, ..., x,} be thetfree algebragenerated by the rc; ou€r a fi.eld of characteristic 0. Let 6 be the dia-gonal mapping of 8, that is, the homomorphism of g into g ggsuch tha t r ;6 : r i81* 18r . Thenaeg is a L ie e lbment i f ando n l y i f a d : a 8 I * 1 8 a .

Proof: We have [a81+ 18 a, b8 t + lBD] : labl&l + l glablwhich implies that the set of elements a satisfying aB -cgl * lgais a subalgebra of 8r. This includes the x;; hence it contains gg.Let yv !2,' " be a basis for $8. Then since g is the universal envel-oqing algebra of 88 the elements yf 'ykz . . . y!#, m arbifirary , h; ]_ 0Oo, :1) form a basis for $. Hence the products

u!'yl' . .- r,:h & ul'yt,form a basis for I I8. We have

(30)

u! 'y! ' . . . y:f f id : ( !rg I + 1 gy,)o'(y, g 1 * 1 gyr)*,

" ' ( Y * 8 1 + L & t ) o ^- y! 'yf; ' . . - y:,rg 1 + k,y! '- 'y!, . . . y!# g y,

+ k r i ' t : z - ' . . . y : fgy , + . . . + k^y ! , . . . y l f * ' @!* + *where I lepresents a linear combination of base elergents of theform yrr 'y l ' . . . -y j '@ylry:r . . . i l , wi th X/, ; > 1. The se(ond throughthe (m * l)-st term do not occur in fh" expressions of this typefor any other base element yltyrr'...!t ' . It follows that in orderthat ad shall be a linear combination of the base elerhents of theform ! l ' . . . i , f e l and lgy i ' . . . . y j ' i t i s necessary tha t in theexpression for a in terms of the chosen basis only bdse elementsy: ' ' . .y f f wi th one &; - 1 and al l the other h:0 occur wi th non-zero coefficients. This means that a is a linear combination of they; ; hence a e 98. Hence a6: a&l + 1ga i f and onl ly i f a e gg.

5. The Campbell-Hausdorff formula

We shall now use our criteria for Lie elements to deritve a formu-la due to Campbell and to Hausdorff for the product of Exponentiaisin an algebra. For this purpose we need to extend the free alge-

V. UNIVERSAL ENVELOPING ALGEBRAS I7I

bra 8 : O{xt, ..., x,} to the algebra I of formal power series in

the vcr. More generally, let !I be any graded algebra with ?Ii 3s

the subspace of homogeneous elements of degree i, i :0,1,2, "'.

Let fl be the complete direct sum of the spaces ?Ii. Thus the ele-

ments of fi ate the expressions }f,a': ao * h I "' such that

l , a i : l ,b i i f and only i f 4t : bt , i : 0, ! ,2, ' ' ' ' Addi t ion and

scalar multiplication are defined component-wise. We introduce a

multiplication in I by ()frar) €i'ail : 23c* where

cn: dobr * arbxt + "' * anbo € lIt .

It is easy to check that I is an algebra. Moreover, the subset of

f l o f e lements Ear such tha t a ; :0 i f i ) m, m:a ,1 ,2" - . i s a

subalgebra which may be identified with lI. The subset E(i) of

elements of the form at * d;+r + "' is an ideal in fr and n [(i): 0'

w e d e f i n e a v a l u a t i o n i n I u y s e t t i n g l 0 l : 0 ' l a l : 2 - i i f ' a + 0 a n d

a eE"', #8"*t'. Then we have the following properties:

( i )

( i i )(iii)

l a l =0 , l a l - 0l ab l < l a l l b l .l a + b l S m a x ( l a l , l A t ) .

i f and onlY i f a:0 .

(31)

and

(32)

This valuation makes I a topological algebra. Convergence of

ser ies xr* xz+.. . , 1, i€[ , isdef ined in the usual way. The non'

archimedean property (iii) of the valuation implies the very simple

cr i ter ion that xr l rz* " 'converges i f and only i f ' l r ; l -0 ' I t is

clear that the subalgebra lI is dense in ?I.

If the characteristic of the base field is 0 and e € [(1), then

are well-defined elements of [ (that is, the series indicated con'

verge). A direct calculation shows that

( 3 3 ) e x p ( l o g ( 1 + z ) ) : L * 2 , l o g ( e x p z ) : 2 .

Moreover, i! 21, zz e 8\L'' and lztzz\: 0, then

(34) exp zr €xP ̂ z2 : €xP @r * z2)

and

172 LIE ALGEBRAS

(35) log (1 * z'Xl * zz) - log (I * z) * log (L + zr)

We consider all of this, in particular, for ?I : I : O{xt, " ', x,}.

The resulting algebra $ is called the algebra of. formal'power series

in the rh. We shall also apply the construction to the algebra

I A 8. Since I : XLo O 8r, where Sr is the subspace of homo-geneous e lements o f degree i ,848:E0(8r88) i We have(8n I 8l) (8n' I 8r') G $i+r' I 8r*.,'. Hence, if we set i (8 I 8)r :

8*880 * 8*- ,88 ' + " '+ 8088r , then (888)*(8 88) , ] -s (888) ' * ,and $88: >0(888)- . I t fo l lows that $88 is graded wi th(8I8)r as subspace of homogeneous elements of de$ree ft. Wecan therefore construct the algebra 8-6R-.

I f a : I . fan, a, ie 8 i , then X(4r81) and:(18dd) are e lementsof 8re=S which we denote as a I 1 and 1 I a, respectively. Themappings a -) a@ l, a-t I I a areisomorphisms and homeomorphismsof 8-- i"to ffi. We have [a 81, 1@ &] - 6 for afy a,b e$.Also, in the characteristic 0 case we have exp (a I 1) = exp a @ 1,exp (1. I a) - 1 I exp a and similar formulas for the log function.

Let S denote the subset of $ of elements of the form b'* b, + . . '

where Dr is a Lie element in Si. It is clear that F it a subalgebraof 8r. The diagonal isomorphism d of I into $ I I h4s an exten-sion to an isomorphism of s into ffi. we note first that if

ai e 8; then ai6 e (8 I 8)t. This is immediate by induction on f,

using the formulas $;: Ei=,4$_rt )cid : n$l + 1 & xt. Hsrce ifa :2f at, a; € $;, then X a;d is a well'defined element of I A 8'We denote this as ad and it is clear that d: a -, a6 is an ibomorphismand homeomorphism of $ into fi e 8. It is clear also from Fried-richs' theorem applied to the ai that in the charactedistic 0 case

a:}a i , a i€ 8; , is in f f i i t and only i f . ad : a& 1 + + 84.We assume now that r : 2 and we denote the generdtors aS f' y

and write 8: a{x,y}. Assume also that the characteristic is 0.

Consider the element exp t exp J of S. We can write this as | * z,

where z : zr * zr+ ., . , zi€ 8;, and we can write

(36) exprexpY : 1 * z: exPLU ,

where w : log (1 + e). We shall Prove

PnoroslrloN 1. The element w - log(exprexpy) is a Lie element,that i,s, ?rl e 8T,Proof: Consider (exprexpy)d. We have

V. UNIVERSAL ENVELOPING ALGEBRAS T73

(exp rexPY)d : (exP rdXexPYd)- exp( r8 1 + 1 8 r )exP(Y I 1 + 1 8Y)- exp (r I 1) exp (1 I r) exp (v 81) exp (1 8v)* (exp r I 1)(1 I exp r)(exp v I lxl I exp v): (exp r I 1)(expy I 1Xl I exp rXl I expv)- (exp .rexpY I 1)(1 I exP r exPY) .

Hence (1 + e)d : ((1 + e) 81)(1 I (1 + e)) and so

(log (1 + e))d - los(1 + e)D- tos (1 + e) I 1)(18 (1 + e))- l o s ( ( 1 + z ) 8 1 ) * l o g ( 1 8 ( 1 + z ) )- tos(1 + e) I 1 + 18 log(1 + e) .

This shows that log (L + z) satisfies Friedrichs' condition for a Lie

element. Hence log (1 + e) e 8E'.Now that we know that ar - log (1 + z) is a Lie element, we can

obtain an explicit formula for this element by using the Specht'

Wever theorem. We have

e : e x p f , e x p y - l : f * q>osr No tcn

i lp IT '

Hence

(38) rog(1 *z)-4A,+## ##,f u *q t>0 '

Since this is aLie element, if we apply the operator a to the terms

of degree k we obtain ft times the homogeneous part of degree k

in this element. It follows that we have the following expression

and

(37) '^:An#### ##' f ' l*q;>o'

of log (1 + z) as a Lie element.

(39) loe (1 * e)

:*,*,a#It is easy to calculate the first few terms of this series and obtain

174 LIE ALGEBRAS

(40) Iog(1 * z ) : r+ y + + I ry l+ $ .U* t l l -+ Ixy)x)+ . . . .

6. Cohomology of Lie algebras.The standard complex

In this section we shall give the cartan-Eilenberg definition ofthe cohomology groups of a Lie argebra and we shall show thatthis is equivalent to the definition given in g 3.10. Tb obtain thecartan-Eilenberg definition one begins with tie field o which oneregards as a trivial module for g, that is, one sets fx :0, E e o,/ e 8. This module and all g-modules can be considened as rightmodules for the universal enveloping algebra ll of g, since everyrepresentation of 8 has a unique extension to a reprdsentation ofu. The term ll'module will mean "right [-module,'ihrtoughout ourdiscussion. we recall that a modure is called free if. it is a directsum of submodules isomorphic to the module lr. one seeks a se-quence of free ll'modules xo, xr, xr,. . . with ll-homomorphism e ofXo into b and ll-homomorphisms di_, of. & into &_, such that thesequence( 4 1 ) Q + - o T X o T X , T X z + . . .

is exact, that is, e is surjective, the kernel, Ker e -- irhage , lm do,and for every i > l, Ker d;-r: Imdi. A sequence (41) i; called afree resolution of the module o. Now let llt be an irbitrary g-,hence 11', module. Let Hom (x;,Tft), i >- 0, denote the set of ll-homomorphisms of X; into llt. The usual definitions of additionand scalar multiplication can be used to make Hom (X;, Ut; a vectorspace over O. If n, e Hom (X;,Wt), then d,;ni e Hom (Xo*,, Dt). Wetherefore obtain a mapping d{ n;-+d;rJi of Hom(X,,UD intoHom (xr*,, tJt), i = 0. This mapping is linear. Moreover, the ex-actness of (41) implies that d*dd :0, '>

0, and this implies thatdrdi*t - 0. Hence Im dr s Ker di+,. One defines the i-th: cohomotogygroup of I relatiue to the module tn to be the fdctor spaceKer dr/Im di-r, i >- L, and Ker do for l' : 0. The Cartan-Eilenbergtheory shows that these groups are independent of the particularresolution (4D for O.

We shall not go into the details of this theory here but shall becontent to give the construction of what appears to be the mostuseful resolution (41). The space X = Xi"Xr we shalll obtain is

V. UNIVERSAL ENVELOPING ALGEBRAS T75

called the standard, compler for 8. It will turn out that the stand-

ard complex has an algebra structure as well as a vector space

structure. This has the consequence that in suitable circumstances

one can define a cohomology algebra in place of the cohomology

groups.We now suppose we have a representation R of a Lie algebra 8

by derivations in an algebra lI. Let tl be the universal enveloping

aigebra of B. Then we shall define a new algebra (!I, U, R), the

atgebra of diferenti,al operators of the representation R of 8. The

space of (lI, u, R) will be ?I8 u. since the mapping of elements

i e A into their left multiplications vr(u -+ au) is an anti'homo'

morphism, the mapping l'-+ - lt is a representation of 8 acting in

U. We now form the tensor product of this representation and the

given representation R in 2I. The resulting representatibn maps /

into the linear transformation sending a@z into al* 8u- aSlu'

We can extend this representation to a representation of U, which

we shall now write as ztL where zr is the anti-homomorphism of 1l

such that tn - - l, I e g, and L is an anti-homomorphism of 11 into

the algebra of linear transformations of lI8 U. We write the

image of a e 1l under Z as Lo. Then we have (a & u)(l(trL)) :

a I " @ u - & 8 l u ; h e n c e ,

(42) ( a @ u ) L r - - a & l u - a l o 8 u .

Next if DelI, we define Latobethe linear mapping in ll81l such

that

(43) ( a 8 u ) L a : b a & u

The mapping D -, Ln is an anti'homomorphism of ?I into the algebra

of linear transformations in ?I8 U. It follows from (42) and (a3)

that(a & u)LoLr -- ba & tu * (ba)t* & u

: ba I lu - blna & u - b(atB) & u

(a & u)LtLo : ba I lu - b(aln) & u

( a 8 u ) L o r a : ( b l a ) a 8 u .

Hence we have

(M) IL6LJ: - LotB ,

which shows that the set of linear transformations lu is an ideal

in the Lie algebra of linear transformations of the form L,, * Lt.

176 LIE ALGEBRAS

It follows that the enveloping algebra fi of this set of transfor-mations is the set of mappings of the form \L"tLor (cf. 92.2).

We have the canonical mapping of lI8 U into I sending 2arg4^ u,into \L*rLot. If I e 8, lln :0, since /8 is a derivdtion. Hence(l& u)Lt : L & lu and, consequently, (1 I o)L" : t&\ua, u,u € "0.

This implies that (1 &t)L"L": a8 z so that, if > Lur\on:0, then

E a;8 u;:0. Thus the mapping X ai& ui -+ ) L"rLdn is a vectorspace isomorphism. Since the set of mappings of the fdrm 2 LotLotis an algebra we can use this mapping to convert ?1811 into analgebra by specifying that our mapping is an a'lgebra anti-isomorphism.. The resulting algebra whose space is lI S ll is thealgebra (?I, U, R) which we wished to define.

It is immediate from our definitions that the subset of (lI, U, R)of elements of the form a & L is a subalgebra isombrphic to ?I.We identify this with ?I and write a for c I 1. Similhrly, the setof elements of the form L&u, u€,LI, is a subalgebrh isomorphicto O. We identify this with tl and write u for I & U. The map-p ing o f ( l I ,U ,R) :UBU in to I sends a :a8 l in to L 'L ' : L "andsends tr. : | & u into LuL, - L*. The formula (44) gives the follow-ing basic commutation formula in (lI,1l, R):

(45) l b l l : b l - l b : b l n , b € U , / e 8 .

Since every element of f has the form > LutLu,, every element of(lI, U, R) has the form l, aiui, a; e \, h Q'/.7. Also I aru, : g ifa n d o n l y i f > a e & u ; : 0 i n U 8 U .

We establish next a "universal" property of the algelbra (?I, U, R)in the following

Pnorosrttorq 2. Izt E be an algebra and let 0r be a hoVnomorbhismof tl into 9, 0z a homomorfhisrn of 11. into 9 such that

(46) [b0bl02] l=(b0)( t0r1 - ( t0,)(b0,)- (bf)0, , b e ?I , / e 8,

holds. Then there exists a unique hornomorphism 0 of (W,lJ, R) intoD such that b0 : b0u u0 : u9z.

Proof: We form (S, [, fi) e E which we consider a$ an algebraof pairs (x, d), .r e (?I, U, R), d e E, with component fddition andmultiplication. We have the homomorphisms zr, rz of this algebra

onto (?1, U, R) and 0 respectively defined by (x, d)rt : JC, (x, d)r, -

d. Let E be the subalgebra of the direct sum generated by the

elements (a, a0 r) and (2, u02), a € ?1, z c 11. The mapping zr1 induces

V. UNIVERSAL ENVELOPING ALGEBRAS L77

a homomorphism of u onto (?I, U, R) and the mapping zz induces

a homomorphism of D onto the subalgebra of D generated by the

elements alrand uTr, ae}I, uerl. By (a5) and (46), we have the

following relation in 9

(47) (a, a0r)(1,10) - Q,l0)(a, a0r) - (aln, (alB)lr)

This permits us to carry out a collecting process such as indicated

in the discussion of the construction of the algebra ({n [, R) to

write the elements of E in the form 2(a;,ai0r)(u;,ui0z), aie \ '

u;e,l l. The property of ?l8U (:(?l,U,R)) implies that we have

a vector space homomorphism of (?1, U, R) onto 9 sending |'aiui

into x (at,a;0)(ui,u;02). The existence of this mapping implies

that nr is an isomorptrism of U onto (U, U, R). Then 0 : rl'r, is

a homomorphism oi (?I, U, R) into O such that a0 : anl'rr:

(a, a\r)rz : a|t and u0 : trrclrftz : (u, u02)rc2 -* u'02' The uniqueness

of d fotlows from the fact that tr and ll generate (u, u, R).

Remark. If tn is a set of generators for tr then the conclusion

of Proposition 2 will hold provided that (46) holds for all b e Tft'

This follows from the fact that the set of D satisfying this for all

/ e B is a subalgebra of ?I-as can be verified directly.

We consider next an extension of the notion of a derivation:

Dpr.rNrttou 2. Let tr and E be algebr?s1 s1 and s, homomorphisms

of ?l into E. Then a linear mapping d of. s into E is called an

(sy s)-deriuation if

(48) (ab)d - (as,)(bd) + (ad)(bs,)

It is easy to check that the conditibns on d are just those which

insure that the maPPing

(4e) o -,(3" :!,)is a homomorphism of ?l into the two-rowed matrix algebra Ee'

(A special case of this remark was used in the proof of 7 of Th'

1.) it U i. a subalgebra of $ and sr is the inclusion mapping, then

we call d. an sz-d.eriuation and if sz is also the inclusion mapping,

then we have a d,eri.uation of 2I into E. One can verify directly,

or use (49) to see, that if d is an (s1, sr)'derivation, then the kernel

of d is a subalgebra. It fotlows that d :0 if. Tftd: 0 for a set of

generators TJt of ?I. Also if. d, and, dz are (sr, sz)'derivations then

L78 LIE ALGEBRAS

d, - d, is an (s,, ss)-derivation. Hence it is clear that dr : d, ifxd1: xdz for every x in a set of generators.

We consider again the algebra (?1, U, R) and we cah prove thefollowing result on derivations for this algebra. ,

PnopostrtoN 3. Let e be an algebra, sr, sz homomorphisms of(2[, U, R) into 9. Let d, be an (sr, s)-deriuation of 2I into D and dzan (st, s)-deriuation of U inta I (s,, s, are the restrictipns /o ?1, 1lresfuectiuely of Sr, sz on (?1, U, R).) Suppose that for D e E, / e 8:

(50) (bs,)(ld) + (bd,)(ls) - (/s,XDd,) - (td,)(bs,) - (bt*)d, .

Then there exists a unique (sr, s2)-deriaatian d of (21, U, R) into D suchthat bd: bdr, ud: udz.

Proof: Consider the mappingssuch that

Since dr is an (s,, sr)-derivation, d, and d, are homomorphisms intothe matrix algebra Qr. A direct calculation shows that the condi-tion (46) is a consequence of (50). Hence Proposition 2 itnplies thatthere exists a homomorphism d of (?I, U, R) into Sz such that b0 :b01, u0 : r,t?z, c € ?I, u € \. It is clear that d has the form

o: , -, (!t' J\\ O x s z / '

Set y - xd; then since d is a homomorphism, d is an (s,, sr)-deriva-tion of (?I, U, R) into D. We have bd,: bdr, ud: udz, aiq required.The uniqueness is clear.

The proof of Proposition 3 and the remark following Proposition2 show that it is enough to suppose that (50) holds for all D in aset of generators lJt of ?1. We shall need this sharper form ofProposition 3 later on.

Let lIt be a vector space over @ and let E(IJI) be the exterioralgebra (or Grassmann algebra) over fi. We recall thAt E is thedifference algebra of the tensor algebra T. : O 1@tlt O (9n I tX) O. . .with respect to the ideal generated by the the elementsi r @ r. Itfollows immediately that if s is ^ linear mapping of $lt into analgebra E and (rs)' : 6 for every t E 9Jt, then s can bb extendedto a unique homomorphism of E into E. The canonic4l mapping

"': (3'' uf:,)'dr and 0zof.2l and 11, respectively,

uor: (us' ud'\ .\ 0 usz / '


of g onto E is an isomorphism on @1 €) 'lt' One identifies aL + tIft

with its image. It is clear that fi generates ^E. We shall denote

the multiplication in E simply as ab (rather than the more compli-

cated e Ab which is customary in differential geometry.) lf x, y € tJt,

t h e n t 1 ' y + y x : ( x * y ) ' - t c ' - J t : 0 ' T h e a l g e b r a E i s g r a d e dwith \Jt- as the space E* of homogeneous elements of degree m.

If. {utlj e I} is an ordered basis for IJ}, then the set of monomials

of the form i l i { t iz ' . . i l i^ , i , < i r a " 'a i^ is a basis for E . In

part icular, i f d imm- i , th.n E*:0 i f . m> nand dimE--( f r ) i f

m S n. Hence dim E :2o. The exterior algebra has an automor'phism 7 such that xq: - N for all , e Uft. lf x^ e E^ then x*T :

(- l ) x*.We are interested in derivations and 7'derivations of E(9]t) into

algebras E containing ,E as a subalgebra. The Z-derivations will

be called anti-deriuations of E into E. We shall need the following

criterion.

PnorosrrroN 4. Let E be an algebra which contains the exterior al'

gebra E: E(Tft) as a subalgebra. Let d be a linear maffing of Ift

into E. Then d can be extended to a deriaation (anti-deriaation) of

E into A if and only if x(xd) * (xd)r : 0. (r(rd) - (rd)x - 0) forall x e ,lt.

Proof: The conditions are necessary since tr' :0 in -E implies

x(xd) + (xd)x:0 or x(rd)- (rd)r - 0 according asdisa derivation

or an anti-derivation. Conversely, suppose the conditions hold. In

the first case we consider the mapping

and in the second, the mapping

o: x-,(x 'l)

o" '- ' (; 4)of TJt into the matrix algebra Ez. In both cases one checks that(r0)z : g. Hence 0 can be extended to a homomorphism of E.The rest of the argument is like that of Proposition 3.

It is clear that when the condition is satisfied the extension isunique.

We apply this first to the following situation. Let 8 be a Lie

algebra and TJt a module for 8. Let E: E(IJ|) be the exterior

180 LIE ALGEBRAS

algebra defined by Dt. lf. r e lll, / e g, rl e llt so that x(xl) * (xt)x -0 in ^8. Hence there exists a unique derivation dr ini^E such thatrd1 : rl, r € TIt.

_We have rdtfttz - xdtl * xd4, rd,t - aldu rd7rrr27:x[d4d4l for x e Tn, a e O, l,lt, 12, e g, sincl tn is A" g_module.Since all the mappings considered here are derivations afid tjt gener-ates ^8, we have d-rr:rr- f,\* dh, d61 : M1, d.prt2J:fdtrdhf in E.Consequently, I - dt is- a represetttatiotr of 8 by derivatlo". i" EOn).

we consider the special case of this in which rlt = g and therepresentation is the adjoint representation. we form the algebra1: (E(8), u, A) where 4 denotes the extension of the adjoint r"pr"-sentation to a representation by derivations in E(g). I There i,

"slight difficulty here in our notations since we now havF two copiesof 8, one contained in ll, the other in ^8. we shall {herefore de-note the elements of the copy in u by l, I e g and we qfrite E: {D.we denote the corresponding elements in E by t anl we denotethis copy as 8. In order to avoid ambiguity we shail denote theLie composition in 8-which does not coincide with the Lie productin E"-by llr"lz7. Then we have the Lie isomorphisrn /--i of Bonto E; hence we have WI - [lrtr] : I,l, - irlr. Since g and ggenerate ll and E, respectively, 8 u E generates X The followingrelations connecting these generators are decisive:

(5t1 U" l'l : [7=o I'zl , 12 : 0 i

l lr, ir l : f l lo 12! , l/; € 8. The last of these is a consequence of (a5). ,

The last relation in (S1) implies that if rr e E;, ther[ [r;, I] e Er.It follows easily from this that f,g, : E[17'i for

-g-Tir -

O L + E ' + - . . + S 7 i n l l . H e n c e l J E r : B r g . I f w e s d t X ; : E ; I I ,then we have X;Xt : E;OEiIJ g E;*ilJ : X;+i. Also since X -E11, - E'8 u it follows from the properties of tensor pfoducts thatX:2 OX, : X O Etl. This shows that X is a grdded algebrawith x;: E;11 as the space of homogeneous elements df degree l.

Pnoposnror* 5. There erists a unique automorphism u of x -(E(8)' a, A) and a unique v-deriuation d of x into x such that

(52) l n : - l , i n : il d : i , i d : 0 , / e8 .

Moreoaer, Td + dT :0 and d' :0.


Proof: Evidently we have a homomorphism 4r of. .E into X such

that 14 - -1, / e B and a homomorphism \z of 11 into X such that

in - i. ' The condition (46) for 6 : Ir, I : ir, 0r-:4r, 0z:42 reads

l i rnr , l rnr l - f l ro lg lzr which is equivalent to [ - l r , i r ) : - l l ro l2) ' This

holds by (51). Hence Proposition 2 can be applied to prove the

existence and uniqueness of T. It is evident that the mapping

d z : 0 i s a n 7 ' d e r i v a t i o n o f O i n t o X . S i n c e t i - i t : U l ) : l l o l l : 0

the condition of Proposition 4 is satisfied and so there exists an

anti-derivation d., of. E(8) into X such that til: i, / e 8' We now

check (50) for b : l r , I : l r , sr : 1, sz: L dt : dr , dz: dz ' The

left-hand side is

/,0 + l,i, - iri, - o(-/,) - [i,irl .

The right-hand side is Ll,ir)d, - U, o lr7d., : WTI : lirir). Hence

(50) holds for these elements, so by the remark following Proposi'

tion 3, there exists a unique ?-derivation d in X satisfying the condi-

t ions (52). We have Xod:0 and Xd tr X;-r where Xt: E;LI , i > l '

This can be established by induction since X;: X;-rX'' Also

it is immediate that r;n: (-l) ixi if. )e e X;. Hence xfl\d: (-L)'xid

and x;dn : (-l) i-tx;d. Hence nd + dn:0. If x,y € X, lx-v)dz :

(x(yd) + (xd) (yil)d : x(td') + (xd) (ydd + @d)(vrtd) + (xd')(vu\ :

x(yd') + (xd'z)(yq'z\. Since T' : I this shows that d' is a derivation.

On the other hand, (52) implies that ld':A -id'- Hence d':0'

This concludes the proof.

Le t d . ; - r , i :1 ,2 , . . ' , denote the res t r i c t ion o f d to X ; . - Then

d;-r mdpS X; into Xr-r and, d,id*r : 0. Since M :0, d and the di

commute with the right multiplications by elements of U. Hence

if we consider X; in this way as a ll-module, theh di is a ll-homo'

morphism. We have X;- E;17-Er8i l and X: l@X;, by theproperties of tensor products. Hence any @-basis for E; is a set

of free generators for Xi relative to U. Thus every X; is a free

ll-module. We have Xo : U. If U' denotes the ideal in U generated

by the I , /e 8, then l l l lJ '=A. l f {u1l i e I } is an ordered basis

for 8, then we know that the elements 1, r7;rl l ;r ' ' ' i i i t ,, i , S ir 3' ' ' 3 i,,

form a basis for 11. lf. a : a) * | a;r...;rr1;rt1;r... . [t i ,, then we de'

fine a mapping e of U into 0 by ae: a. This mapping is a ll'

homomorphism of Xo : 1i into O considered as ll-module wherert17' :0, al-_ d, a e A. Evidently e is surjective.

Let b e Xr. Then D: X up1, {u} the basis for 8, Di € U. Thenb d o : b d : l i l p 1 e ) J ' a n d S f l o e : 0 . C o n v e r s e l y , l e t A e X o a n d

182 LIE ALGEBRAS

a s s u m e Z e : 0 . T h e n a : 2 a , ; r . . , ; r i i ; r . . . f i ; , € U ' a n d , d = b d o f o fD : X rr1,() a;, ir i l iz. .. u;r). We have therefore shown thatS +- O *; & *tu X, is exact. It remains to show that Ker d;-r :

Im di, i > 1. For ihis purpose we consider first the case in whichI is abelian; hence 1l is the polynomial'algebra in the 17; whichcommute and are algebraically independent. Also in ithe abeliancase [ ' ' l i ] --lmorl : 0, ttr,/ e 8. Hence the elements of '11 commute

with those of E(S). Consequently, X is the tensor pfoduct of. E

and tl in the sense of algebras.

Now let 8, - llt O Tl where 9ll, yl are subspaces, hence:subalgebras

of the abelian Lie algebra 8. Let F: E(9lt), S : tl(ili); Y : X(n)'

G : E(Tt), [8 : U(Yt), Z: X(\l). It is easy to see that F can be

identified with the subalgebra of E generated by Dl, E with the

subalgebra of u generated bV fft : {m e glt} and Y with the sub-

algebra of. X generated by llt + ft. Simitar statemen]ts hold for

G,L\,Z. These results follow easily by looking at bfses. Simi'

larly, we leave it to the reader to check that X : YZ where Y

and Z are the subalgebras we have indicated and that we have a

vector space isomorphism of Y&Z onto X sending y&z intoyz,

I e Y, z € Z. This result implies that if p is a linear mapping in

Y and y is a linear mapping in Z then we have a unique linear

mapping i, in X such that (yzil - Q1t)(zv)-We now regard the augmentation e of. Xo into O as, a mapping

of Xo into the subalgebra OL of. X and we extend this: to a l inear

mapping in x such that x;e:0, i> 1. This is an algebra homo'

morphism of. X onto the subalgebra @. We now provel

PnopostttoN 6. There exists a linear mapping D in X: X(8), 8

abelian, such that Dd + dD = 1 - e.Proof: Suppose first that dim 8 : 1 and (u) is a basis for 8.

Then X has the bas is l , i l , i i z , I t ' , " ' i t t ,L6 i , i l i l ' , " ' . We have

{ r n d : 0 , u i l i d : f i i ' | , i Z 0 ; l e : l , t l i ' t e : 0 , u u i e : 0 - I i l e n c e i f w e

def ine D to be the l inear mapping such that LD:0, i l i t 'D-- t t i l i ,

uuiD :0, i 2 0, then one checks that dD + Dd:1 - e as required'

We note also that TD + Dn -- 0 if A is the automorphisnri previously

defined in X (ilin : {ii , utliq : - uu'). Moreover, we have tfre relations

e d : 0 : d e a n d e T : e : n e . N o w s u p p o s e , 8 - l l l @ I t i w h e r e T t i s

one-dimensional. Then X : YZ, Y = X(9ll), Z : X(tt) wihere Y and

Z are che algebras defined before. It is clear that Y and Z are

invariant under d, e and T, and that their restrictiorls are just

V. UNIVERSAL ENVELOPING ALGEBRAS I8:}

the corresponding mappings in X(St) and X(gt). Let Dz be the

mapping just defined in Z and let Dr be any mapping in Y suchthat dD, + DLd: 1 - e in Y. There exists a unique linear map-ping D in X such that

( 5 3 ) ( y z ) D : y ( z D ) * ( t D ' ) ( z e ) , v e Y , z e Z .

Then we have

(yz)Dd, - y(zD2d) + (vd)(zD*i * uD)(zed) + (tDd)@e7t)

- y(zDzd) + (td)QDni + UD'd)(ze)

(yz)dD-(y(z i l+Ud)kd)D: y(zd'D,) * (vD)(zde) + (td)(zvD) * (vdD')(zr1e)

: y(zdD) + UA@rtD) * (ydD,)(ze) .

(yz)(Dd + dD): y(z(\- e)) + (y(1 - e)Xze)

- yz - y(ee) + y(ee) - (ye)(ee)' - yr( ! - e) .

The inductive step we have just established implies the result in

the finite-dimensional case by ordinary induction and in the infinite'

dimensional case by transfinite induction or by Zorn's lemma.

This proposition implies the exactness of Xo T Xr

7 X,

( - . . . . Thus le t r € X ;=rX isa t is fy xd :0 . Then r - - r ( l -e ) :

r(d.D + DA - @D)d e lm d. This implies that Ker d;-, -- Im dr i > L.

We consider now the general case of an arbitrary Lie algebra 8

and we introduce a filtration in X using the space of generators

I t - 8 + E . T h u s w e d e f i n e y r i t - o L + ! t + s t s + " ' * T t j :

1o*tstEngl. Since (Ebg\ds Ei-,3&+r, ytit is a subcomplex. Also

it is clear that X(i) is a sum of homogeneous subspaces relative to

the grading in X. We can form the difference complex Xtit lyti-rtwhich has a grading induced by that of X. lf. {uili e I} is an

ordered basis for 8 then the cosets relative to Xtr-t) of the base

elements u\ . . . u io i l t1. . . i l ; * , i , < i r .h + k: j form a'basis for Xrit lyti-tt. We identify these cosets

with the base elements. Then we can say that d in X titl;gtJ-r) is

determined by the rule

(u t r ' ' ' t t '5o i l t , ' ' ' i l t * )d(54)

184 LIE ALGEBRAS

where the ^ indicates that u,o is omitted, and the ordbr of the sub-

scripts in the product of the z's is non-decreasing. This differenti'

ation does not make use of the structure of 8. Thus it is the same

as that which one obtains in the abelian Lie algebra. In the latter

case the space spanned by the monomials on the left.hand side of(54) sat isfy ing h+k- j f ixed form a subcomplexof Xand Xis

a direct sum of these subcomplexes for j :0,1, ' ' ' . It follows

that if r is in one of these subcomplexes and is in Xr+t , i > 1, and

tcd:0, then x:!d, ! e Xi and where y is the given 0omplex con'

taining r. The result we wished to prove on the exadtness of (a1)

will now follow by induction on the index j in x (i)

by dhe following

Lpuu.r, 3. Let Y be a subcomplex of a graded com\lex X :27ae Xd

such that ts: xlo( y n xr). set z - xlY - l.::o'.s^ Zo, Zt:Xil(Y n X,). Suppose Ker dn-r: Im dr i2l, holds in"Y and i,n Z.Then this holds also in X.

Proof: Letx € Et>'Xisat isfy rd:0. Then (x* Y)d:0. Hencet h e r e e x i s t s x ' + Y s u c h t h a t ( r t + Y ) d : t * Y . T h u S N ' d : x * ! ,y e Y. Then yd: x'd'- rd:0 and so there exists a, ! ' e ts suchthat y : !'d, Hence r : r'd - y - r'd - !'d : (r' * y'rd.

We have now completed the proof of the following

THponpru 10. Let X: (E(8),11, A) be the algebra of diferentialoperators determined by the exterior algebra of E(8) Iand the er'tension of the adjoint re|resentation of 8 to E. Iat d be the anti.'deriuation in X defined in Proposition 5 and let X; j E;17. ThenA* , Xo* ^ Xr+ "' l's a free resolution of the nlodule tD-

It remaini"to show that the cohomology groups dedned by this

resolution coincide with those of S 3.10. Let there he a ll-homo-

morphism ep of Xt into a (right) ll-module llt. Let (lr,,lr, ' ' 'r /r) h

an ordered set of i elements of 8 and define a nlapping f of

8 x . . . x 8 ( i t i m e s ) i n t o l l t b y f ( | r , " ' , l ) : ( l J r " ' 1 , ) p w h e r e

lrl, . . . l; € Ei tr X;. It is clear that this / is multilinef,r and alter'

nate and it is easy to see that if / is any multilindar alternate

mapping of 8 x ..' x 8 into fi then there exists a linear mappingcp of E;(8) into l}t such that (lJz "' l;)tp : f (l', '",lr)- t This lp has

a unique extension to a ll-homomorphism of Xi into IIt. We there-

fore ti"ve :l 1 : 1 linear isomorphism of the space of ill'homomor-

phisms of. Xi into !]t and the space of multilinear alternate map-pings of the e-fold product set I x ... x 8 into 1Il. Ifr/ is a map-

(55)

We have


ping of the latter type we define fB by

( f l ) ( l r , " ' , l ; * r ) : ( ( / ' " ' l * r )d)g .

6+r( l r . - - l r * r ) d : t ( - l ; t * ' - o r , . - . i o . . ' l ; * r ,

and, by (45),

iolo*r " ' l r*r : lq+r " ' l r*r lo

: lq+t . . . l r * r ln+ X 1" . r .1)p; ,1 . . . l ; * ,_

r>.t

: lq+r. . . l r*r ln+ > (-1)t* t* ' ro+r . . . t r . - . l i+r f lq l r i .

Hence

( l r - - . l * r )d : H( - l ; i * ' -o r r . - . io . . . l r * r lo

, + 1

q = l

* "F=rt-1)T*n/,

" ' in ' ' ' t , ' ' ' l ;+rt lql , l .

It follows that

( f | ) ( t t , . . . l ; * , ) : t i

( - ! ) i * ' -o f ( l r ,

which are the same definitions as given in $ 3.10.

7. Restricted Lie algebras of eharoateristie p

In many connections in which Lie algebras arise naturally one

encounters in the characteristic 2 * 0 case structures which are

somewhat richer than ordinary Lie algebras. For example, let ?l

be an arbitrary non-associative algebra and let O(?l) be the set of

derivations of ?I. Then we know that E(U) is a dubalgebra of theLie algebra of linear transformations in ?I. We note also that onehas the Leibniz formula

(56)

, tr,

(57) (ab)Dk: * Oao'>(bDo-)

186 LIE ALGEBRAS

for any derivation D. This can be established by induction on fr.Now assume the base field of [, hence of S(?l) is of characteristicp and take k - p in (57). Then the binomial coefficients (f) : 0if 1 < i = p - l. Hence (57) reduces to

(58) (ab)DP : (aDe)b * a(bDP) ,

which implies that De is a derivation. Thus S(U) is closed underthe mapping D -, De as well as the Lie algebra compositions.Similarly, let ?I be an associative algebra and a + a an anti-automorphism in ?1. Let 8 be the subset of skew elemOnts relativeto a -+ a. Then we know that 8 is a subalgebra €Er,r Moreover,i f the character ist ic isPthen a - - a impl ies that ae:d.P:(-a)n:-ao. Hence ao e 8, so again we have a Lie algebra closed underthe p-mapping. The systems S(?l) and 8, just considpred are ex-amples of restrlcted Lie algebras which we shall defind abstractly.

For this purpose we need some relations connecting ap with thecompositions in a Lie algebra ?[2, U associative of characteristic p.We recall first the following two identities in 0U, pl, f,, tt alge-braically independent indeterminates, @ of characteristic p:

(5e)( l - p ) ' : l P - t t e

. xt-l(t - p)o-' -f I oo-r-i

t '=O

The first of these is well-known and the second is a fonsequenceof the first and the identity ),e - pe:(i - p)(>tr:t^tpot'-'). Theserelations imply corresponding relations for commutirtg elementsa, b in any associative algebra. In particular we may take a : bn,b : bz the right and left multiplications determined by an elementD e ?I. These give

(b" - br)o : f" - b!,: (bn)" - (bn)"p - l p - l

(b* - br)o-' :'>_- b'*b?-'-t : t (bn)*(bo-'-')" ,C = u t , = 0

p

L' ' ' lab lb l ' ' ' b l : fabol ,

:5 be-L-iabi

O f t

(60)

(6t1 t . . . t a b l b l . . . b l


Also it is clear that

(62) (aa)o : aeaP ,

and we shall use (61) to Prove that

(63) @ * b)e : aP * b ' +Tsi(a,b) ,

where is;(a,o) is the coefficient of ,'-: ,"

(64) a(ad,Q,a + r))n-' ,

I an indeterminate. To prove this we introduce the polynomial

ring ll[.i] and we write

(65) Q,a -l b)' : ),PaP * D" +5 s;(a,b)]i ,

where s;(a,b) is a polynomial in o,i='* total degree 2' If we

differentiate (65) with respect to i we obtain

p - l . o - 1

Tflo * b)ta(la * b1o-i-r - ! is;(a, b)^'-''

By (61), tfri, girr..

(66) a(adQ,a * b17n'r: 5 is;(a, b)^'-' .t.:r

Thus we see that is{a,D) is the coefficient of ii-r in a(ad (}.a * D))o-'.

on the other hand, substitution of I : 1 in (65) gives the relation

(63). It is clear that s,i(a,b) is obtained by applying addition and

commutation to a,b and so is in the Lie subalgebraof.\Ir. generated

by a, b. For examPle,

s'(a, D) - labi if f :2 ;

s,(a, D) - llablbl , Zsr(a, b) : llablal if f : 3 ,

s,(4, b) : fillablblDlrl ,Zsz(a,b) : llllablalblbl + tlltablblalbl + ll[lablb]blal ,

3ss(a, D) : [t[tcb]alalbl + Llllablalblal + llllablblalal,4s,(a, b) : tt[tab]alalal if 0 :5 .

These considerations lead to the following

Dsr,ll,rtrtoN 4. A restricted Lie algebra f] of characteristic p + 0

is a Lie algebra of characteristic p in which there is defined a

188 LIE ALGEBRAS

mapping a -t srnr such that

(oa)l'l - ooolnl

(a + b)rot : 6rn! a 6tr.' + 5 s;(a, b),i : r

where is;(a,D) is the coefficient of ,ld-' in a(ad(ta * 6))1i-' and

(ii i) [obc'11: a(adb]' .

If ?I is an assrciative algebra of characteristic p, then theforegoing discussion shows that ?I defines a restricted Lie algebrain which the vector space compositions are as in A, Wbl: ab - ba

and at't : a?. We use the notation ?Iz for this restricted Lie alge-

bra (as well as for the ordinary Lie algebra). A homomorphism S

of a restricted Lie algebra into a second restricted Lie algebra is,

by definition, a mapping satisfying (a * b)s : 4s * bt, \(aa)t : aas,

fab]s - laubul, (arn:ru - (as1tol. Ideals and subalgebras lare definedin the obvious way. A representation of 8 is a homofnorphism of8 into the restricted Lie algebraEz, @, the algebra of linear trans'formations of a vector space llt over O. If R is a representationacting in the space tjt, thln !ft is an 8-module relative to xa = x&n,

reTlt, a€,8,. The module product ra satisfies the usual Conditions ina Lie module and the additional condition that ,otil - ( " (xa)a) "'a,

f a 's.We consider now the following two basic questionsf (1) Does

every restricted Lie algebra have a 1 : 1 representationl (2) What

are the conditions that an ordinary Lie algebra be redtricted rela-

tive to a suitable definition o1 oIorT In connection'\,tiith (2) it is

clear that a necessary condition is that for every a e 8, the deriva-

tion (ada)P is inner; for, in a restricted Lie algebra (ada)p : ad'ar'l.

We shall show that this condition is sufficient. In fact, we shall

see that it will be enough to have (ad u;)e inner for every u; in abasis {u;} of. 8,. We note also that if I is restrictedl relative totwo p-mappings a-+4rn\ and a-'+sferz, thenf(a) - rtn\l- otnJz is inthe center of 8 since lb,-f (a)l : lbafttrl - lbat'tz1- 6(ad a)'i- b(ad a)p :

0. It is clear from (ii) and (i) that r

(67) f (a+b) - f (a )+ f (b ) , . f (aa) : ao f (a ) .

A mapping of one vector space of characteristic P + 0 into a secondone having these properties is called a p-semi-lincw mapping-Conversely, if 8 is restricted relative to a-trtrh apfl 4-+f(a) is a

( i )

( i i )


2-semi-linear mapping of 8 into the center G of 8, then 8 is alsorestricted relative to Q -+ qtnJ, - otrh + f @). The kernel of. a p-semi-linear mapping is a subspace. Hence if f(u;) -0 for u; in abasis, then / : 6. It follows that if two p-mappin gs a -, arr}, anda -+ srn72 making 8 restricted coincide on a basis, then they areidentical.

Suppose now that 8 is a Lie algebra with the basis {uli e I}where ^I is an ordered set and let ll be the universal envelopingalgebra of 8. The Poincar6-Birkhoff-witt theorem states that the"standard monomiats" u!]ul] -.. u::, i, I i,basis for U. We have the filtration of U defined by 11tlcr -O L + 8 + 8 ' + . . . + 8 & . I t i s e a s y t o s e e ( i n d u c t i o n o n f r a n d t h eusual "straightening" argument) that the monomia.ls

"fi"i; ...

"f:such that h * hz *... + h, < k form a basis for l l 'e). We assumenow that for each base element ui there exists a positive integerfli ?\d an element ze in the center 0 of tl such that

(68) at : u\ t - z;

is in U(ni-r). Then we have the following

Lpuul 4. The elements of the form

(6e) ,i;"ii... ziiui:ui:... "i:

such that L 1 iz la basis for 1J.

Proof: We show first that every element "f:...

,ry, fu > 0 is alinear combination of the elements (6g). rf. k - k, * kz + ... a k,,then we employ an induction on h. If every li 1. /tt1, then theresult is clear- Hence we may assume ki 2 nt, for some /. Thenwe may replace uTlt 6t h1 * zt, ?Dd obtain

"f : . . . u!; -

", ,uf l- .- u!1-"" . .- "f : + ui i . . . u!1-"o,rn, .

"f : .The elements

"f: . .. u?,1-"" . . .

"f: and the second term on the right

are in qt&-t). Hence these elements are linear combinations of theelements (69). Since the set of elements in (69) is closed undermultiplication by any zi, the result asserted is clear. We show nextthat the elements (69) are linearly.independent. we replace zr, b!Ztr: uTlt - ut, in ,! i ... , i i , .t i . ..ui:. This gives

189

190 LIE ALGEBRAS

,ii--. ,i;"i;.-- "i: : ,!;"i;"!;"i: -.- ,?:,i:

(70) - 1uT!, - u,,)o'"i: ... fuT), - a,|n'uli= u!]"tr*\ulzntz+\z . . . y!,n!t) ' , (modtU,ft

-r, ;

i f h :2 i - , (h i t4t1* t i ) . Hence the element t?: . . . t i ! " i i - . . u l ; -

u!fltf xt .. u!;"',*^" + * , where * is a linear combination of stand-

ard monomials belonging to U('c-r). It follows that if we have a

non-trivial linear relation connecting the elements in (69), then wehave such a relation for terms for which the "degree" fu -

Iii(h;tu1* ii) is fixed. This gives a relation with the same coef-

ficients in tlie corresponding standard monomials el|'"it tr' ' ' ' u!:"t'*^'.

Since the standard monomials are linearly independent we musthave hyhl * li - ht, i - 1, . . ., rt for the elements of (69) whichappear in the relation with non-zero coefficients. Sitrrce ),i 1 rh1this implies that hi, f,i are determined by the equation hifttl * i1 :

h1. Hence there is just one term in the relation. This isimpossible in view of (70). Hence the elements of (69) are linearlyindependent and so form a basis for 11.


Tnponpu 11. Iat 8, be a Lie algebra of characteristic f + 0 withordered basis {u;} such that for euer! u;, (ad u;)p is an inner deriaa'tion. For each u; let utot bn an element of 8, such that (adu)p - adu?r.Then there erists a unique mapping a --> rrnt of g into'g such thatulo' it as giuen and I is a restricted Lie algebra relatiue to the ma\'

bing a -+ sttr.Proof: Let 11 be the universal enveloping algebra ff 8. Since

(adu)e: ad ulor, zr: u! - rrln] com*utes with every t e g. Hencezi is in the center of tl and u'{ : z; * ut't where ul't e g. We cantherefore conclude from the lemma that the elements

(7r) ,i;ri;... ,i;"ii .. . "::such that ir < iz <U. Let E denote the ideal in tl generated by the z;. Then it isclear that the subset of our basis consisting of the elements (71)with some hi > 0 is a basis for A. Hence the cosetS of the ele-ments ul i . . . u i l ,0 ( i i < p - 1, form a basis for the plgebra [ , :U/4. Since thecanonicalhomomorphism x-+E:x* $ is a homo-morphism of 11, into [r, the restriction to 8 is a homo{norphism of


8 onto E : (8 + 8)lg. Since the ili:u; * E are linearly inde'

pendent, t-rl:/+E is an isomorphism of 8 onto E. We note

next that E is a subalgebra of [z considered as a restricted Lie

algebra. Thus we have alt : (u;+ bf : al * E: ul'r + I e E' It

follows from (63) and (62) that (f,aiili)e e E which proves the asser'

tion. The isomorphism of 8 and E permits us to consider I as a

restricted Lie algebra by defining |tt\ fiy 1tt\ - 1c. Then we have

ulotr - ul +E:uf,or *E so that ulpt'-uI't as required. We now

write l/tnr -;trlr 41d the result is proved.

We recall the result of $ 3.6 that if 8 is a finite'dimensional Lie

algebra with a non-degenerate Killing form, then every derivation

of 8 is inner. It is clear also that the center of I is 0. Hence if

the characteristic is 2, then for every @ e I there exists a unique

element atol such that the derivation (ada)p --ad.at't. It foltows

that we can introduce a p-operator in 8 in one and only one way

so that B is a restricted Lie algebra. We therefore have the

following

Conor,r,.lny. If I is a finite'dimensional Lie algebra of characteris'

tic p + 0 with nan-degenerate Killing form, then tlwre is a uniquc

p-mapping in 8, which makes I restricted.We suppose next that 8 is an arbitrary restricted Lie algebra

and we prove the following

Tnponpu 12. Let 8 be a restricted Lie algebra of characteristic

P + O, 17 the uniaersal enueloping algebra, E the ideal in 1I gener-

ated by the elements ap - ar'l, a € 8, [ : 1l/l8. Then the mapping

a --+ a : a * E fs an isomorphism of 8, into the restricted Lie alge'

bra fr.". If S is a hornomorphism of 8, into the restricted Lie alge'

bra \1, W an atgebrg (associati,ue with L), then there exists a uniqrc

homomorPhism of tr into ?I such that a -- at. It 8, is of finitedimensionalitY n, then dim-Ll. : f".

Proof: Since otnt - ap : ap * E, a --+ a is a homomorptrism of the

restricted Lie algebra I into the restricted Lie algebra [r. lf. {u;}is a basis for 8, over O, then it is clear from the rules for p-powers

and the operation a --+ nEor that ap - ottt is a linear combination of

the elements af - ulot. Hence E is also the ideal generated by the

elements ut - uE'\. The proof of the preceding theorem shows that

the cosets il;: ui * E are linearly independent; hence a -+ d is an

isomorphism of I into [2. Now let S be a homomorphism of I

192 LIE ALGEBRAS

into a restricted Lie algebra ?Ir, U an algebra. Thqn we have ahomomorphism of U into ?I sending a e 8 into a8. r Under thismapping a.p -- (au)o : (ot't)t. Thus oo - orot is in the liernel and soA is in the kernel. Consequently, we have an inducbd homomor-phism of [ : 1l/E into ?I such that a -- a,u. If 8 has the finitebasis ut, xtz, . . . , u* then we have seen that the cosets of the ele-ments ul'ul ' -.. uh", 0 3,ir < p -1 form a basis for [. ] This provesthe last statement of the theorem.

The algebra [: WE of the theorem will be called the u-algebraof the restricted Lie algebra 8. It plays the same fole for 8 asrestricted Lie algebra as is played by 11 for 8 congidered as anordinary Lie algebra. In particular, a representation of 8 asrestricted Lie algebra defines a representation of tl and conversely.Since I has a faithful representation it follows that evdry restrictedLie algebra has a faithful representation. Moreover, iif 8 is finite-dimensional, then I is finite-dimensional and so has a faithful re-presentation acting in a finite-dimensional space. Consequentlythis holds also for 8.

8. Abelian restricted Lie algebraa ]

An abelian restricted Lie algebra is a vector spabe 8 with amapping a -+ ap (we use this notation for the p-operatpr from nowon) in 8 such that

(72) (a * b)e -- ao + b' , (aa)o : dPaP .

The theory of these algebras is a special case of the theory ofsemi-linear transformations. This is equivalent to ,a theory ofmodules over certain types of non-commutative polynorhial domains(cf. Jacobson, Theory of Rings, Chapter 3). In the prepent instancethe polynomial ring is the set of polynomials do * tar + . -. * t*a^,di€O, / an indeterminate such that at:tap. If @lis perfect itcan be shown that the ring has no zero divisors an[ every leftideal and right ideal in the ring is a principal ideal, The studyof this ring and its modules is a natural tool for studying abelianrestricted Lie algebras. However, we shall not undertdke this here.Instead we shall derive one or two basic results on lthe algebraswithout using the polynomial rings.

TnsonpM 13. Let 8, be a finite-dirnensional abelian testricted Liealgebra oaer an algebraically closed field of characteristif F. Assume


that the p'mapping in g i 's L:1. Then t has a basi's (ar,ar, " ',a*)

such that alt : a;.Proof: Let a + 0 be in 8 and let m be the smallest positive inte-

g";'J; $;; d* : &ta * azap + '" * d^so^-'. Then a,an, "',ao^-'

are linearly independent and every aek is a linear combination of

these elements. If dr -- 0, set Ft : ailp, i : 2, " ' , ffi' Tfen

(oo*-' - Fza + - - - - pn-rap*-t)' : 0 which implies that aF*-' :

Bza *. . . + B^-ran*-' contrary to the choice of. m. Hence ar * 0.

This implies that a is a linear combination of ao,ao', "'. .Weshalt now show that there exists a b : 9fi * Bzae *... + g*6e*-r +0

such that bP : b. This will be the case if the Fr satisfy the follow-

ing system of equations:

9r: Be*a,

B;: Bo*a; * Bl-t ,(73)

(74)

i : 2 , , . . t f f i ,

and not all the B; are 0. Successive substitution gives

g^: go#ol^-' + go#-'al^-' + ... + \o*a^

Since dr * 0 and O is algebraically closed this has a non'zero solu-

tion for F-. Then the remaining Fi can be determined from (73)

f.or iSm- 1 and the equation for i:rn wil l hold by Q$. Now

suppose we have already determined au az, . ", a' which are linearly

independent and satisfy alt : ai. Then 8, : E Oai is a subalgebra

of 8. Suppose a is an element of 8 such that ae € 8r. Since a is

a l inear.combinat ion of ao,aot, " ' i t fo l lowsthat a€8,. Thus we

have shown that ,8/8r is a restricted Lie algebra satisfying the

hypotheses of the theorem. Hence if 8 + 8,, then we can find a

b 4 g, such that be = b (mod 8'). Thus b' - b - Z\T;a;. We can

determine di so that df - di * T;: 0, i -- L,2, " ', r. Then ar*r :

b + > dia; satisfies 4"+r Q 8r, a?+t: ar+t. Hence the result follows

by induction.

Tnnonpu 14. Izt 8 be a commutatiae restricted Lie algehra of

f.nite-dimensionali,ty oaer Gn algebraically closed rt'eld. Suppose thep-mapping a *+ ap is 1 : L and tet TIt be a fi,nite-dimensional module

for B. Then !)t rc completely reducible into one'dimensional sub'modules. If (ar, ctzt .. -, ao) is a basis such that al : a; and a -

2 f,;a;, then euery weight in Wt has the form A(a) : \ m;tr;, m; inthe prime fi.eld.

Proof: Let the basis (ar, ar, . . -, ao) be as indicated and let

194 LIE ALGEBRAS

a; + A; in the given finite-dimensional representation. Then AI :

A; so the minimum polynomial of At is a factor of )p - f,:

flfi:'r(f - m). Thus the minimum polynomial of Ai lhas distinctroots and these are in the prime field. It follows that ithere existsa basis (xr, xr, . . . , xn) for D? such that xiAi - t'nrixi, rnri inr the prime

field. Since the A; commute we can find a basis which has thisproperty simultaneously for the Ai, i = 1,2,' ' ' ,n. Then l*i(X f,;A) :

(1, m;itl.).f,; so that 9lt is a direct sum of the irreducible invariantsubspaces Ox5 and the weights are Li: l1rft;il;.

Remarh. It is easy to extend the first part of rheorem 14 to

arbitrary base fields of characteristic 2, that is, complete reduci-bility holds if the p-mapping is 1 : 1. On the other hand, it hasbeen shown by Hochschild (t4l) that if all the modules for a te'

stricted Lie algebra (everything finite-dimensional) are completelyreducible, then 8 is abelian with non'singular p-mapPtt*

Exercises

1. Let 8 be a Lie algebra over a field of characteristic zero, U the uni-

versal enveloping al{ebra. Show that every element of tl is a l inear combi-

nation of powers of elements of 8.

2 (Witt). Let 38 be the free Lie algebra with r (free) generatois ot, t2,' ' ' , xr

over a field of characteristic 0, 8 the universal enveloping algejbra of 88. Let(Bg)o = S8 fl 8n, 3n the space of homogeneous elements of dfgree n in 3-

Show that

dim (8l l )n = ! r p(d,)r,,ran d l n

where p is the Mtibius function.3. Let ! be finite-dimensional of characteristic 0, Il the

Show that the collection of linear transformations exp (adz),nrl radical of 8.

z e Tl, is a group

under multiplication.

4. Show that if Z is a nilpotent l inear transformation in a finite dimen-

sional vector space over a field of characteristic 0, then exp Z [s unimodul ar

(det exp Z : 1). Show that if Z is skew relative to a non'de$enerate sym'

metric or skew bil inear form, then expZ is orthogonal relativelto this form.

5. If ?I is an algebra there exists a unique automorphism r pf period two

in !I @ lX such that (o I b)t : b I o. Show that if 1l is the universal en'

veloping algebra of a Lie algebra, then lld' is contained in the bubalgebra of

r-fixed elements of tl @ 11.

6. Let ll be the universal enveloping algebra of a Lie algebra and let 11*

be the conjugate space of tl. lf e,* € l1*, then there exists a nrnique linear

func t ion s@* on 11@1l such tha t ( s I * t f>u i$ tu t ) :Z , t@d*h t i ) . De f ine


eV € U* bV (,p*Xa) : (e @g)(ud), d the diagonal mapping of 1l into U E U'

This makes ll* an algebra. Show that this algebra is commutative and as-

sociative.7. prove the following analogue of Friedrichs' theorem in the characteristic

p+O case: an e lemento o f S sat is f ies ad--a81+1@c i f and on ly i f c

is in the restricted Lie algebra generated by the rt.

B. Let 8 be a finite-dimensional Lie algebra, 8r and 8s ideals in I which

are contragredient modules for I relative to the adjoint representation. Let

(ut,. . . ,%n), (ut, . . . ,?ro) h dual bases for 8r and 8z as in the definit ion of a

Casimir element of a representation ($ 3.7). Show that 7 = lulut is in the

center of the universal enveloping algebra tl of 8.g. Let the notations be as in 8. Let R be a finite-dimensional representa-

tion for I (hence for tl) and let f(a) : tr 7t'R ' a € tl' Show that the element

t -fQur*tr'" utr)ulrutz " ' u'trN 1 , . . , l t : I

is in the center of lI. More generafly, sbow that if e 'is

a permutation of

L ,2 , - . . , f , t hgn

n2 f(uqeary "' utre)utrutt "' ttr6'

tt...trr-L

is in the center of Il.10. Determine ttre cerrter of ttle u,niversal enveloping algebra of the three-

dirnensicrat split sirnple algebra over a field of characteri:stirc aero'

ll. lret o be a field, altil= 1frfh,tz, .. -,,t L be the_algebra af polynomials

in indeterminates te, Q 1ta)} : il 1 tt, tz, "', tr') : c:[fl ttre ctoeure of' @ltil

regarded as a g.raded algebra in the ustlal way (an e*cr;remfi is' hsmogeneous

of degree k if. it is a homogeneous polynomial of dcgree & in ttrc usual sense).

O < tt > is called the algebra of forrnat puwer sefies in the tc md its quo'

tient field P is the f,etd of fwmol putir series in the tr.. o'1 tt ) has a

valuation as defined in $5. Thus if a: $x * ar+t * ar+z * -", atr homogene'

ous of degree i, a* + 0'then lal : 2-n. This valuafirm has'a t'miqUe extension

to a valuation in Psatisfyinglobl: iol l 'D' | . LetPnbethe a' lgebra of nXn

matrices with entriesinP. Define l(ou)l -maxlaUl. Shsqrthat this defines

a valuation in P" and show that if Dr, Dz, -.., Dr are matrices with entries

in o, then there exists a unique continuous homomorphism of the algebra

g : @ { r r , " ' , f r r ! o f $ 5 i n t o P o m a p p i n g i l t - + t t D u i : I , Z , " ' , T '

lZ. Let I be a finite dimensional Lie algebra over a field @ of character-

istic 0 and let P be defined as in 11' Let Dr' Dz' " "

Dr be derivations in

[! over 1l and denote their extensions to grover P by D.1:,D2,";,D, again.

Show that G - exp trDr exp tzDz" ' exp trD, is i1' $rGIl defined automorphism

in 8r and that G : exp D where D is in the subalgebra gencrated by the Dl

in the Lie algebra of derivations of 8r.

196 LIE ALGEBRAS

l3- Let ti be a finite-dimensional restricted Lie algebra in iwhich everyelement is nilpotent; apk: 0 for some b > 0. show that the #algebra of ghas the f.orm 1[1* S where S is the radical.

f4. A polynomial of the form f,pm +a1f,nm-r +... +amlid cal led a p-polynomia,l and it is called regular if. an * 0. Let g be a restficted Lie al_gebra (possibty infinite-dimensional) with the property that foi every o € gthere exists a regular p-polynomial po(i) such that pq(a) : e. Show thatif c is an elernent of ! such that all the roots of po(t) are in dhe base field@, then c is in the center G of 8. Hence show that g is abellan if o is al-gebraically clobed. Show that any finite dimensional nonabelian hestricted Liealgebra over an algebraicalry crosed field contains an ereme$t a # 0 sucht h a t a p : 0 . l

15- Let 8 be restricted with the property that ap : ae. a fixed and +0.Prove that I is abelian.

16. use 14 and 15 to prove that if o,oz : a in a restricted Lie algebra,then lJ is abelian. conjecture: lf apn@) - a, n(a) > 0, then g is abelian-

17. Prove that if I is restricted of characteristic three andior:0 for alla, then any finitely generated subalgebra of g is finite-dimensiional.

Conjecture (probably false but probably true under addition]l hypotheses):If u is finitely generated and every element of g (restricted of characteristicp) is algebraic in the sense that there exists a non-zero p-pdlynomial p6(l)such that p,^(a) - 0, then g is finite-dimensional.

18. Call a derivation D of. a restricted Lie algebra rcstrli,cted, if. aeD =@D)(ada)n't- Note that every inner derivation is restricted,r Show that aderivation is restricted if and only if it can be extended to I derivation ofthe. u-algebra. Show that if I has center 0, then every defivation is re-stricted.

19- Let I be an abelian finite-dimensional restricted' Lie hlgebra over aperfect field. Show that I : !o O,[Jr where 8o is the space o[ nilpotent ele-ments (an* - 0) and [!1 : flI]ar.

20. ! as in 19, base field infinite and perfect. Assume go 4 0. show thatll is cyclic in the sense that thereexists an element b such ttiat g : ZOwi.

2l' Let !l be the group algebra over a field of charact.j,.istic p of thecyclic group of order p. Then !I has the basis (1, &, az, .. ., nn-ry with cp :I ' Show that the derivation algebra D of r{ has a basis D1, i :0, l , . . . , p -1,where 0D1: st+r . Ver i fy that

lDDi : ( i - j )Dt+t

D t : D , , , D ! : 0 , i > 0

Prove that S is simple. S is called the Witt algebra.22. Generalize 2l by consideringthederivation algebra of ithe group alge-

bra ?I of the direct product of r cyelic groups of order p,i0 of character-istic p. These derivation algebras are simple too.


23. Prove the following identity f.or Lie algebras of characteristic p * 0:

p - 2

b-I (ad a)o-z- t (ad c)(ad oXt = 0

p - 2: c'rZ_r(ado,)n-z-;(ad DXad o)d .

ZL. Let I be a nilpotent Lie algebra of linear transformations in a finite'

dimensional vector space over a fietd of characteristic p I 0 such that 8p : 0. Show

that if /, B €8 then (/ + Do : le + Be. Use this to prove that if the base field

is algebraically closed, then the weight functions are linear'

Xt. Let to be the Lie algebra of n X n triangular matrices of fface 0 over a field of

characteristic p * 0, p X n. Prove Eq complete. (Hintz Show that every derivation

is restricted if fo is considered as a restricted Lie algebra and study the effect of a

derivation on a diagonal matrix with distinct diagonal entries. ( O may be assumed

infinite.))

L97

CHAPTER VI

The Theorem of Ado-Iwasawa

In this chapter we shall prove that every finite-dimensional Lie

algebra has a faithful finite-dimensional representation. We shall

treat the two cases: characteristic 0 and characteristic p separately.

The result in the first case is known as Ado's theorem. For this

we shall give a proof which is essentially a simplification of one

due to Harish-Chandra. For the characteristic p case the result is

due to lwasawa. The proof we shall give is simpler than his and

leads to several other results on representations in the character-

istic p case.

7. Preliminary reeults

If R is a homomorphism of a Lie algebra I into ?I1, where ?I is

a finite-dimensional algebra (associative with 1), then wq know that

R has a unique extension to a homomorphism R of ll into 1I. If

X, is the kernel, lllt,: Un g U so UIX is finite-dimensional. Ingeneral, if It is a subspace of a vector space 9Jt, then the dimensionof ltt/t will be called the co-dimensi,on of Tt in Tft. Thus R deter-mines an ideal f in 1l of finite co-dimension. The homomorphismR is an isomorphism of I if and only if I n I * 0. Conversely,let I be an ideal in U such that f n I : 0 and I has finite co-

dimension in ll. Then the restriction to I of the canonical homo-morphism of lt into [: IiI is an isomorphism of lJ into the finite-dimensional algebra [". Since any finite-dimensional algebra hasa faithful finite-dimensional representation it is clear that a Liealgebra I will have a faithful finite-dimensional representation ifand only if the universal enveloping algebra U of IJ contains anideal t of finite co-dimension satisfying * n I :0.

We recall that an element a of. an algebra lI is called algebraicif there exists a non zero polynomial g()') such that s(a) - Q. Thisis equivalent to the assumption that the subalgebra lI generated

by a is finite-dimensional. Consequently, every element of a finite'

[1es]

2OO LIE ALGEBRAS

dimensional algebra is algebraic. If X, is an ideal in l?I we shallsay that a e W is algebraic modulo X. if there exists ]a non-zeropolynomial tp(),) such that rp(a) e X". This is equivalentl to sayingthat the coset d. : a * I is algebraic in fl - U/I. It follo*vs that if Iis of finite co-dimension, then every element of lI is algebfaic modulof. We now state the following criterion for the univer$al envelop'ing algebra 1l of a Lie algebra.

Lpuur 1. Izt 8, be a fini,te'dimensional Lie algebra, (Iru 0az, ''', ilo)a basi.s for 8,17 the uniuersal enaelopi,ng algebra of 8, X, nn ideal inU. Then X, is of finite co'dimension in ll if and only if \eaer! ui isalgebraic modulo X,.

Proof: The necessity of the condition has been establi{hed above.Now let ei(D be a non-zero polynomial such that cpt(ur) € f and letni:dega{D. Then every ul is congruent modulo f [o a linearcombination of the elements 1, ua rlt,, " ', %Ti-t. The set df standardmonomials uftut'...uh*, ki2 0, is a basis for 1l and the rlemark just

made implies that these monomials are congruent modrllo I to alinear combination of the monomials u!'ul'"'ul* with 0 < tn 1 n;.Since this is a finite set, lUf; is finite-dimensional.

Lnnur 2. Same assumptions as Lemma !, X, and E ideals in 17.

If X, and P are of fi.nite co-dimension, then Ig is of finitv co'dimen'sion in lI.

Proof: Let p;U), </'r(J) be non-zero polynomials such thalt gi(u) e f;,

*{u) e E. Then p;(l)*,(l) has the property that cpi@){.r@;) e XV

and the result follows from Lemrna 1.

Lnnrur 3. Let 1l be an algebra, B a set of generatorr fo, II andD a deriaation i.n A. Sufbose that for eaery u e B thelre exists apositiue integer n(u) such that uD"t') : 0. Then for eilery a e A

there exists a positiae integer n(a) such that aD"'o' :0. If \ is

finite-dirnensional then D is nilpotent. I

Proof: Let S be the subset of elements b such that bD"(D) - 0for some integer n(b) > 0. If bb D, e E and brD"r : 0 - brD"z then(br + br)D" : 0 for n: max(nr, n). (ab)D"r :0, a e @tand

(b,b)DN: t(T) @,D\(b,D'-') : o

if N : lh * nz - l. Hence E is a subalgebra andl

sinpeE= .B ,

N. THE THEOREM OF ADO.IWASAWA 2OI

E : l[. This proves the first statement. The second is an im-mediate consequence.

2. The characteristic zero case

The key lemma for our proof of Ado's theorem is the following

Lsurul 4. Let e be a finite-dimensional soluabte Lie algebra ouera field of characteristic zero, n the nil radicat of 6,lr the uniuersalenueloping algebra of g. sufbose x, is an ideal of lr of fi,nite co-dimension such that eaery element of n is nilpotent modulo *,. Thenthere exists an ideal 3 in lI such that: (l) B g X, (Z) g is of finiteco'dimension, (3) 3D g 8 -for eaer! deriaation D of E (extend,ed toU), (4) eaery element of n is nilpotent modulo g.

Proof: Let D be the ideal in ll generated by r and !t. ThenU = f; and Etx is the ideal in u/ff generated bv (gt + x)lx. since(tt + f)/r is an ideal in the Lie algebra (6 + x)lx and the elementsof this ideal are nilpotent in the finite-dimensional enveloping as-sociative algebra lUX. of (g + X)IX. it follows from Theorem 2.2 t}rrt(tt + I)/f is contained in the radical of ElX,. Hence E/f; is in theradical. This implies that there exists an integer r such that3=U'g f . l f . le 6 and D is a der iva t ion in€ then we knowthat lD e n (Theorem 3.7). It follows that llD g E. Hence tDD g D,which implies that 3D:v'D s g':8. Hence (3) holds forBandwe have already noted that (1) holds. since p contains r, U is offinite co-dimension and 8 -- V' is of finite co-dimension in ll. Thisproves (2). rf z e fr, z" e f; for some positive integer z. Hencez" e V and z"' € 9' : 3. This provei (4).

Trponnu 1. Let g:g@ gr wh.ere e is a soluable ideat and grisa subalgebra of the finite dimensionat Lie atgehra g of characteristic0. suppose we haue a finite dimensional representation s of 6 suchthat zs is nilpotent for eaery z in the nil radicat rt of @. Thenthere exists a finite-dimensi.onal representation R of g such that:(l) if NR :0 for r in 6 thcn .rf :0, (Z) y" is nilpotent for eaeryt of the form ! : z * u wlure e e gt and u € gr is such that ad,guis nilpotent.

Proof: s defines a homomorphism of the universal envelopingalgebra u of 6 whose kernel r is of finite co-dimension. Also ife e rt then (e")' : 0 so z" e fr and f satisfies the hypothesis of

202 LIE ALGEBRAS

Lemma 4. Let 3 be the ideal in the conclusion of ithis lemma.We shall define the required representation R in 1l/8. We firstdefine a representation R' of 8 - 6 @ I' acting in the space 11.

If s e 6 we set sa' : snr the right multiplication in llr determined

by s. If / e ,8, w€ define lR' to be the derivation in 11 which extends

the derivation 5 -+ [s/] of. g. The R"s defined on 6 and on 8r

define a unique linear transformation R' on 8 which is a represent-

ation of 6 and 8, separately. To prove that R' is a representa'

tion for [! it suffices to show that [s/]8' : [so', /t'], t e g, / e 8,.

Now [s/] e 6 so [sI]"' : [s/]n. On the other hand, if D is a derivation

in 11 and a e'n, then the derivation condition gives [anDl: (aD)n-

Hence we have [s/]t' : [s/]n : (s/8')" : [sn, l*'l: ls*' , l*'f , as required.

Since 3 is an ideal in 11 such that 3D tr 3 for any derivation D of 6,

3 is a subspace of ll which is invariant relative to the representation

R' of I acting in tl. Hence we have an induced represeiltation .R in

the finite-dimensional factor space ll/3. Let r e 6 satisfy xe :0.

This means that r8 maps 11 into 3. Hence x e g, r € f ar[d so .trs = 0.

Let e € It. Then, by Lemma 4, z is nilpotent modulo 3. Hence e" is

nilpotent. Since Tt is an ideal in I (Theorem 3.7) it follows that

z* is in the radical ft of the algebra of linear transformationsgeneratedbyS". Now tet y:z* a where e € f t , u€'Et ' andadga

is nilpotent. Since e* € Dt, in order to prove that yB ls nilpotent,

it suffices to prove that uR is nilpotent. By definitiod u,*' is the

derivation in 11 which coincides with adgu on 6 and ladga is nil-

potent. Since 6 generates tl it follows from Lemmh 3 that for

every a e lJ there is an integer z(a) such that a(un')"'o' = 0. Hence

for every a eall we have z(z) such that a(u*)"' i ' :0' Since 1113

is finite-dimensional this implies that u* is nilpotent. Thus R

satisfies the conditions (1) and (2).We can now prove

Ado's theorem. Euery finite-dimensional Lie algebra 8' of charac'

teristic zero has a faithful f.nite'dimensional representation.Proof: We recall that the kernel of the adjoint representation

A is the center G of 8. It will therefore suffice to prove the ex-

istence of a finite-dimensional representation R of 8 wtnich is faith'

ful on the center O. For then we can form the directl sum repre-

sentation of R and ,4. The kernel of this is the interse[tion of the

kernels of R and of A. Hence this representation is faithful as

well as finite-dimensional. We proceed to construct R. Let g

VI. THE THEOREM OF ADO.IWASAWA 203

be the rad ica l , I t the n i l rad ica l . Le t T t r :6c l tz c . . . cT tu :T twhere each lt; is an ideal in the next and dim lti+r : dim lti * 1.Such a sequence exists since rt is solvable and contains G. Ifdim 6 : c then in a c * l-dimensional vector space there existsa nilpotent linear transformation e such that e' + 0. Then G isisomorphic to the Lie algebra with basis (t,z',...,2'), so G has afaithful representation by nilpotent linear transformations in a finite.dimensional space. Since each Il; is nilpotent and }ti+r : IL @ Ou;+twhere ou;*, is a subalgebra, the preceding theorem can be appliedsuccessively to obtain a finite-dimensional representation ? of !t bynilpotent linear transformations such that ? is faithful on 6. Nextwe obtain a sequence of subspaces, Er : It c 6g c ... c 63 : @such that 6i+r is an ideal in 6; and dim 6i+r : dim 6; * 1. Then€;+r : 6; @ Ou;+r. Also It is the nil radical of every 6i (Theorem3.7). Hence the theorem can be applied again beginning with r toobtain a representation S of 6 which is finite-dimensional, faithfulon G' and represents the elements of 9t by nilpotent linear trans-formations. Finally we write 8 : 6 @ 8r, 8r a subalgebra (Levi'stheorem). Then we can apply Theorem 1 again to obtain the re-quired representation R of 8.

Remark: The R constructed has the property that el is nilpotentfor every e e tt. The same holds for the adjoint representation.Hence the direct sum has the property too. we therefore have afaithful finite-dimensional representation such that the transform-ations corresponding to the elements of !t are nilpotent-and henceare in the radical of the enveloping associative algebra.

3. The charaeterietie p eqse

we recall that if o is of characteristic p then a polynomialof the form dolo* * arlon-r + ... * anl, a; e O is called a p-nrlly-nomial. lt p(]) is a polynomial of degree m, then we can write

( 1 ) Joo = p(t)q{t) * r;(}) , , fll,

where the rt(i) are of degree < rn. since the space of polynomialsof degree <mis ze-dimensional there exist a; , i :0, . . .s/rr t notall 0 such that xair;(l): O. Then (1) implies that >;a;]rr' :p?')(2a,tr,{^)). we have therefore proved that every polynomial isa factor of a suitable non-zero F-Wlynomial.

Now let I be a finite-dimensional Lie algebra over o, u the

204 LIE ALGEBRAS

universal enveloping algebra. Let a e g and let p(t) be a non-zero polynomial such that p(ad a) : O. Such a polynomial existssince the algebra_o! transformations in g is finite-dimerrsional. Letm(1) - ô* + at^pn-t + ... * a*r be a Apolynomial divisfible by p(r).Then we have

(2) (ada) ' * * a r (ade) r * - '+ . . . * a^ (ad ,d :a .JIn other words, for every 6 e Il we have

--ry- Pn-r. r _ |( 2 ' ) [ - . . I b a ) . . . a ) * a r [ . . . t b ; ] . . . " 1 * . . : r d ^ [ b a ] + 0 .

on the other hand, we know that [ ..wF:;1 - [bao). Iterationof this gives

( 3 )

Hence (2') implies

( 4 ) I b , a o ^ * a r 6 o * - r + . . . * a * a J : 0 ,

D e 8, which irnplies that the element

( 5 ) z = a ' ^ * a , r s n * - r + . . . * o * a

is in the center G of 11. we have therefore proved thp following

Lpuur 5. Let 8, be a finite-dimensional Lie atgebra buer a fietdof characteristic p + 0 and let o be the uniuersal enuelop{ng atgibra.Then for euery a G 8, there erists a porynomiat m,e) tutyrlhol*"61is in the center E of \.

The result just proved and Lemma 5.4 are the mdin steps inour proof of

Iwasawa's theorem. Euery finite-dimensional Lie algefua of char-acteristic p + 0 has a faithful finite-dimensional represenfation.

Protif: Let (ur, t4z, . . ., u) be a basis for I and let m;(l) be ap-polynomial such that m;(ur) : zi e 0, the center of thE universalenveloping algebra. If deg mre) : pî then zi : ulni 1l ut whereu;6l loî - t Hence, by Lemma 5.4, the elements z?tz lz. - .zf"q1' . . .uf ;o,h;20,0 s in < p*o form a basis for t l. Let E be the l ideal in Ugenerated by the a;. As in the proof of rheorem b.ll,i the cosetsof the elements ul ' . . .utr" ,0 < f , i < pî f .orma basis for l l /8. Hencethis algebra is finite-dimensional and the canonical mapping a -+ a :c * E, a e 8,, is an isomorphism of lJ into [r, [ : ll/U. It follows

pk

f "WTlal:1ba'*)

vI. THE THEOREM OF ADO-IWASAWA

that there exists a faithful finite-dimensional representation of 8.We shall show next that in the characteristic p + 0 case there

is no connection between structure of Lie algebras and completereducibility of modules. In the following theorem we shall needa result proved in Chapter II (Theorem 2.10) that an algebra ?t oflinear transformations in a finite-dimensional vector space whichhas a non-zero radical cannot be completely reducible. We shallneed also a result which is somewhat more difficult to prove,namely, that if z is an element of a finite-dimensional algebraand a does not belong to the radical of the algebra, then thereexists an irreducible representation R of the algebra such thatzR + 0 (See, for example, Jacobson [3], Theorem 3.1 and Definition1 .1 . )

THnonnu 2. Euery finite-dimensional Lie algebra ouer a fi.eld ofcharacteristic b * 0 has aL:L finite-dimensional representation whichis not completely reducible and a r: r fi.nite-dimensional completetyreducible representation.

Proof: Let the u; and zi: tni(ut be as in the proof of Iwasawa'stheorem. Let Er be the ideal in ll generated by (z?,22, ...2n). Thenthe argument shows that zr * E, ;a 0 in Ul8, but (2, + E,)' :0.Hence zr * Er is a non-zero center nilpotent element in the finite-dimensional algebra 11/E'. The ideal generated by such an elementis nilpotent. Hence 11/8, is not semi-simple. Hence any 1: l re-presentation of this algebra is not completely reducible. Since(8 + E'yEr g€n€rates 11/E' this representation provides a represent-ation for I which is not completely reducible. The argument usedbefore shows that the canonical mapping of I into u/8, is anisomorphism. Hence the representation we have indicated is 1 : 1for 8 and this proves our first assertion. Next let a be any non-zero element of 8 and take h : a in the basis (ur, trr, . . ., uo) for8. Let a, +0 be in 0. Then mr(l) -a is not divisible by I andthis is the minimum polynomial of. a * Ez in ll/Ez where Ez is theideal generated by m(u) - d, ?/tz(az), . . .,r/tn(un). Thus a * Eg isnot nilpotent and so it does not belong to the radical. It followsthat there exists a finite-dimensional irreducible representation oftl/Ez such that a * E' is not represented by 0. This gives a finite-dimensional irreducible representation R" of 8 such that aR" + 0.Let S, denote the kernel of R, (in 8). Then flreg,Fo:0. Since

205

$ is finite-dimensional we can find a finite number (rr, az, , Q ^ O f

206 LIE ALGEBRAS

the a's in 8 such that O r'S"r : 0. We now form the module !!ltwhich is a direct sum of the rn irredacible modules fi1 correspond-ing to the representations R r. Then evidently llt is completelyreducible and the kernel of the associated representatiorf is 0S,r-0.Hence this gives a faithful finite-dimensional completely reduciblerepresentation for 8.

Exercises I

l. Show that any finite-dimensional Lie algebra of charactbristic p hasindecomposable modules of arbitrarily high finite-dimensionalitiies.

Exercises 2-4 are designed to prove the following theorem: I Let lI be analgebra over an algebraically closed field of characteristic 0, 8 a finite-dimensional simple subalgebra ot Vt which contains a non-zero algebraicelement. Then the subalgebra of ' { generated by 8 is finite dimensional.We may as well assume that this subalgebra is ![ itself and:it suffices toshow that 8 has a basis consisting of algebraic elements. ]

2. Show that I contains a non-zero nilpotent etement e. (Hintz\Use Exercise3.11. )

3.in some Cartan subalgebra 0 of 8. (Hintz Use Theorem 3.17, Fnd Exercise

3.13. )4. lf e, have the usual significance relative to 6 show that there exists a

root a * 0 such that ho, ao, a-a are algebraic. Then show that this holds

for every root a and hence that I has a basis of algebraic elements. Use

this to prove the theorem stated.5. Extend the theorem stated above to 8 semi-simple under ithe stronger

hypothesis that I contains a set of algebraic elements such that the ideal in

8 generated by this set is all of 8.

6. Extend the result in 5 to the case in which the base fieldt is any field

of characteristic 0.7. (Harish-Chandra). Let 8 be a finite-dimensional Lie algebrh over a field

of characteristic 0 and let B be a faithful finite-dimensional rQpresentation

of g by linear transformations of trace 0 in lllt. Let 80, i: t,?,, "' denote

the representation in llll I xlt €) . .. I ![4, i times and let Ie denote the kernel

in ll of .Br. Prove that flifre = 0.8. Show that every finite-dimensional Lie algebra has a fafthful finite-

dimensional representation by linear transformations of trace 0.

Show that 8 contains a non-zero algebraic element I which is contained

CHAPTER VII

Classification of lrreducible Modules

The principal objective of this chapter is the classification of thefinite'dimensional irreducible modules for a finite-dimensional splitsemi-simple Lie algebra 8 over a field of characteristic 0. Themain result-due to Cartan-gives a L: 1 correspondence betweenthe modules of the type specified and the "dominant. integral"linear functions on a splitting cartan subalgebra e of 8. Theexistence of a finite-dimensional irreducible module correspondingto any dominant integral function was established by cartan byseparate case investigations of the simple Lie algebras and so itdepended on the classification of these algebras. A more elegantmethod for handling this question was devised by Chevalley andby Harish-Chandra (independently). This does not require caseconsiderations. Moreover, it yields a uniform proof of the existenceof a split semi-simple Lie algebra corresponding to every Cartanmatrix or Dynkin diagram and another proof of the uniqueness (inthe sense of isomorphism) of this algebra.

Harish'Chandra's proof of these results is quite complicated.*The version we shall give is a comparatively simple one which isbased on^ an explicit definition of a certain infinite-dimensional Liealgebra 0 defined by an integral matrix (A;t) satisfying certainconditions which are satisfied by the Cartan matrices, and thestudy of certain cyclic modules, "e-extreme modules,, for 0. Theprincipal tools which are needed in our discussion are the Poincare-Birkhoff-Witt theorem and the representation theory for split three-dimensional simple Lie algebras.

7. Definition of eertain Lie algebras

Let (A;) , ' i , j :1,2, . . . ,1, be a matr ix of integers A; i havingthe following properties (rn'hich are known to hold for the cartanmatrix of any finite-dimensional split semi-simple Lie algebra over

* Chevalley's proof has not been published.

12071

208 LIE ALGEBRAS

a field of characteristic 0)

(a) Ar; : 2, A;i * 0 if. i + i, A;i :0 implies Ah : 0.(p) det (A;.r) + 0.(r) If (ar, dzt ..., a,t) is a basis for an /'dimensional {ector space.bi over the rationals, then the group W generated byl the / linear

transformations S,, defined bY I

( 1 ) a i S a l - d ' i - A i i a t , i - 1 , " ' , 1 ,

is a finite group. i

Let O be an arbitrary field of characteristic 0. We rshall define

a Lie algebra 5 oner o which is determined by the rhatrix (A;)'

we begin with the free Lie algebra 88 ($ 5.4) generfted by the

free generators oi , f r l t ; , i , : I ,2," ' , / , and letS be thb (Lie) ideal

in 88 generated by the elements

( 2 )

lh;hil

le$il - driht

leihil - Ai$i

[f+hil * Ai,f,l

tet S - &8/n. Let 0 be the subspace of 88 spanned by the ht

and let a; be the linear function on 0 such that

( 3 ) a { h ) - - A i o , i - 1 , " ' , 1 .

The condition (9) implies that the / at form a basis fop the conju-

gate space b* of 0.Since S8 is freely generated by the ei,frh'i, i -1,4,"',1, any

mapping €t+Et, ft-Ft, hi--+Hi of the generators ]into l inear

transformations of a vector space defines a (unique) representation

of 88,. In other words, if f is any vector space with basis {ui},then f can be mdde into an fi8-module by defining the module

products u&t, utfr, utht in a completely arbitraty mafiner as ele-

ments of f. We now let I be the free (associative) al$ebra gener-

ated by I f.ree generators xr,l'z, . .., fit. Then X, has the basis

L , x r r " ' x t ,y t i i : I r2 , " ' r l , f : L rZ , ' ' " Le t 1 : l (h ) 'be a l inear

function on 0. Then the foregoing remark implies that we can

turn f into an S8,-module bY defining

( 4 )

VN. CLASSIFICATION OF IRREDUCIBLE MODULES 2W

L h : / , 1 ,

(x\ " . rt,)h - (1 - 4\ - a+)x\. .. tci, iLf; : x;, fr \ . . . xqf; : X\ . . . Xt,X6 ile;- 0 ,fr\. .. Xtr€i -- (nr. .. xir_r€;)X4

- 6;ro(A - 4rL - air-)(hr)ror. - . rir-t .

In these equations and in those which will appear subsequentlywe abbreviate /.(h) by A, etc., but write in full A(h;), etc. Let S'be the kernel of the representation of 88 in f. we proceed toshow that s' = s, the ideal defining 5. This will imply that f canbe considered as defining a representation, hence a module, for 5.The linear transformation in fi corresponding to & has a diagonalmatrix relative to the chosen basis. Hence any two of these trans-formations commute and so [h;hil e S'. We note next that thelinear transformation corresponding tofi is the right multiplicationri* in f determined by re. The last equation and fourth equationin (a) imply that

(x " =" _'' {j,,il,:Z,

; ::,rY,u, ;) xn, n i r _,

This implies that [e;-fi - diihi is in the kernel S'. We have

rrr;ht =Y'!,; !-t! ,ti, ,

n*': (/t - a;)h - *x;

xrr " ' rr,tf;hl: ttr... r4(fih - hfr)- It\ . .. Xtrh;h - U - &ar - a,1r)I4 . .. ItrX;: (A - dq - &+ - ai)rq .. frr,r;

- (A - str - a+)rq ". rr,ri- - d.irq .. XtrX; - - tra1... Xrr(difi) .

Hence lfthl + aif; e, fti. This implies that if r is an element of f;such ttrert xh - M(h)r : Mtr, then (rf,)h : (M - a;)(tf), or (rx;)h:(M - a)(rr;). We now assert that (rr,,... n,€t)h : (l - 4tr -

dr, * a;)xq. - - frt,€;. This is clear for r: 0 if we adopt theconventio'n that the corresponding base element is 1. We now as-sume the result for r - L. Then

210 LIE ALGEBRAS

(x\. .. xqe)h : (Qc\ "' rrr-r€;)x4)h- 6;,t(tl - &rL - sit)(hr)xrt

: ( 1 - s \ d i r _ t * a t - a , 6 r ) ( ( n r . . . x ; r - r o r ) n r )

- Bnro(tl - &rt - dir-)(hn)U - utr d;rrt)x\'" x;r-r

: (A - s\ - dir_t * a; - a6)(xq

This proves our assertion and it implies that

!c\ . . . tci , leihT - r\ . . , n,(e,th - he;)

: (A - 4\ - 4tr * a;)xq.. . xtr4;

- (l - s\ - aq)xtr "' xtrQ;

: d;lCt1 " ' Xtr0;, .

Hence le;.hl - a+€,t € fii . Wegenerators of S are contained

have therefore proved I thatin S/. Consequently $ g S/

all theand soprovefr can be regarded as a module for E - 88/4. We ca4 now

Tuponpu 1. Let 88. denote the free Lie algebra gdnerated by 3l

elements e;,fr, h;, i :1,2, "',,!, let R' be the ideal in 88' generated

by the elements (2), and let E - 88/n. -Thenz (il The cananical

homornorphism of 8S onto ! maps 2laet + Zloft I 2toh iso'

morfrhically into i,, so we can identify the corresponding subspaces.

(ii) The subspace g; = ae; t afn * ah; of 8 is a subalgebra which

is a s\tit three-d,imensional simple algebra. (iii) The bubalgebra i-

of i, generated, by the .f; is the free Lie algebra generh(d bt these

elements and a similar statement hotds for the subalgebrd 8+ generated

by the er Q = \lfih is an abelian subalgebra and we h4ue the uector

space decomposition

0:0OE-g>0* .( 5 )

(iv) If iI is the uniuersal enueloping algebra of i, then tl - Sfl*[-

where E is the subalgebra generated by b,lI* the subalgebra generated

by i* and iI- the subalgebra generated by 8--

Proof: For the moment let i: r * ft, t e 88. Consider the

representation R of ! determined in f by -the linear fqnction I : 0

o n 0 . I f . h e Q a n d h : 0 , t h e n f r . * : h * : 0 , w h i c h i i m p l i e s t h a t

(at , * . . . + a4)(h): 0 for a l l choices of the aor. Sincei dt , dz, ' ' ' ,dr

form a basis for the conjugate space this implies that h = 0.

rt"i"-n-.I is r:r. we have [eth;-l:26;, l f,fr l- -2f0, ldi,f;]:fi', hence odt * of t, + oltt is a subalgebra of s whi4r is a homo-

VtI. CLASSIFICATION OF IRREDUCIBLE MODULES 211

morphic image of the split three-dimensional simple Lie algebra.Because of the simplicity of the latter the image is either 0 or isisomorphic to the three-dimensional split simple algebra. Sinceh.i, + 0 by our first result, we have an isomorphism. This proves(ii). We have Wn'\ - O for any h, h' e 2 Aht, and ldthl : a;(h)6;,

If ofi: - ai@)f;. Since the linear functions 0, * a; are all differentthe usual weight argument implies that a relation of the form

2 l i r 6 o + Z l r h f , t + h ' - 0 , h ' e Z o h t i m p l i e s f i : 0 , T i : o f o r a l tf and fr,' :0. Then h' :0 by our first result. Thus we see thatx --+ fr is an isomorphism on 2 Ae; + 2 O.ft * 2 Oht We make theidentification of this space with its image and from now on wewrite ot, fr, hi etc. for d6, fr fir etc. We write Q : \lhh; and w€have seen that this is an /-dimensional abelian subalgebra of 5.We have noted before that in the representation R of E acting infr, ff : rcint the right multiplication in f determined by rr. Let [-denote the universal enveloping algebra of 5-, the subalgebra of Egenerated by the f;. Then we have a homomorphism of [- intothe algebra of linear transformations in I mapping ,fr into rrn. Ifwe combine this with the inverse of the isomorphism a - an ofI (the regular representation) we obtain a homomorphism of [-into I sending f, into xr. On the other hand, since fr is freelygenerated by the fri we have a homomorphism of t into Il mappingn into f;. It follows that both our homomorphisms are surjectiveisomorphisms. Since the free Lie algebra is obtained by takingthe Lie algebra generated by the generators of a free associativealgebra it is now clear that 0- is the free Lie algebra generatedby the fr and fi- is the free associative algebra generated by thefo. Also the Poincar6-Birkhoff-Witt theorem permits the identifi-cation of fi- with the subalgebra of fi generated by 0-, hence bythe f;. The basic property of free generators implies that we havean automorphism of $8 sendiag e; -.f;, .f; + €;, hi n - h;. Thismaps the generators (2) of S into ,ft and so it induces an auto-morphism in 8 - 88/n which maps f; into ei. Since the subalge-bra generated by the fi is free it follows that the subalgebra E*generated by the e; is free and its universal enveloping algebral1* is the free associative algebra generated by the e1. This alge-bra can be identified with the subalgebra of fr generated by the€.i. It remains to prove (5); for once this is done then the relationfr. : Sfr.*fr- follows from the Poincar6-Birkhoff-Witt theorem bychoosing an ordered basis for E to consist of an ordered basis for

212 LIE ALGEBRAS

0 followed by one for 0* followed by one for 0-. To prove (5)we show first that E, = O + E* + CI- is a subalgebrb of 5. Theargument is similar to one we have used before ($ 4.3).: We observefirst that every element of E* is a linear combinatidn of the ele-m e n t s f e r r e q . " e r , l : [ . . . l e t r e 6 J . . . 0 n , ) a n d e v e r y e l e r n e n t o f 0 - i sa linear combination of the elements [/c, . .. .fn,]. The Jacobi identi-ty and induction on r implies that

( 6 ) I lenr "' eqJh) : (atr * a6z+ "' I a6)fetr " ' un,l

\ - / l l fn r " ' . fn , lh ) : - (a t r l aq+ " ' * a t , ) t f t r " ' f r , l .

Hence tE,0l S !,. We have Ie;fi e 0 and by inductibn on r 2 2we can show that [[er, .. . en,)fil e E*. It follows that, Erad .f t =ir.Iteration of this and the Jacobi identity implies that [0,CI-J s 0,.Similarly, [0,0*] tr 0,. These and [0,0] s 0, imply thatr5, is a sub-algebra. Since e;,fr,hie i, i t follows that E, -S, {hat is, 0-0* + O + E-. Equations (6) imply that 5 is a direct sum of rootspaces relative to A and the non-zero roots are tlle functions+ (at, + . . . I an,). It is clear that E* is the sum of ther root spacescorresponding to the roots et, * - . . * aq and E- is the sum ofthose corresponding to the roots - (ar, + ... I otr). It follows that0 : 0* O 0 O E-. This completes the proof .

2. On certain eyclic modules for i

A Lie module llt is cyclic with generator x if. Wl is the smallestsubmodule of !)t containing r. If U is the universal envelopingalgebra of the Lie algebra, then vll - {rulu e U} is {he smallestsubmodule containing x. Hence llt is cyclic with r as Senerator ifand only if !n - A1. The module f; for E - 88/A which we con-structed in $ 1 is cyclic with 1 as generator since lfq .- . .fr,:r\ . .. xt, ?r7d these elements and 1 form a basis for ff.

We shalt call a module !)t for i, e-extrerne if. it is cyclic and thegenerator r can be chosen so that xh - A(h)x and fi€r:O, i -

L,2, . . . ,1. I t is .apparent f rom (4) that f is e-extreme with 1 asgenerator of the required type. Thus we see that for every linearfunction A(h) on b there exists an e-extreme module fof which thegenerator r satisfies xh - 1(h)r, x€i: Q. We shall now considerthe theory of e-extreme modules for 5. A similar theory can bedeveloped for /-extreme modules which are defined tO be cyclicwith generator y such that yh - A(h)t, lft:0, i, : I,2, .- .,1. We

v u . C L A S S I F I C A T I o N o F I R R E D U C I B L E M O D U L E S 2 1 3

shall stick to the e-extreme modules but shatl make use of the

corresponding results for /-extreme modules when needed.

Let llt be e-extreme with generator r satisfying rh = ,l(hy, 1c2i:

0. We know that the universal enveloping algebra ll of 8 can be

iactored as ff. : St*fi- where S, [* and f- are the subalgebras

generated by O, E*'and E- respectively. Then Dt - ril: rSfi.*[-.

Sin." rh * 1(h)x, rE : Or and since r01:0, r11* -- Or. Hence !y1 :

r8fr.*fr.- - ril-. Moreover, [- is generated by the elements fi.

Hence every element of llt is a linear combination of the elements

x f \ . . . f , , ,

where we now adopt the convention that fi, "'fr,: L if. r - 0'

We have lfor "'fn,, hl - - (ao, + "' + a.-)ftr'" fn, (induction on

r) and xh : 1r and these relations imply that (rfrr "' f,,)h :

xl f tr- . . f0,, h|+'@h)fq"' . fr , - (A- ctdL - a4)rf4" ' . fr , ' Hence

( 7 )

( 8 )

(e)

Thus Dt is a direct sum of weight spaces relative to 0 and the

weights are of the form

l - ( a ; r * a q + " ' * a q ) : At,

-Zk ;a t ,I

where the fti are non-negative integers. Also it is clear that the

restriction to a weight space of the linear transformation corre'

sponding to any h is a scalar multiplication by a field element. It

is clear also that the weight space llt,r corr€sponding to I has r

as basis and so is one-dimensional. The weight space l}tr corf€-

sponding to the weight fu[ : A - > k;ai is spanned by the vectors

(7) such that d;r *' ' ' + 4tr l, kia;. Clearly there are only a

finite number of sequences (lr, ir, ''', a) such that a;r + ' ' ' * da,:

2 k;ai where the ki, are fixed non-negative integers. Hence lltr is

finite-dimensional.The weight. l can be characterized as the only weight of 0 in

fi such that every weight has the form A - > h;ai, hi non-negative

integers. We shall call A the higltcst weight of 0 in fi or of fi.

If !t is isomorphic to fi, then !t is also e'extreme and has the

highest weight l. It follows that two e'extreme 0-modules having

distinct highest weights cannot be isomorphic.Let gt be a submodule of Dt : > e 'ftr where

module corresponding to M. ll y e It, J e lltr, *is the weight* llltrp where

Str

2t4 LIE ALGEBRAS

{Mr, . . ., Mo} is a finite subset of the set of weifhts. Hencey e tt fl (TJt', + ... * Iltr*) and this is a submodule iof the finite-dimensional F-module lf=,I}tyr. Such an Q-module ib split and isa direct sum of weight modules whose weights are ini the set {M}(cf. $ 2.a). This means that It n (> Tft* 1): X (It n Tlta ). Since yis any element of Tt we have also that Tt : X O Ttrr where ltr :Il n lll,u. If ltll * 0, M is a weight for 0 in It. At any rate, itis clear that It is a direct sum of weight modules and the weightsof b in Tt are among the weights of 0 in llt.

Now let [Jt' be the subspace of m spanned by the Dt'( withM + I and, assume that the submodule !t +Yft. In fhis case, wemust have Ttt :0 since, otherwise , Iln -- IJlz which iis one-dimen-sional. Then r € Tt and Tt - )tI: IJt contrary to assurfiption. Thuswe see that any proper submodule Tt : Z*+oftn g IJl/ c l]l. It fol-fows from this that the sum S of all the proper subfirodules of TJIis contained in TJt' c TJt and so this is a proper subrnodule. Thisproves the existence of a maximal (proper) submodule lB of !]t.Moreover, T is unique.

We consider again the module I constructed in $ 1 *hich we shallnow show is a "universal" e-extreme module with highest weightI in the sense that every module fi of this type is a homomorphicimage of-I. For this purpose we define 0 tobe the lihear mappingo f t o n t o 9 f t s u c h t h a t ( x ; , . . . n r ) 0 : r f h - - . f n , ( x n r . . . t c c r : 1 i 1r: A). Then

tctrtc;)0 - rfnr "' frrf,.

,xq)o(11)

: (x f1 r . . . f i , )h : ( (xo ,

(xnr ' ' ' x t ro6)0

n,)0)h

(10) (r t ' ' " xt ' f )0 : (x" '

,-- ((xr, "' xu,)0)ft

(r;,

- ((r,, "' xi,-t€i)n)0 - 6o,dA

: ((r;, . .' x;r-tet)for)0 - Brrr(l

: ((x4. .. x;,-t€i)0)fn, - dr,u(A

_ &ir

_ gir

rir-r)0

t/,ir-r)0

xl,r-1)0

If we use induction on r we can use this to establislr the formula

(12) (xir "' th,ei)0 : (x\ " ' h,?Pi .

Since the elements e;,f;,,ft; gen€rate E, equations (10), (11) and (12)

VII. CLASSIFICATION OF IRREDUCIBLE MODULES

imply that d is a module homomorphism of f onto fi.

The result just established shows that any e'extreme 0-module

with highest weight I is isomorphic to a module of the form I/Tl,

It a submodule of the module I. If lDt is irreducible we have

[!l - I/$ where S is a maximal submodule of f . We have seen

that there is only one such submodule. Hence it is clear that any

two irreducible e-extreme modules with the same highest weight

are isomorphic. The existence of an irreducible e-extreme module

with highest weight A is clear also, for, the module gJl : I/!S

satisfies these requirements.We summarize our main results in the following

Tuponpu 2. Let the notations be as i,n Theorem I and let A(h) be

a linear function on b. Then there exists an irreducible e-extreme

F,-moduti with highest weight A. The weights for such a module

are oJ the form i - >, krai, h; a non-negatiue integer. The weight

space corresyonding to A is one-dimensional' and all the weight

iporn, are finite-d,imensional. Euery h e b acts as a scalar multi'

plication in eaery weight space. -

Two irred'ucible e-extreme E'

modules are isomorphic if and onty if they haue the same highest

weight.

3. Finite'dimensional irred'ucible modulea

We shall call a linear function A on .b integral if l(h;) is an

integer for every i :1,2, . . ' , / , and we shal l cal l an integral l inear

function dominant if A(hi)>:Q for all i . In this section we es'

tablish a I : I correspondence between these linear functions and

the isomorphism classes of finite-dimensional irreducible modules

for 0. In view of the correspondence between the isomorphism

classes of irreducible e-extreme E-modules and the highest weights

which we established in Theor em 2 it suffices to prove two things:

(1) Every finite-dimensional irreducible module is e-extreme with

highest weight dominant integral, (2) Any irreducible e-extreme

module with highest weight a dominant integral l inear function is

finite-dimensional. We.prove first

THponnu 3. Let i be as before an1 let Dl be a finite'ditnensional

irreducible i-module. Then Dt is e-extrente and its highest weight is

a dominant integral linear function on' S)'

Proof: 9Jt is u finit"-aimensional module for the subalgebra 81 :

2t6 LIE ALGEBRAS

Oet * Af; * Oh; which is a split three-dimensional sir{rple Lie alge-bra. Hence lll is completely reducible as 8;-module. The form ofthe irreducible modules for 8i ($ 3.8) shows that there exists a basis(x!" , rl" , . . ., ,1'l') for IJt such that rtt h; : llti*frt) where the m;* arteintegers. Since [hrhi] - 0 the linear transformations associated withthe different hi commute; hence, we can find a basisi(frt, fr2, - . . , JCr)such that x*h : f lt ikgk, i : l , ..., l , k : 1,..., N. lf. A* denotes thelinear function on 0 such that l*(ht) : ,n;*, then 1*iis integral and

(13) x*h: lrr* , h : L , 2 , . . . , N .

Then 1* alre the weights of S in !]t. Let 8i be the fational vectorspace spanned by the linear functions c;. It is easy to see that alinear function d e 0f if and only if the values a(h) are rationalfor i : t,2, . . .,1 (cf.. the proof of XIII in g 4.2). He4ce the weightslk e Qt and so we may pick out among these weights the highestweight I in the ordering of ,pi which is specifiediby saying that

Zllra > 0 if the first li * 0 is positive. In this lase it is clearthat I * a; is not a weight for any ai. Let x be d non-zero vectorsuch that xh -- tlx. Then (xe;)h - (1 a ar1@e;) ,2ci xe;:O by themaximalftY of. A. Since llt is irreducible and rll iis a submoduleof l}t it is clear that Dt : ril. Hence fi is e-extreme with I asits highest weight also in the sense of the last section. The re-sults of $ 2 show that'every weight is of the form I - > k;a,;, kta non-negative integer. On the other hand, the proof of the repre-sentation theorem, Theorem 4.1, (applied to S *lAe; + Af) showsthat if M is a weight for 0 in l)1, then M - M @)ai is also aweight. Hence for each i, ,l - A(h;)a; is a weigh[ and so has theform A - 2 kia;. It follows that A(ht): ft; 2 0. Thus we see thatI is a dominant integral function. This completds the proof.

Next let l be any dominant integral linear function on 0 andlet Dt be the irreducible module furnished by rTheorem 2 withmaximum weight 1. The weights of A in !n have the form/1 - 2, k;a;, k; integral and non-negative. Hence these are integraland so they can be ordered by the ordering in Qfr. We shall provethat ![t is finite-dimensional. The proof will bd based on severallemmas, as follows.

Le t 0 ; i : f {ad f ) -a i i+ t , i + j -L ,2 , " : ,1 .. . .,1, and Tft?;i - 0 for any e'extrvme

Lsurtra, 1.0 , k : I , 2 ,module TJt.

Then lfl;ierl:irreducible E-

VII. CLASSIFICATION OF IRREDUCIBLE MODULES 2I7

Proof: lf. k + i, [fie*]: 0; hence ladf, ad e*l : 0. Then

l0iie*) : f /61d f )-aii+t ad, et, : f 1 ad e*(adf ;)-ait+r

: lf ptlkrd f i)- ^ti*' - - 6 i*h t(ad f )-

a ii+r

If. k +j this is 0. lf. k :, we obtain l0;grl: - Ai"fr(ad/t;-rt i ' 11

A;t :0, At, - 0 by the hypothesis (a) of. $ 1, so the result is 0 in

this case. Otherwise, -A;r ) 0 and i(ad 7'1-tti - 0' Next let

h:i,. Then l|;ie*l:f1@d,fi)-'{ii+r ad,ei. We recall the commuta-

tion formula: akr: xak -(!)r 'ao-t + (!)r"ok-z - " ', wher" rt -lxal,

x" : lx'a\, . . . . (eq. 2.6). If we make use of this and the table:

leii - hi, lIe1;Lfi:2f;, llleJif;lf;l : 0, we obtain

f t@dfr)-^i;+r ad et -- fi adet(adf)-a,i+l- 1- Au + L)fi ad, h;(ad f)-^it , if A;i : 0

- f 1@d et)@df;)-a;i+t - 1- Au + L)fi ad hi(adft1-^;t

* 21-A;r + lX- Adfi@d2f;)(adf)-tu-t , if -Au > 0 '

The first term in both of these formulas can be dropped since

f r a d s ; : f f p ; l : 0 . I f A ; t : 0 w e h a v e f i a d l t - f f f i i ] : 0 . I f--A;t > 0 we use thesecond formula and fiadhi-fff i i : - Auft

to obtain - q-A; i* lx-Ar ) f i@df i ) -a; i

* q-A;t + 1X-4r r)f {adft)-/ ' i : 0 .

This completes the proof of l0;p*l:0. Now let llt be an irre'

ducible e-extreme E-module and, as before, let III' be the subspace

spanned by the weight spaces corresponding to the weights other

than the highest weight A. Consider the submodule \ll?;;lJ":

!ltdriS[*[- where E,'[* and fi- are as in Theorem 1. If we use

the fact that ad h is a derivation and that lfthl - - aif; we see

that0 ; 1 h : h ? u * [ 0 u h l

- hoit + (-at t, (Au - L)a)oit ,

which implies that IJld;.rS: Wt2u. Alry 0;i0tc:.erdrr and this im'

plies thaf Tftflriil* - T!10u. Hence Dhzufr : Tftl;tll-. If. x is a canoni-

cal generator of f i such that rh- t lx , x€; :0, i , : t ,2," ' , / , then

every element of IJt is a linear combination of the elements of the

form xf\...fr, and every element of. wl' is a linear combination

of these elements for which r Z l. It follows from the definition

of 0;1 that Thliis IJt/. Also it is clear that lft'fi- g tJt'. Hence

218 LIE ALGEBRAS

TftLrt g llt' and so Tftarii. is a proper submodute of ItJt. Since l)tis irreducible we must have yll?;t:0 and the proof ,is complete.

Lpuur 2. Let Tft be an irreducible e-extreme module for i whosehighest weight Li.s a dominant integral linear functiun. Then forany y e tlt there exist positiue integers /;, s; such that yeii :0 -

y f i i , i : 1 , 2 , . . . , 1 .Proof: Let x be a generator of IJt such that rh)- Ax, xei : 0,

i: L,2, '. -,1. Then every element of l l t is a l inedr combinationof elements of the form xf;rfir. .. fr,. It suffices to prove the re-su l t fo r every t : x f ;J t r . . . fn , . Wehave lh : My, M- A-Zh iarh an integer 2 0. Then (ye!i+')h - (M * (h; + I)aDlteli+'. SinceM + (h * l)ar is not a weight, yeli*' - 0 which provds the assertionfor e;. (This argument is valid for arbitrary e-extreqne 0-modules.)Let 1(h): /n, i20. We show next that x. f { i * '=0. We havexhi: lltflc, x€i : Q. Hence if we apply the theor! of e-extrememodules to the algebra g+: Oe; * O.f, * Oh; (in placb of 5) we seethat the 8;-submodule fii generated by r is the spa0e spanned byx,xf i , tcf ! , . - ' . Suppose thishasaproper submodulelTt . r + 0. Sincekf!)h; - (m, - 2ilxf!, the spaces oxf! are the weight spaces rela-tive to Ohr Hence lii is spanned by certain of the subspaces Axf!and k>- L since Sti c TJti. Moreover, if. xf! € Tte then xfl e Ttr forall q > h. It follows that It; : Zqzk>r0xf: where & iS the least posi'tive integer such that xf! € }ti. Evidently Tti S !Jtl. We now ob'serve that TIih I Tti since (xf{)h: (A - qa;)(xff) andl Tl;e*9 Tti sincethis is clear for h: i and it holds for h * i since

"/f eo : rer,f,iq : A.

It follows now that Tt,E - Iti and It,fi* - ftr. Then nlf - ft,8fi*fi- -

nrf- g !:tt'. Thus Tti[ is a proper E-submodule oi TJt + 0. Thiscontradicts the irreducibility of fi and so provesi that IJtr is 8r'irreducible. Now in g 3.8 we constructed a finite-dimensional irre-ducible 8.i-module with a generator N' such that xthi,: mixt, x'e.r:0and x'f{i*t : 0. It follows from the isomorphisrnl result on irre-ducible e-extreme modules (Theorem 2) that this module is iso-morphic to IJti. Hence we have rfli*' - 0. Now sqlppose we havean integer m20 such that (xfnr" ' . f+-) f i :0. l f i , : I th is im-plies that (rfi, ..-f;,)f{ - 0. lf i,: j + i we use lhe relation

fi.f{-"ot = f!"-a;i f, + ( m -.An'\ff-^,,-' [email protected])- r t J r ' \

L ) t t .

+ + (* - ^o"\frf o@of,)-o" (mdd dri)

\ - / rd i /

v I I . C L A S S I F I C A T I O N o F I R R E D U C I B L E M O D U L E S 2 1 - 9

where d;r. is as in Lemma 1 and the congruence is used in the

sense of the ideal in ff. generated by |ri. It follows from Lemma

1 that

xfrr " 'f,,-rf Ji-aii : xfir " 'frr-rf{-^' if t

+ ''' * (:t,oo)rrn "' f,,-,f{fi@df,)-a;i - 0 '

This proves the assertion on f; by induction on r'

Lnuur 3. Let \It be as i,n Lernma 2. Then if M is a weight of

Q in In, M - M(h)ar, is a weight for i : L,2, ' " ,1 '

Proof: Let y be a non-zero vector such that yh - M(h)y' If

M(hr )20 , then we choose q so tha t z :yeq i *$ , ye l * ' :0 and ?n

so that zf{ + 0, zf{*' - 0. This can be done by Lemma 2. Then

the determination of the finite-dimensional irreducible modules forg; : Oe; * Af n * Ohi (S 3.8) shows that Ef= obzf! is such a module

and zh,t: mz. On the other hand, yh: U(h)y implies zh: (yeon)h -

(M+qd)(h)z and. Qfhn - (M + Qdi - na)&)zf!. Hence (M + qa)(h;):

M(h) * 2q -- rn and

M + q a ; . , M + ( q - l ) a t " ' , M + ( q - m ) a ;

are weights (corresponding to z, zf ;,. . . , zfr). 'we

have M -M(h;)ar--

M + (2q - rn)ar and q - rn 3 U - m S 4 since M(h) - tn - 2q ZO

and q Z 0. Hence M - M(h)at, is in the displayed sequence. If

M(ht) < 0 we reverse the roles of. ei, and fi and argue in a similar

fashion.We can now prove

Tnponuu 4. Let Tft be an irred,ucible e'ertrerue module for E such

that the highest weight /l is a dominant integral linear function'Then Dt els fi'nite-dimensional.

Proof: Let s"o denote the linear mappinE f -' E - E(h+)a.; in the

space 0i of rational linear combinations of the linear functiorls 4t

such that a;(h) -- Ain. We have a'iSal - d,i - A;td; so that S", is

one of the linear transformations specified in our axiom (r) of $ 1.

This axiom states that the group W genetated by the S,n is finite.

On the other hand, Lemma 3 implies that the set of weights of Q

in fi is invariant under 17. We now consider the set J of images

under W of. the maximal weight /1. This is a finite set, so it has

a least element M in the texicographic ordering we have defined

220 LIE ALGEBRAS

in 0f (at the beginning of this section).* Let y be a non-zerovector such that yh - My. Since M - M(hi)ar e E, M,3 M - M(h;)a;so tha t we have M(h t )<0 , i :1 ,2 , . . . ,1 . One o f t l te l inear func-t i o n s M - d i , M * a c i s n o t a w e i g h t . T h u s f u l - l S , S e W a n di f . M-d i , M+d l a ; t l we igh ts then u- * 'S- ' : (M -ar )S- t and1* arS-' are weights. However, one of these is greater than Iin the ordering in 0i which contradicts the maximality of A. IfM+ cr is not a weight we have !o; :0 as wel l a$ Jh;-M(h)y.As in the proof of Lemma 3, y generates an irreducible 8i-moduleof dimension M(h) + L Hence M(ht) > 0. Since I M(h;) S 0 wehave M(h):0. If M - at is a weight we would have also thatM - d ; - ( M - a ) ( h ) a . i - l l [ - e ; * 2 a i : M + a i i s a w e i g h t c o n -trary to assumption. Thus M - oei is not a weigtlt if M * ar isnot a weight. Hence in any case, M - av; is not a weight. Conse-quently, !f;:0, i : 1,2, - . .,1. Since !)l is irredubible W : yD-,Thus we see that rJt is an /-extreme module. It follows that theweights of 0 in l l t have the form M+), j ;dt i r integer>0. I ffu[ - I - 2 her it is now clear that the weights have the form/l - > k&; where 0 S k; 3 li. Thus there are only a finite numberof different weights and since every weight space is finite-dimen-sional we see that Dt is finite-dimensional.

4. Existence theorern and isomorphism theoremfor semi-simple Lie algebras

Let F denote the collection of finite-dimensional irrbducible repre-sentations of 0 and let So be the kernel of the collection f', inatis, the set of elements b e E such that bR :0 forl every R e F.Then ,Qo is an ideal in 0. If R e F we can define a representationR for 8 - E/So by setting (, + So)o : /o. Since the r.t of repre-senting transformations in the corresponding moduile is the sameas that furnished by the representation or E it is dlear that R isa finite'dimensional irreducible representation for l]. iWe shall nowshow that 8 is a split semi-simple Lie algebra and thd given matrix(Ad is a Cartan matrix for B. Thus we have the following

THponpu 5. Int E - $g/n where 8S rs the free Lie algebragenerated by e; , . f i ,h i , i . :L,2, . . . ,1 and R, is the ideal in $8, gener-ated by the elements (2). Let Ro be the hernel of all the finite-

* This is the only place in our discussion in which axionl (1) is used.

v U . C L A S S I F I C A T I o N o F I R R E D U C I B L E M O D U L E S 2 2 |

d,imensi,onal i.rred.ucible representations of i and' set g= 0/So. Thcn:

(1) B is a fi.nite-dimensionat spli.t semi-si,mple Lie algebra. (2) The

canonical mapping of Q + Zl'Oe; + >l0fi into I is an isornorphism

so ue can id.entify this subspace with its irnage. (3) .b is a splitting

Cartan subalgebra, the a,; form a simple system of roots and thp

hi,2;,f; form a set of canoni'cal generators whose associated Cartan

matrir is the giuen matrix (A;,). (4) Let E be any finite-dirnen'sional sptit semi.-simple Lie algebra wi'th l'dimensional splitting

Cartan subalgebra -Q

and canonical generators Er, dr, f , wh,ose as'

sociated Cartan matrix is (A;). Then there exists an isornorphism ofg onto E send, ing hr- f r i , a i -+ d; , f i - f - r , i : L,2, ' ' ' ,1.

Proof: Let ,l; denote the linear function on Q such tl:p,t ^i(h) -

di.r. Then ,ii is a dominant integral linear function and so it. corre-

sponds to a finite-dimensional irreducible representation Rt in a

module llh for 5 (and for 8). The ]; form a basis for 0* and

every dominant integral linear function I has the form 1:}nt;f,i,

mi integral = 0. We shall show first that .Qo C"n be characterized

as the kernel of the finite set of representations Ri. Thus let IIt

be any finite-dimensional irreducible module and let A : 2 mi),; b'

its highest weight. Let $ be the module which is the tensor prd-

uct of ,nr copies of tJtr, ,nz Copies of lJtz, . . ., fnt copies of lltr. Let

ft b a generator of llt; such that rlh : ]iri, h e Q, ri0r:0, i :

L , 2 , - . . , / , a n d s e t

(14)

f/tl rrt2 nl.t-r

r : x r A . . . 8 ; , I r z & . . . @ r r A . . . @ n 8 " ' 8 r r .

In this definition and formula we adopt the convention that if all

the mi:0, which is the case, if and only, it A:0, then TJt is the

one-dimensional module with basis x and, xl:O, / e 8. Then, in

any case, we have xh: tlx, trer: 0, so "[

is an e-extreme module

with highest weight /1. Consequently, the irreducible module fi

corresponding to tl is a homomoiphic lma ge of xit. Suppose / e 5

a n d l R i : 0 f o r i : 1 , 2 , . . ' , 1 . T h e n f i ; I : 0 ; h e n c e $ l : 0 a n d

r|J/ - O. This implies that IJtl : 0 or lR :0. We have therefore

proved our assertion that So is the kernel of the finite set of repre'

sentations Ri, or equally well, the kernel of the single representa-

tion S which is the direct sum of the Rr. Evidently S is finite:

dimensional completely reducible and B - E/So o'8s a finite-dimen-

sional completely reducible Lie algebra of linear transformations.

Since x;h: ).;(h)x;, hed, hs :0 implies that ),r(h) - 0 for all i-

LIE ALGEBRAS

Since the & form a basis for Q* this implies that k :0. Hence0 is mapped faithfully by S and consequently by 1 the canonicalmapping of 0 onto 8. It follows from this that the simple Liealgebras 8r zre mapped faithfully. Then the argument used toprove the corresponding assertion in Theorem 1 shows that (2)holds. Since 0 is a direct sum of .p and the spaces 0o corre-sponding to the roots llrl -:r (: ks;), h integral * 0, the sameholds for 8. It follows that Q is a splitting Cartan Subalgebra for8,. Consequently, the center G of 8 is contained in b.

l f . ho e 6 we must have lethol : a;(ho)ei :0. Thenladh):0 forall i and ho - 0. Thus 6 - 0. Since 8 is isomorptlic to a finite-dimensional completely reducible Lie algebra of linear transfor-mations and has 0 center it follows that 8 is semi-slmple. Hencewe have established (1). We have [erh) - d;er ?nd eVery root of 8relative to 0 has the form :t (> kid;), k; integral and non-negative.Hence the a; form a simple system. We have [e|i] = 6i,th;, lethtl:A60r [.fthA- - Airfr. This means that et e8rr, ',fre 8.-*r, thegenerators €r,.f;, hi are canonical and the associated Oartan matrixis (A;r) (cf . $ 4.3.) This completes the proof of (3). iNow let E beas in (4). Then it is clear that there exists a hombmorphism of0 onto E such that e,i->0-t,.fr-fr, h-+8,i. Since E ls semi-simpleit can be identified with a completely reducible Lie algebra oflinear transformations in a finite-dimensional vector space and thehomomorphism of E onto E can be considered as a rppresentation.Since So is mapped into 0 in any finite-dimensionjal irreduciblerepresentation, the homomorphism of E maps ,Qo indo 0 and so wehave an induced homomorphism of I onto E sucti that 0r -> dr,ft-ft h;- E.t. Since 6 is /-aimensional the homombrphism mapse isomorphically. If ,Rr is the kernel of the homurilrorphism of ,Bonto E, since S, is an ideal, it is invariant under ade 0. It followsthat if St ;E 0 then it contains a non-zero element of 0 or it con-tains one of the (one-dimensional) root spaces 8,,, a * 0. The firstis ruled out since Sr fl 0 : 0. If S, 2 8r, S, 3 [8"8 -rf * 0. Since[8,8-"] s Q this is ruled out too. Hence Sr :0 ahd the homo-morphism of 8 is an isomorphism. This proves (4).

Theorems 3 and 4 establish a 1: 1 correspondence between theisomorphism classes of finite-dimensional irreducibls modules forthe (infinite-dimensional) Lie algebra E and the colledtion of domi-nant integral linear functions on O. Also it is clbar from thedefinition of 8 that any finite-dimensional irreducible E-module is

V I I . C L A S S I F I C A T I O N o F I R R E D U C I B L E M O D U L E S 2 2 S

an 8-module. The converse is also clear since 8 is a homomorphic

image of !. Hence we see that Theorems 3 and 4 establish a I : 1

correspondence between the isomorphism classes of finite-dimension-

al irreducible modules for the finite-dimensional split semi-simple

Lie algebra and the collection of dominant integral l inear functions

on a splitting Cartan subalgebra Q of 8.

We recall that a set -X of linear transformations in a vector space

llt over A is called absolutely irreducible if the corresponding set

(the set of extensions) of linear transformations is irreducible in

lJlr for any extension P oI the base field. This condition implies

irreducibil i ty since one may take P: 0. The term "absolutely

irreducible" wil l be applied to modules and representations in the

obvious way. It is clear from the definition that a module llt for

a Lie algebra 8 is absolutely irreducible if and only if Dlp is

irreducible for 8r for any extension P of, 0. Now let ll be a split

semi-simple Lie algebra over 0 as in Theorem 5 and let l l t be a

finite-dimensional irreducible 8-module. We assert that l l t is abso'

lutely irreducible. Thus we know that 9jt is an e'extreme module

wi th genera tor r such tha t xe t :0 , i :1 ,2 , " ' ,1 , and xh : l (h )x ,

h e b where A is a dominant integral linear function on sJ. we

know also that the weight space 9J? r corresponding to I coincides

with Ax. Consider lltr as module for 8p. Since this is finite'

dimensional and ,8r, is semi-simple, lltp is completely reducible.

Every irreducible submodule tt of gltp decomposes into weight

modules relative to 0r. Hence every weight module for Q. in 9ltp

can be decomposed into submodules contained in the irreducible

components of a decomposition of 9lt" into irreducible 8p-modules'

In particular, this holds for (9Jt.r )r : Px. Since this is one dimen'

sional it follows that r is contained in an irreducible submodule

It of !ltp. Since r generates lJtp we have Tl : lJlp and lltr is irre-

ducible. This proves our assertion on the absolute irreducibility

of !lt.

5. Exietenee of Et and Eg

In $ 4.6 we established the existence of the split simple Lie alge-

bras of the types corresponding to every Dynkin diagram except

E, and Es. (Our method-an explicit construction for each type-

admittedly was somewhat sketchy for the exceptional types Gr, F,

and Er.) An alternative procedure based on Theorem 5 is now

LIE ALGEBRAS

available to us. This requires the verification that the Cartanmatrices (Au) obtained from the Dynkin diagrams sati$fy conditions(a), (F), (r) of $ 1. We shall now carry this out for the diagramsE and Er and thereby obtain the existence of these Lie algebras.We remark first that in any such verification (a) iis immediateonce the matrix is written down and (P) is generdlly easy. Toprove (7) one displays a finite set of vectors in Si *trictr span 0iand is invariant under the Weyl reflections S",. This will provethe finiteness of the group W generated by the S"n.

Ea. The Cartan matrix is

(15) (Ai) -

One can see that det (A;1) : l, so (9) is clear. A[so (a) is ap-parent. We introduce the vectors

f,, : 3(a, * a, + ds * dt * rrr) * 2au + dt *',.de

f,z : 3(az * a, + dt * at) * Ztut + dz * de

f,s :3(ds * a, + a) * 2*u+ dt * de

h : j(ar.1 ar) * 2au * at * ae

l r : 3 d r l Z a o * a r * a a

] a : Z a e * a r * a a

f , t : - d e * d z * d e

. l e : - d o - Z a z * a e

where (dr, dr, . . ., a,a) is a basis for an 8'dimensional vector spaceover the rationals. It is immediate that the (ft form a basis.We write Si for the Weyl reflection S', defined by (1). We canverify that Si, L < i 3 7, permutes li and li+' and leaves the other

, l ; S e : i r t * ( i u * , l r * , l e ) , 1 < i < 5

. i c s e : t ( , t . - 2 h - z A E )

2 - L 0- 1 2 - t

0 - 1 2 -0 0 - 10 0 0 -0 0 00 0 00 0 0

0 0 00 0 00 0 00 0 0

- 1 0 - 12- r 0

- 1 2 00 0 2

0 00 01 02 - LL 20 - 10 00 - 1

li fixed while

VII. CLASSIFICATION OF IRREDUCIBLE MODULES 225

irS, : +(-2),6 + ^7 -U,B)

i rs r : * ( -2^6-2 f ,2 * , la )

Let 2 be the following set of vectors:

l i - h , t ( t i * \ * ) * ) ,

: t ( i r * t i * l * * h * l ' ' * l " ) ,

*.(2),i* ir ' * lr * h+ i- + l" + lp + lq)

where the subscripts are all different and are in the set (1, 2,''',8).

It is easy to see that these vectors generate the space. Since the

Si, i = ? , are permutation transformations of the I's it is clear

that these Sr map J into itself. One checks also directly that Ss

leaves .5 invariant. This implies that the group W genetated by

the S; is finite.Et. Let a,1, dzt '' ', ao ba a simple system of roots of type Ea in

the split Lie algebra Es. The matrix (Aii - (Z(ai, al)l(ai, a)),j, k : 2, . . ., 8 is the Cartan matrix Ez. This satisfies (a) and (0).

Moreover, (r) holds since the group generated by S"r, '' ', S", is

finite; hence the group generated by the restrictions of these map-

pings to the subspace of 0f spanned by ar, ''', dt is finite. Thisproves the existence of ^8.

A similar method applies to Ee. We remark also that it is easy

to see that if e;,f;,h;, i:1, " ',8, is a set of canonical generators

for Es, then the subalgebra generated by €i,f i,ht, i :2, " ',8 is

Er. (Exercise 1, below.)

6. Basic irredueible modules

Let 8 be a split finite-dimensional semi-simple Lie algebra over

a field of characteristic 0, and, as in the proof of Theorem 5, letllli be the finite-dimensional irreducible module correspondjng to

the dominant integral linear function li such that l;(h): dii,

i , i :1,2, . . . ,1. I f r ; is def ined as in the proof of Theorem 5 and

r is as in (14), then the argument used in $4 to prove absolute,tLl

irreducibility shows that the submodule of ffit, Om I

. . .4 f f i r8 . . .4 l ] t r genera ted by r i s the i r reduc ib le modu le cor re -sponding to A - 2 m;Ar. The problem of explicitly determiningthe irreducible modules of finite-dimensionality for 8 is thereforereduced to that of identifying the modules fi6, which we shall

226 LIE ALGEBRAS

call the basic irreducible modules for 8. We shall now carry this outfor some of the simpte Lie algebras. Others will be indicated inexercises.

At. As we have seen in $4.6, Ar is the Lie algebra of (/+1) x (/+1)

matrices of trace 0 over O. lf (e;) is the usual m4trix basis forthe matrix algebra A*, and the Cartan subalgebrh and simplesystem of roots are chosen as in S 4.6, then a se{ of canonicalgenerators corresponding to these is

€ t : e z t , 0 z : € g 2 , " ' r 0 t : € I + r , ,

( 1 6 ) f t : - o n , f r : * o z s t ' ' ' , f t : - o t , t + t

h r : € t - € z z , l l r : € z z - € g s t ' ' ' , h t : € L L - V I + r , L + r ,

The Cartan matrix (A;r) is given by (38) of Chapter IV and we

have A; t :2 , A ; * r , ; : - ! : A i , * r , A ; i :0 OtherWise ; Th isCan bechecked by calculating lethil: A1$r, using (16). The simple rootd.i is specified by: at(hi): An. The Weyl reflfction $at is

E - E - € ( h ) a t iWe consider first a representation of At by the IJie algebra of

linear transformations of trace 0 in an (/ + l)'dimdfrsional vector

space !lt. We can choose the basis (ur, ur, '' ', th+r) so that %;hi:Lt.;, tJt+rht= -ui+rt t l ih;:0, j + i, i + L. Then we havg uth - lt(h)ut

where the weights l; are given by the table:

' l r ( h ) : 0 , i +(17)

t , (hr) - 1

t l r (h+r ) :

1 ,

tr,(h)- l , i

- 1 lr(h) : I

1 + 'A* r (h ) - - 1 , A t * r (h ) : 0 ,

It follows from this and the table for the

A' i .Sq : A i+ r A*rSr,t -- tli , ltSq - A5 if j +

- + I

that

+1

Thus s"o interchanges A;, and, lr+r 'od leaves fixed lthe remaining

weights li. The group of transformations of the weights A;generated by these mappings is the symmetric grodp on the I + I

weights.Now let r be an integer, L S r s /, and form thp z-fold tensor

product U[ I tn I ... I iln. Let 9Jt, denote the subspace of skew

symmetric tensors or r-aectors in tn I ' ' ' I tX' By definition, this

is the space spanned by all vectors of the form

= }

t ! i r 8 - t r , A . . . I ! ; , , ,(18) lyr , yr ,

(2t)


where the summation is taken over all permutations P - (iiz "' i,)

of (1, 2, .. ., r) and the sign is * or - according as P is even or

odd. It is easy to see that if (ur, ttr,' ' ', zr+r) is a basis for fi :

fir then the (tl') vectors

( 1 9 ) f u i r , u t r " ' , u t r l , i t l i r <

form a basis for IJl". lf. a e 8 : .A,, then

(20) Ur , !2 , " ' , ! ' 7a : l ! ' a , !z ' " ' ' y " l * I ' y "y 'a ' i s ' " " ! ' l

+ " ' * [Y t , !2 , " ' , !&7 ,

which implies that !Il" is a submodule of !n I . . . I st. we may

suppose that ufu * ii(h)ui. Then (20) implies

l u l r , u l r ' ' ' , u l r ) h

: ( L i r * A i r + " ' * t L ) f t 4 i r , c t i r , " ' , u t r l '

Hence the basis given in (19) consists of weight vectors. We have

trh:0, which impl ies that At+r: - (A, , + " ' , - 1) . Onthe other

hand, (17) imp l ies tha t h* . . . *Ay- ] r where ) , * (h ; ) :d ik , i f

I < k < 1 . H e n c e t l u " ' , 1 1 i s a b a s i s f o r 0 i . l f i , : l i 1 , t h e n

l i r + " ' + t l t r - Ql*r+ " ' * 1*rr"-rr) whet l - -kv " ' , kr- ,"- r , is the

complement of (i ,, . . l, i,-r) in (L,2, ' ' ', '). It follows from this

t h a t i f l i t + " ' + [ t , A l r * " ' + A i ' , w h e r e i r < " ' 1 i ' a n d

ji < "' 1 i i ,, then ir : i 'r, ir: iL, " ', i , - i ' , . We now see that the

weight space corresponding to the weight Att* /7iz+ "'+ 4!, i"

spanned by fr4ir, etir, ''', uirl and so is one'dimensional' Let It be

a non-zero irredlucible submodule of fi'. Then It contains one of

the weight spaces corresponding to, say, n - lir + "' a 1i,' On

the other hand, if A' : Ai', *''' + Ait, is any other weight, then

there exists an element S of the group generated by the S'n such

that At -- /15. Then, by Theorem 4.1, At is a weight of F in Tt

also and lud, - . ., u1l € Tt. Thus It : Sl" and so !il|' is irreducible'

We have seen tha t h* . . .+A, : ,1 " and th is i s a dominant in te -

gral linear function. It is clear from the definition of lyr, yr,' '', !,7that [yr, !2, . ' ' , !,f : 0 if two of the y; are equal. This can be

used to check that lur, ttr, ' ' ' , u,le; : Q, i -- 1,2, ' ' ' ,1' Since

fur, " ', urfh : (A, * 1z + "' + l,)fur, " ', x4,7, srl" is z'extreme with

maximal weight i,. We therefore have the following

Tuponnru 6. Let At be the Lie atgebra of (/ + 1) x (/ + 1) matrices

ouer A of tuace 0 and let Q be the Cartan subalgebra af diagonal

228 LIE ALGEBRAS

m a t r i c e s i n A t , h t : e u - z i + r , i + r , i : L , 2 , . , . , 1 . L e t W r , t S r S l ,be the space of r-uectors. Then m" is an irreduciQle module forAr and it is the irreducible module corresponding to the dominantintegral linear function ),, on b such that ^,(h,) - \ l,(h) :0 ifj + r .

Bu lZ2. If we use the simple system of:roots of $4.6 f.or Btwe obtain the canonical generators

a i : ? i . r z , i + t - € t + t + l , r + d + z , i : 1 , . . . , / : 1

0 I : 0 r , t + r - e z t + r , r

( 2 2 1 f ; : € t + ; + 2 , t + i + r - € i + r , i + 2 , i - 1 , " ' r l - L

f 1 - Z ( e t , z r + r - e * r , r )

h t : € i+ r ,d+ r - €a+ i+ r , t+ i+ r - e i+z , i+z

* € t + i + z , t + i + 2 , i < l - l

h t : Z ( e m , r + r - o z t + r , * + r ) .

For the Cartan matrix (A;) - (d.i(h)) we have the values Aii - 2,A t , i * t - - 1 : A 1 + r , t , i : 1 , ' . . , 1 - 2 , A u r . t - - 1 , A t , r r : - 2 , a l lother Ait : 0. In the representation of Bt by the Lie. algebra ofskew symmetric linear transformations in the n :21* 1 dimension-al space sl we have a basis (ur, ur, . . ., un) such that wrh : 0, ttzh :1 t (h)u2, " ' r t t * rh : Lt(h)u + i tJ*zh : - 1 1(h)u*1t . . . t t tz t+tk : - l t (h)u*+t ,

where the linear functions Lt(h) satisfy i

A r ( h ' ) : 1 , t l r ( h ; ) - 0 , i + l( 2 3 ) r l i ( h i ) - - 1 , A i ( h ) : l , t l i ( h ) : 0 ,

k + j - L , i , j - , 2 , . . - , 1 - L .

L t ( h t - t ) - - l , / 1 ( h t ) : 2 , L t ( h * ) : 0 , ' , k + l - 1 , 1 .

W e h a v e 1 r * . . . + / 1 , : f , , , r = l - 1 , / . r * . . . + 1 r 1 - 2 1 r , w h e r e]t(h,): dri, i, i : L, ' ' ' ,1. Hence the l; form a basis for 00.. TheWeyl reflection S"y i - 1, . . .,1 - 1, interchanges l7 and lis a;nd,leaves the remaining ,Cr, invariant. The reflection Ss, leaves fixedevery tl i , j - 1,...,1- l and maps h into -th. Hence the groupgenerated by these includes all permutations of the li and all themappings which replace any subset of the 4 by their negativesleaving the remaining ones fixed. It is easy to see that the groupgenerated by the S', is of order Ztlt and we shall shdw in the nextchapter that this is the complete Weyl group.

Letr be an integer, L3r<l-L, and let Sl , bq the space of

vII. CLASSIFICATION OF IRREDUCIBLE MODULES 229

/-vectors. It is easy to see that fuz, "', u,+rlh - l,(h)[urt "', ch+rl

and fur, " ' , u,+rlei : 0, i : L,2, " ' ,1. We assert that the cyclic

module generated by \ur, . . ., Lrr+r) is irreducible. Hence this is the

required irreducible module corresponding to the dominant integral

linear function ,1". In general, we note that if It is a finite'dimen'

sional a-extreme module for a finite-dimensional split semi-simple

Lie algebra 8, then Il is irreducible. Thus It is completely re'

ducible and the generator t corresponding to the highest weight

I spans the weight space of 0 corresponding to A. Hence r is

contained in an irreducible submodule It/ of tt. Since t generates

It we see that !t : It' is irreducible.The result we have just indicated on the determination of the

irreducible module corresponding to f,,, r : L, ''',1 - l, can be

improved. As a matter of fact, this module is the complete module

SL, of r-vectors. To prove this one has to show that !Il' is irre'

ducible. We shall not give a direct proof here but shall obtain

this result later as a consequence of Weyl's formula for the di'

mensionality of the irreducible module with highest weight .1,

which we shall derive in Chapter VIII. For the representation

with 1: f,, we shall obtain the dimensionality: ("'*'). Since this

is the dimensionality of !trt" it will follow that !Ul, is irreducible.

If we consider the space of /'vectors a similar argument will

show that this is the irreducible module for Bt corresponding to

the highest weight h * . . . + it :2k. Hence this does not provide

the missing module corresponding to the linear function )t. In

order to find this one, one has to consider the so'called spin repre'

sentation of. Bt. To give full details of this would be too lengthy.

Hence we shall be content to sketch the results without giving

complete proofs.we recall first the definition of the clifford algebra c(flL, (r, y))

of a finite-dimensional vector space IJI over a field O of. character'

istic not two relative to a symmetric bilinear form (x, y) on m.

One fo rms the tensor a lgebra T . :OIOm@f i rO " 'O! f t tO " ' ,

l lh:m8" '8m, r t imes, and one lets S be the ideal in E

generated by all the elements of the form

(24) x @ x - ( x , x ) L

Then C(m, k, 9): t/,F. (For details on Clifford algebra the readershould consult Chevalley [3] pp. 36-69, or Artin [1], pp. 18G193.)It is clear that S contains all the elements

230 LIE ALGEBRAS

( 2 b ) x Q J ^ Y + t O r - 2 ( x , Y ) l : ( r + r ) 8 ( r + J )- ( r * ! , n * y ) 1 - x & x t ( x , r ) l - y Q s ^ t * ( y , y ) 1 .

It is known that the canonical mapping of Dt into C1: C(Wt, (x, y))is an isomorphism, so that one can identify llt wit[ its image inC. We do this and we write the associative product in C as ab.lf (u, uzt . ., uo) is a basis for Dt, then it is known ithat dim C :

2" and, the set of elements

ul tu i ' . . . u l^" , € i : o , 1

basis f.or C. Since the elements (25) are in S it follows thathave the fundamental "Jordan relation"

(26)

i s awe

(27) r.! = L@y + !x) : (x, y)L

for any x, y e sln.

The algebra C is not a graded algebra. However, it is a gradedvector space in a natural fashion which we shall now indicate. Letxr, fr2,.'., r, e St. Then we define anr'f.old product f,trr, xr, "', x,leCinductively by the following formulas

(28)

where [ab]

r xzk-rt xzrf : llxt,

, JCzk, fizk+t1 - ffu

' ' ' , xzn-tfxz*f

" ' r lCzn)'|(zk+t

txLllxulxu

- X t

and L@b + ba) f.or a,b e C.

Lpurur 4. The product lxu ' ' ', r,7 - 0 if any two',of the x; ale

equal. Hence for any permutation P : (it, ir, " ' , i ,) of (I,2, " ',/),

lxrr , . . . ,x;r f : t f , icr ," ' ,h l where the s ign is + or accordingto wheth.er P is euen or odd.

Proof: Since lxy . .., hl defines a multilinear fqnction of itsarguments the second statement is a consequence of the first. Weprove the first by induction on r. We have [xr) =0. Let r > 2a n d s u p p o s e ) c 1 s : f , : t , w h e r e L < k 1 l { r . I f l < r t h e n

lxr ," ' ,x t ) :0 by the induct ion hypothesis and [rr , " ' ,h l :0 fo l -lows from (28). Hence we may assume that I = r. Also the induc-

tion implies that Lxr, " ', x,-t l: t [ " ' x] so we have to show that

Ix, , . , rr-z)cxf : Q. There are two cases: even r a$d odd r andthese will follow by proving the following relations: la'x, rl : 0and lax).x:0 for a€C, x e m. For the first of thbse we have

VII. CLASSIFICATION OF'' IRREDUCIBLE MODULES 231

lL@x * ra), xl: l@x' + xar - xax - x'a)

: l(axz - r'a) : *(r, tc)a - *(x, x)a - 0 ,

and the second follows from

laxl. r - \,(ax - xa)x * Lzr(ax - ra) - hor' - L"'o - 0 .

This completes the Proof.

Louu.r, 5. Let fl|, denote the subspace of c spanned by all the

e l e m e n t s [ r r , . . . , x , f w h e r e 0 3 r 3 n - d i m f i L a n d $ [ o = 0 1 . T h e n

d i m l l l , . : ( D a n d C : O L O ! f t , O i l l ' O " ' O D t " .Proof: If. (n,4, . ., c4,) is a basis for !fr|, then the skew symme'

try and multilinearity of fx', ..., h) imply that every element of

flL,, r 2 1, is a linear combination of the elements [z;r, 14i2, ''',24irl

where i, 1 i,orthogonal: (ui, ut) : O if. i + i . (It is well known that such bases

exist.) The condition (u;, u): 0 and (27) implies that utui -- -tliui.

Hence (urr " ' uir-t)tt i , - ttt;r(t '4ir " ' u;r-r) if i t < i, 1 " ' < i, and the

sign is l- or - according as r - ! is even or odd. The relationjust noted and the definition of lxr, " ' , r,l imply that fu4, " ' , u+f :

,rcuit.,. qir for (u;) an orthogonal basis. Since the 2" elements of

the iorm uir "' c4;r, it 1 iz 1that for a fixed r the elements furr, "', %;rf, i, 1 " ' 1 i,, form a

basis for m". Hence dim fi, - (l), and it is clear that c -

at@s l ,ou t ,G) . . .os1" .Let x, y, z Q, fl|. Then

(r .y) .2 - x.(y 'z) : (x, y)z - (Y, z)x .

On the other hand,

(x-y).2 - tc.(y. z) - I@y * yx)z + *z(xt + ytc)

- Lx(yz * zY) - *(Y, * zY)x- l@yz * yrz * zxy * zyx - ryz - xzy - Yzr - zlr)

- lfy[xz)l .

Hence we have the relation

(29) *tylxzll: (x, y)z - (Y, z)x .'We

can now prove

Tnponuu 7. The space ffiz ,s a subalgebra of the Lie algebra

Ct. If (x,y) is non'degenerate, then this Lie algebra is isomorfhic

LIE ALGEBRAS

to the orthogonal Lie algebra determi'ned by (r, y) in fl|.Proof: The space !ft2 is the set of sums of Lie products lryl,

r, y € Sl. By (29), we have

(30) , IlttJlxzll : lfy[rz]l, ll + lv,[tlxz)]l: 4(x, y)[ztl - 4(v, z)lxt) * 4(x, t)lyzl - 4(t,,2)[yxl

Hence !illz is a subalgebra of Cr This relation and @9) imply alsothat St * Sl' is a subalgebra of Cz and the restiliction of theadjoint representation of !ft * !il}, to the subalgebra Tl, has lfll as asubmodule. If R denotes this representation, then (X9) shows that

lxzlt is the mapping

(31) y -+ 4(x, y)z - 4(y, z)x

If. (r,y) is non-degenerate, then (31) is in the orthogpnal Lie alge'

bra. Moreover, every element of the latter Lie algebra is a sumof mappings (31). Hence the image under R of St, is the orthogonalLie algebra. Since both of these algebras have thel same dimen-sionality (il, R is an isomorphism.

The enveloping algebra of sl, in C(st, (r' y)) will lbe denoted as

C+: C*([?, (x,y)) . l f . (u, . . . , t4n) is an orthogonal ]basis for ! f l ] ,

then the elements 0t;1x1, i < 7 constitute a basis for !ilf2. Since zl :

(u;,ui)! and uiul:-tritrt if i+ j, it is easy to see that the space

spanned by the elements uar14iz "' ztizrt i, I i,

0,L,2, . . . , ln lz) is a subalgebra of C. Since i l ; ( ;z ' " t t izr :(uiru;r) "' (rr;r,-rtrizr), this subalgebra is contained irt C* and sinceit iontains ![t, if coincides with C+. It is now clear rthat C+ is the

subalgebra of even elements of C, or the so-called Second Cliford

algebra of. (x, y). Its dimensionality is 2"-r. The structures of C

and C+ are known. We shall state only what is rieeded for the

representation theory of. Bt, and Dt, For this the sym{netric bilinear

form (x, y) is non-degenerate and of maximal \ryitt index. If n -

2l + I, then C+ is isomorphic to the complete algebra G of linear

transformations in a 2'-dimensional vector space It over O. The

isomorphism can be defined explicitly in the followihg way.Let (n,4zt . . ., t4*) be a basis for lll of the type tused to obtain

the matrices for Br ($ a.6). Thus we have (ur, ut) = l, (ili, ut+;) :

| : (u*i, or) if i :2, ' ' ' ,1* 1 and all other products are 0. Set

u.: ttf iJi4. t iu.: tt( l i+r+r, i :2, " 'r l . Then it is eAsily seen that

the ai and w; generate C+, that oil)i : -t)ioi, a? :0, It);11)t : -tVirDit

w? :0 if t * i, and that the subalgebras generated by the a's and


the eu's separately are isomorphic to the exterior algebra basedon an /-dimensional space. A basis for C+ is the set of elementsgiL ' O;rlDit " ' tfiis Where i, 1 i,

have the relations

(32)

The

(33)

0; l l ) ; * U);U; : - f ,

subspace Il sPanned bY the vectors

U t " ' U t l 0 i t " ' l V i s , i r 1 i r 1 " ' < i "

U ; U ) i : - N i D i , i f i + i

is a 2t-dimensional right ideal in C*. The right multiplications

by a e C+ in Tt give all the linear transformations of Tt and the

correspondence between a and the restriction to It of an is an iso-

morphism of C* onto the algebra G of tinear transformations of 9t.

The representation a-)a'*, ah the restriction to Il of aa, induces

a representation of the subalgebra ![ts of CI. Since St, is iso-

morphic to Bt this gives a representation of. & acting in It. The

Br-module Tt is irreducible and this is the irreducible module with

highest weight ir which we require. We shall sketch the argument

which can be used to establish this.We note first that the matrix e;, i : 1,2, ' ' ',1 - l, in (22) can

be identified with the linear transformation r -'+ (x, ttr++z)tti+r -

(x, u;*r)u;+r+s relative to the basis (t6r, r4r, ' ' ' , iln) which we have

chosen. Similarly, 0tc?Yr be identified with the transformation r-+(xru1)ut+r - (x, t t t+t) t r r , hr , i - - 1r2, " ' ,1- 1, wi th t r - -+(xr0a*t+r)1r;+r -

(r,u;*r)ut+i+r- (x,ut+inr)ur+z* (x,lt;+z)ttt+;+2, h with r- '2(r,u*+r)tL+t-

2(r, unt)uzt+r. In this isomorphism with Slz (cf . (29)) we have

(34)

e i - + * l u * ; + 6 l t i + t | : b u ; u t i * r , i - 1 , " ' , / , - l

et - *lur, ur+rl : brr*,

h6 --+ llulai+r, tri+r) - llut*r*r, ?t*zl: $(u*i - nt+(t;+r) ,

hr , -b lur t+r , th+L) - L * uwt

L e t z : t ) r . . . u e T t . T h e n z € 1 : 0 , i : L , " ' , 1 , z h ; : Q , i -

t, . .- ,l - ! and zht -- z. It follows that the cyclic Br-module gener-

ated by z is the irreducible module corresponding to ,lr. It can be

shown that this is all of Yt. We shall call Tt the spin module for

Br We can summarize the results on Br in the following

Tsnonpn 8. Let Bt, I 2 2, be the orthogenal Lie algebra in a(21 + t)-dim.ensional space defined by a non'degenerate syrnmetri,c

294 LIE ALGEBRAS

bilinear form of maximum Witt index. Let the basis)for a Cartansubalgebra b of Bt consist of the h; , i :L,z, . . . ,1 of (22) and let)'i be the linear functi.on on 0 such that ),i(ht): d,it. Then theirreducible module for h with highest wei,ght h, i, - 1,...,1-tis the shace Yti of j-aectors. The irreducible module with highestweight )v is the sfin module Tt defi.ned aboae.

Gr. If (at, az) constitutes a simple system ofthe Cartan matrix is

(35)

so that we have

a z ( h r ) : - t , a z ( h z ) - 2 .( 3 6 ) a r ( h ' ) : 2 , a r ( h r ) - - 3 ,

We have seen in $ 4.3 that the positive roots are d1, dz, dr * dz,dr * Zatz, ar * Saz, 2a, * 3a* The highest of these is Zal * 3otz.We have (2a' + Satz)(hr) : L, (2a' + 3a)(hz): 0 so th1t Zaq * 3az isthe linear function ,lr such that ),'(hr) - l, lr(h) : Q. Since G, issimple the adjoint representation is irreducible. Thus I itself isthe irreducible module whose highest weight is l'. lWe shall shownext that if. G, is represented by the Lie algebra ofi derivations inthe split Cayley algebra G then the representation [nduced in theseven-dimensional space Oo of elements of trace 0 is the irreduciblerepresentation corresponding to the linear functiorl ^z such thatAz(h') - g, ]r(h) : L. We shall show this by prdving that thedimensionality of any module ![t for Gz satisfying frLG, * 0 is 2 7,and if the dimensionality is 7, then the module is irreducible andcorresponds to iz.

We first express the roots in terms of the basis l[, ]r. Thus wehave ah : 2h - 3)r, dz : - h * 2]2, by (36). Hende the positiveroots are

+ 2 4 2 ,+ 3,,12 ,

If 1 : TtrL * mzf,z € Oil and Sr = Srr, S, = S", are the Weyl reflec-tions determined by a, and 42, r€sp€ctively, then lSr = A - l(h)ar:A - rnt(zlt - 3,ir) : -oorf,r * (m, + sm)]zand lSz : (mp * mz)),r-r/tz)2.If I is the highest weight of 0 in the module Ut, then the mt arenon-negative and not,all 0 and the following lineart functions areweights of 0 in ![|:

(-3 -'r)'roots for Gr, then

j

(37)) ,1


I : t f l r f , t * m212, 151 : - lr tr f , t * (mz + 3rn)]2,

l52: Qn, + m)L - 'tftzf,z, lS,S2 - (Zmr * m)),, - (m, + 3m)lz,

lszsr : -(mr * rn)h * (3mr * 2mz)),2,

lSrSzSr - -(2mt + rnz)h I (3mr + 2rnz)f,2,

lSrS,Se - (Zmr + /nz)^L - (3mt + 2m)f,2,

l(S'Sr)' : (m, + nh)h - (3mr + Zrnz)f,2,

l(S'S')' - -(Zmt * mz)),, + (3mt + rnz)^z,

l(S'Sr)'Sr: -(//h * m)1,, * mzf,z,

l(SrS,)'Sz: I,rttf,r - (3mr * mz)f,2,

l(S,Sz)t : -tltrf,r -- lrtzf,z: l(SrS,)t .

lf. m, > 0 and wz ) 0, then this list gives twelve distinct weights

which means that dim !n > 12. If. mz: 0 the list gives six distinct

weights: frtrJr, -rrtrf,t, Zmr)r - 3mrAr, -lltrf,r * 3m1A2, -2mJt * 3mJz,

lt;r|t - 3mJz. If mr: 1, then St contains a Submodule isomorphic

to 8 and so dim !n = 14. If lrh ) L, then since lr is a root and

the ir-string containing l/tJr co\tains also -tthf,t there are at least

2m, + 1 = 5 weights in this string. This adds three new weights

to the list and shows that dim m = 9. Now let mr: g. Then the

following six weights are all different: ?,rtzf,z, mzf,t - frtzf,z,-Il lz\t * 2m2)'2, l l lzAt - Zmzf,z, -tl lzJr * mzf,z, -fl lzf,t. lf mz: t the

,iz-string containin g tflJz contains 0 as weight so that dim flt Z 7 '

lf. mz > 1 then the ,lz-string containing fttzf,z cotrtains at least five

weights and this implies that dim tn > 9. We have therefore shown

that dimln = 7 and dimfi :7 can hotd only if the highest weight

A : f,2. Since there exists a module 6o for Gz such that dim Oo : 7

it follows that Go is irreducible and its highest weight is l'.

Tnponpu 9. The i,rreducible module for 8, : Gz with highest weight

,1, is 8 itsetf. The irreducible module for Gz with highest weight ),2

is the seaen-dimensional module 6, of elements of trace 0 in the split

Cayley algebra 6'.

Exercises

In these exercises we follow the notations of this chapter: I is a finite'dimensional split semi-simple Lie algebra over a field of characteristic 0, 0a splitting Cartan subalgebra, ar., ft,lar canonical generators for 8, tl the uni'versal enveloping algebra, etc.

l

236 LIE ALGEBRAS

1. Let e t , f t ,h i , i :1 ,2 , - . . ,1 , be canonica l generators fo i tJ . Show thatt he suba lgeb ra 8 r gene ra ted by t he e lemen ts e . s , f J , h i , j : 1 ,1 . . . , k , kS l i sa split semi-simple Lie algebra. Show that .E6 is a subalgdbra of Ez, and.Ez is a subalgebra of .Ee.

2. Show that lJ has only a finite number of inequivalent itreducible mod-ules of dimension 5 N for N any positive integer.

3. Let I be the kernel in tt of the representation determined by a moduleIJI and let l(') denote the kernel of 9Jt @ . . . I tJl, s-times. Show that afinite-dimensional lJl can be chosen so that if S is any ideal in ll of finiteco-dimension, then there exists an I such that X(s) g 9.

4. Let S be the orthogonal Lie algebra defined by a non-{egenerate sym-metric bilinear form in an n > 3 dimensional space !Jl. Provt that the space9 J l " o f f - v e c t o r s i s $ - i r r e d u c i b l e f o r L = r S l i f n : 2 1 * l a h d l S r S l - 1if. n = 21.

5. i)etermine a set o{ basic irreducible modules for Ct, I > 4.6. Show that the minimum dimensionality for an irreduclble module $?

for .Eo such that NtEa * 0 is 27. Hence prove that .&'o and ,Bo, and .oo andG are not isomorphic.

7. Prove that the spin rep;esentation of Dt in the second Clifford algebra(of even elements) decomposes as a direct sum of two irredpcible modules.Show that these two together with the spaces of r-vectors I S r 3I - 2 arethe basic irreducible modules for Dt, I > 4.

8. Show that a basis (gr, gz, ..., gt) can be chosen for S sdch that [e$i=6 i i e i , I f c l t l : - du f i . Le t h :2Z lg t - Zp&r whe re t he p ' LeO. Le t e=}Tpr where every 7i;* 0 and let / : Zft 'ptf t Prove that (e,, f ,h) is acanonical basis for a split three-dimensional simple Lie algpbra 0. Provethat adg$ is a direct sum of I odd-dimensional irreducible representationsfor S. (The subalgebra S is called a "principal three-dimensional subal-gebra" of 8. Such subalgebras play an important role in the cohomologytheory of !. See Kostant I3l.)

9. Determine a S as in 8, for .4r. Find the characteristic roots of adghand use this to ob.tain the dimensionalities of the irreducibld components ofadg0.

f0. Let Sl be a finite-dimensional module for 5J, let !+ be the (nilpotent)subalgebra of 8 generated by the et and let I be the subspdce of Dl of ele-ments a such that el =0, I € !+. Show that dim8 is the r iumber of irre-ducible submodules in a direct decomposition of lJl into irreducible sub-modules. (This number is independent of the particular ddcomposition bythe Krull-Schmidt theorem. See also S 8.5.)

ll. Let lll and Il be two finite-dimensional irreducible mddules for 8 andlet !Ji* be the contragredient module of $Jt. Let .E and S, fespectively, bethe representations in [Jl and !?, 8* the representation in FJI*. Show that

VtI. CLASSIFICATION OF IRREDUCIBLE MODULES 237

the number of submodules in a decomposition of Sl* @ It as direct sum of irre-

ducible submodules is dim 8 where I is the subspace of the space G(m' n)

(= fi* I m by $ l) of linear mappings of 9[t into [t defined bv

8: {Z lZ e G(Vt ,n) ,FZ : Z ls} .

tZ. (Dynkin). If Ilfu and $lz are finite-dimensional irreducible 8'modules

with highest weights h and Az and canonical generators sl and rr, respec'

tively, then Sls is said to be subord,innte to Wt if nru:0 implies ozll=A

for every a in the universal enveloping algebra 1l- of the subalgebra 8- of

8 generated by the /a. Prove that $?z is subordinate to $h if and onlj' if

Ir: tlz * M where M is a dominant integral linear function on s*. (Hi'nt:

Note that if tt is the finite-dimensional irreducible module with highest

weight M and, g is a canonical generator then II|1 can be taken to be the

submodule of Sts@It generated by rz8A.)

lg. Note that the definition of subordinate in 12 is equivalent to the fol'

lowing: there exists a ll-homomorphism of IIh onto ![lz mapping cr onto tr2.

Use this to prove that if a finite-dimensional irreducible module St with

Stg + 0 has minimal dimensionality (for such modules), then III is basic'

CHAPTER VIII

Characters of the Irreducible Modules

The main result of this chapter is the formula, due to Weyl, for

the character of any finite-dimensional irreducible module for a

split semi-simple Lie algebra 8 over a field of characteristic 0. If

the base field is the field of complex numbers and R is a finite-

dimensional irreducible representation, then the character of R is

the function on a Cartan subalgebra O defined by

( 1 ) x ( h ) - t r e x P h & ,

where , as usua l , expa:1* a+ az lz ! + " ' . I f 6 i s a connected

semi-simple compact Lie group, then (1) gives the character in the

ordinary sense of an irreducible representation of O. Thus in this

case b corresponds to a maximal torus and it is known that any

element of 6 is conjugate to an element in this torus. Then (1)

gives the values of the characters for elements of the torus. We

know that hn acts diagonally, that is, a basis can be chosen so

that the matrix of lzR is

(2) diae {A(h), M(h), ' ' '} ,

where /l(h), M(h),. . . are the weights of b in the representation.

Then the matr ix exphRis diag{exp A(h),expM(h)," ' } ; hence

( 3 ) x(h) : Znn exP M(h)

where r4p is the multiplicity of the weight M(h), that is, the

dimensionality of the weight space !ItN.One obtains a purely algebraic form of the definition of the

character x(h) by replacing the exponentials exp M(h) of (3) by

formal exponentials which are elements of a certain group algebra.

Weyl's formula gives an expression for the character Xt of the

finite-dimensional irreducible module with highest weight A as a

quotient of two quite simple elementary alternating expressions in

the exponentials.Weyl derived his formula originally by using integration on

i23el

240 LIE ALGEBRAS

compact groups. An elementary purely algebraic rqethod for ob-taining the result is due to Freudenthal and we shhll follow thisin our discussion. A preliminary result of Freudenthal's gives arecursion formula for the multiplicities zr. Weyl's formula can beused to derive a formula for the dimensionality of the irreduciblemodule with highest weight l. It can also be used to obtain theirreducible constituents of the tensor product of trl,'o irreduciblemodules.

7. Some propertiee of the WeUl group

In this section we derive some properties of thd Weyl groupwhich are needed for the proof of Weyl's formuli and for thedetermination of the automorphisms of semi-simple Lie algebrasover an algebraically closed field of characteristic 0 (Chapter IX).As usual, 8 denotes a finite-dimensional split semi-sijmple Lie alge-bra over a f ie ld @ of character ist ic zeto, e; , f ; ,h; , , i - -L,2, . . . ,1,are canonical generators for 8 as in Chapters IV and VII, and Ois the splitting Cartan subalgebra spanned by the &r. Let O* bethe conjugate space of 6, OI ttre rational vector spdce spanned bythe roots of S in I]. If. a is a non-zero root therl the Weyl re-flection is the mapping

( 4 )

( 5 )

S,: t -+ { -z-(E' al old, a)

in 6i. Here (E,d is the positive definite scalar product in Oidefined as in $ 4.1. The mapping S' is characterized by the pro-perties that it is a linear transformation in OJ whigh maps a into- a and, leaves fixed every vector orthogonal to a. The reflectionS, is an orthogonal transformation relative to (t, z) aild S' permutesthe weights of 6 in any finite-dimensional module for 8. The S"generate the Weyl group 17 which is a finite group. lf. T is anorthogonal transformation such that aT is a root for some roota * 0, then a direct calculation shows that

Sro

In particular, this holds for every T e W. We noti also that if ais a root then - a is a root and it is clear from (4) ithat S': S-o.

The roo ts d , i t i :L ,Z , . . . ,1 , such tha t a i (h ) : A i r , (A i theCartan matrix, form a simple system. The notion of a simple

VIII. CHARACTERS OF THE IRREDUCIBLE MODULES 247

system of roots is that given in $ 4.3 and depends on a lexicographic

ordering of the rational vector space OJ. We recall that / roots

form a simple system if and only if every root a : -t- ()&ta;) where

the k; are non-negative integers. This criterion implies that if

ft : {ar, dzt . . ., a,t} is a simple system and ? is a l inear transform'

ation in SJ which leaves the set of roots invariant, then rT:

{arT, " ' ,dt ?} is a s imple system of roots. The set P of posi t ive

roots in the lexicographic ordering which gives rise to the simple

system z is the set of non-zero roots of the forrn Dk.a;, h >- 0.

On the other hand, if the ordering is given, so that P is known,

then z is the set of elements of. P which cannot be written in the

form 9 + r, 3,r e P. Thus P and rc determine each other. Any

simple system of roots defines a set of canonical generators et,fthi(cf . S a.3).

Ler r lM.c , 1 . Le t n : {d r ,d2 , . " , * , } be a s imp le sys temof roo ts and

let d be a positiue (negatiue) root. Then aS", > 0 (<0) i.f a #-at

a n d a S . . , < 0 ( > O ) L l a : a ( o : - c , ) .

Proof: If a is a negative root, then - a is a positive root and

the assertion on a will follow from that on - a. Hence it suffices

to assume a > 0. We have a - lhpi, ki Z 0 and

(6) ogai Eu,o,* (u, -*#)* ' .

lf. a + a;, then since no multiple except 0, t a of. a root is a root,

a * k;ai and so k1 + 0 for some j + i. Then the expression for

aS,. has a positive coefficient for some a1 and hence aS'o > 0. If

a: &it then aS, . : - a; 10. This completes the proof.

Our results on I;7 will be given in two theorems. The first ofthese is

Tnponpu 1. If rc : {dr, dz, . . ., at} is a simple system of roots,then the Weyl group W (relatiue to O) is generated by the reflectionsS';, dt€tr. If a is any non-zero root, then there erists ai€r andS e I7 such that a5S: a.

Proof: Let W' be the subgroup of 17 generated by the $cr.We show first that any positive root a has the form aiS, ai € n,S e W . We recall that if a - }k;a; then X&r is the level of a,

and we shall prove the result by induction on the level. Theresult is clear if the level is one since this condition is equivalent

242 LIE ALGEBRAS

to a e tt. If Xft; ) 1, a 6 r, and there exists ani, ar such that(a, a) > 0. Otherwise (d, o) { 0 for all i and sinlce a - Zhrai,h i>0 this gives (a,a) <0 contrary to (a,a) > 0.r Choose 4; Sothat (a, a;) ) 0. Then I : a,Sq ) 0 by Lemma 1 land we have

B : Zi+ikp1 * @t - l}(a, a)l(a;, a;)f)a;. Since (a, a,r| > 0 it is clearthat the level of F is lower than that of. a. Hencer the inductionshows that F : d iS' , d i e r , S' e W'. Then a: dtSlsdi as re-quired. Since a: atS where S e W', S' : S-tS'rSte W' . SinceS, : S-, this shows that every S, e W' and so Wl - W. It re'mains to show that if a is any non-zero root then at: a;5, dl € tc,S e 17. This has been shown for a > 0. Since &So: - a theresult is clear also for a 10.

Tsnonpu 2. Let T and, TE' be any two simple systerns of roots.Then there exists one and only one element S e W such thatzrS : r '.

Proof: Let P and P', respectively, denote the sets of positiveroots determined by r and, r'. It is clear that P and P' containthe same number 4 of elements, since this is half the number ofnon-zero roots. It is clear also that P : P' if and lonly if 7r : rc'and if P + P', then z Z P' and n' * P. Let r be the number ofelements in the intersection P n P'. If / : q, P 4 P' and S : 1satisfies zrS : z', so the result holds in this case. We now employi n d u c t i o n o n q - r a n d w e m a y a s s u m e r < q , o r P + P ' . T h e nthere exists a; € n such that at d P' and hence - di -- atS't e P' .I f 9 e P n P ' , F S , n € P , b y L e m m a 1 . H e n c e 0 S [ , e P n P ' l " r .Also er: (- ar)Sr; e P n ,rSoi. Hence P n P'Sr. contains at leastr * | elements. The simple system corresponding td ,r Sor is z'Sr.so the induction hypothesis permits us to conclude lthat there ex-ists a T e W such that rT- rrSar Then S: ?S"t satisfies zS:rc'. This proves the existence of S. To prove uniquFness it sufficesto show that if S e W satisfies nS : r, or equivalently, PS: P,

then S: 1. We give an elementary but somewha,t long proof of

this here and in an exercise (Exercise 2, below) we shall indicatea short proof which is based on the existence theofem for irreduc-ible modules. We now write S; : S". and by Tfreorem 1, S -

SnrSor " 'S0 . , i i :1 ,2 , " ' ,1 . We cannot have m: ! s ince adrs t r :- air (, tt and if rn :2, then drrSlr : - a\ ?r'td sir[ce (- a;r)St, :

a;,S;,S;, ) 0, iz : i, by Lemma 1. Thus S : S3, ! 1' We now

suppose rn > 2 and we may assume that if T: SirSir"'Si", / 1ffi,

JVIII. CHARACTERS OF THE IRREDUCIBLE MODULES

and, rT: zr, then T: !. Let S' : S;,Srr' ' 'Su,'-r, so that S : S'St-'

S i n c e S + S r * , S ' + 1 a n d s i n c e S ' i s a p i o d u c { o f m - l S i , P S ' + P

by the induillon hypothesis. Then there exists an di € n such

that arS, < 0. Sincl aiS/Sr- ) 0, Lemma 1 implies that *iS/ :

- ai,n Similarly, if g is any positive root such that FS' < 0, then

FS,S- > 0 implies that FS' : - airl: alS'i hence I : at. Thus

S' has the following two properties: o3st : 'd;m alfld PS' > 0 for

every positive I + ai. Set S;o - 1. Then aiSiq > 0 and

aiSioSr. ' "So--r : a iS ' - - c t ' in t< 0 '

Hence there exists a k, L<k<rr t -L, such that ar 'ScoS,r" 'S;*- , )0

and alSroSnr ' ' 'Sr* (0. Since a;S;s ' ' 'Sr*-r>0 and (au1S6' ' 'Si"-)Sr*:0 '

arS;0"'S,*-, - in*. Hence if we set ?: SroStr"'Sn*--r' then

T-tSr? -*So* or S;T - ?Srr. Then, if T' :,Sr**t"'9t-: l for

h < m - r a n a T ' : 1 f o r k = m - 1 , S ' - ? S ; r T ' : S i T T ' ' H e n c e

TT ' :S lS ' . I f a i s a pos i t i veroo t *d i , then9-aSi isapos i t i ve

root * ott. Hence a,TT': aSrS' : PS/ > 0. Also aiTT : cr'SiS' :

(- ar')S' : d;*) 0. Hence nTT' : z and since TT' is a product

of m - | Sr;t the induction hypothesis gives TTt : I' Then

Sr.S' : 1 and so S/ : Sr and S - StrSi. Hence S: 1 by the case

m:2 which we considered before.

2. Freudenthal's formula

Let llt be a finite-dimensional irreducible module for 8 with highest

weight A: 1(h) a dominant integral linear function on 6. We

know that llt is a direct sum of weight spaces relative to 6 and

that the weights are integral linear functions on 6 of the form

1 - Zkiat, k; ? non-negative integer. lf. M - M(h) is any integral

linear function on 6 we define ttre muttiflicity np of M in !]t to be

O if. M is not a weight and otherwise, define rau : dim ![1,r, where

str is the weight space in IIt of b corresponding to the weight M-

We recall that for the highest weight A we have fl,t:\. Weshall

now derive a recursion formula due to Freudenthal which expresses

np ifr terms of the ftp, for M' > M in the lexicographic ordering

oiOil determined by the simple system of roots 't : {d,, dz, " ' ' a''}'

Let a be a non -zero root of .b and choose eo e 8a,, 0-a e 8-, so

that (€,,0-n): - 1 where (x,y): trad xady, the Kil l ing form on

8. Then we know ($ 4'1) that le'e-'\ : h' where' in general for

p e b*, ftp is the element of 6 such that (h, ho) - p(h). As in S 4.2,

244 LIE ALGEBRAS

let 8(') be the subalgebra 6 + Ae" * Oe-,. Then we know thatllt is completely redtrcible as 8'''-module (Theorem 4.1). we con-sider a particular decomposition of fi as a direct sum of irreduc-ible 8'''-submodules and we shall refer to these ast,the irreduci,ble8t't -constituents of Dt. Let rt be one of these. Thed we know that9t has a basis (!0, yr, . . ., !^) such that

l ih: (M - ia)y, , i : 0 , L , . . - , t f l

(7 ) ! ; o -o j

! ;+ r ,

We know also that

( 8 ) m : 2(M, a)l@, a)

where (p, o) - p(h,) : o(hp), as in $ 4.1. Equations (?) imply that

we note also that rt is a direct sum of weight spaces, that theweights are M, M - d., M - 2a, ..., M - md and the weight spaces\t*-r, : \Jlx-p, n Il are one-dimensional.

Let M be a weight of s in Dt such that M + d i$ not a weight.Thenthea-s t r ingof we igh ts in l J t con ta in ing M is M,M-d , . . . ,M-ma, where 7n :2(M,a) l@,a) . Le t 0 < p<m. tThen M- pais a weight and Wly-p, is a direct sum of the I[ - pa. weightspaces of those irreducible 8(''-constituents which have this asrveight. If 0 < p = (M,

") / (d, d), then these are the irreducible

8'"'-constituents having maximal weight M, M - &, . . ., M - pa.Let f f i i , 0<i <(M,a) l (a,a), denote the number rof i r reducible8''''constituents of highest weight M - ja. Then it is now clearthat

(e)

(10)

so that,

(1 1)

l l n - j o : l l l o * m , + * m i ,

l ' f l i : l la- ia - nN- 0-ru

If 0 S j S p s (M, a)l(a, a), then the weight space correspondingto M - pa in an irreducible 8'')-constituent having highest weightM - ja is spanned by the vector !p_i in the notatioh of (Z). Thedimensionality of this module is 2(M-ja,a)l(a,a)+l -(m-Zj)*L.

VIII. CHARACTERS

Hence we may rePlace I bY

lp-Plre| :

OF THE IRREDUCIBLE MODULES 245

to obtain

(12)

We can use this to compute trgqr-, ,e!neY, the trace of the

to T!tr-r, of. e!,e!. We remark that it is clear that

Wt"-rrLre, E Wtr-,r*r, ne| g lltr-po,

so that yftr_p, is invariant under e!,e|. By (12), the contribution

to trg4r- ,,e!re| of the m1 itreducible 8'(')-constituents with highest

w e i g h t M - i a i s

* ( f - i + t ) L f -m+ i ) ( a ,a ) ;t re t 2

hence,

trss, -n,e!,g* = *^*

o W (a, a)

: f,tn"-io - rtr-.i-,,') W (a' a)

i--o

: i rtr-io Qi = m) (a, a) 'i=O t '

Since mlT : (M, a)l(a,. a) we have the formula

(13) try1"-rne!,o2 - - i nx-n(M-ia,a),

f o r o = P = ( M , a ) l ( a , a ) -Next let (M, a)l(a, d) < p 3 m. If we apply the weyl reflection

S, to M - fa we obtain M - pd - 2((M - 0d, a)l(a, a))a --

M - ( m - 0 ) a , s i n c e m : 2 ( M , a ) l ( a ' a ) . W e h a v e 0 3 m - P <

(M, a)l(a, c) and we see that M - pa, is a weight for the irreduc-

iUt" 8'"'-constituents having the following highest weights:

M, M - dt " ', M - (m - ila and only these. The reasoning used

to establish (13) now gives

(14) trg1"-o'elneT : -^f'n"-in(M - ia, a)i = o

f.or (M, a)l(a, d) < p < fit. we recall that if. M is a weight, then

p - i a n d m b Y

LIE ALGEBRAS

MS, is a weight and /tN: nilso (Theorem 4.1). Hence nx-ia:rtx-w-ita. On the other hand, (M - ja,a) + (M - (m - j)a,a):(2M - ma,, a,) : Q. Hence

(15) n r -n (M - ja ,d )+ / tx - rm- j to (M-(m- j ) * ,d ) - 0 .

This and (14) imply that (13) is valid also for p > (M, a)l(a, a).we recall that M was any weight of 0 in TJt such lhat M * a isnot a weight. Hence M - pa can represent any w€ight of O inDt. We now change our notation and write M f.or IWt - po. If werecall that np, : Q if M' is not a weight, then we can re-write(13) as

(16) trgl*e2,e3 : -in**t,(M + ja, a)j :0

for any weight M and any root a * 0.we consider next the casimir element defined by the Killing

form. This is /' - ZiuludR where (u;), (ui) are duhl bases for 8relative to (x, !). We know that laell : 0 for all o b g. Since Ris absolutely irreducible it follows from Schur's lemma that r : TL,T e A (cf. Jacobson, I*ctures in Abstract Algebra II, p. 2Z6). Wechoose dual bases in the following way: (hr, . . ., h), (h,, . . . , h') aredual bases in 0 relative to(x,y). Then since (er,e_d) - -l ande,is orthogonal to 6 and to every €p, g * -d, the follor,ying are dual:

( h t , . . . , h r , € - o , a - g , . . . )

( h t , . . . , h ' , - € o , - € F r . . . ) .

Then

d*0

(17) r : i h f h i "

If we take the trace of the induced mappings in llt1, M a weight,we obtain

(18) Tna -- Irtry1*hf;h'* - \trry*e!,u* . l

Since hn is the scalar multiplication by M(h) ir, iUt, we havetryi*hf hiR : naM(h)M(ht), so the first term on the iight hand sideof (18) is ny2;M(hi)M(h'). We proceed to show that ErM( h)M(hi) -(M, M). T'hus we write lru : Zp,ihr. Then (M, M) = Z(hr h)p;ptand M(h):(hn, h):Zipi(hi, h;) and M(ht):(hr, ht):Epi(h1, hi): pr.

VIIT. CHARACTERS OF THE IRREDUCIBLE MODULES

Hence ZM(hr)M(h;1 : l(h;, h)prpi : (M, M). It follows

2;try17*hf h'^:no(M,M). If we use this result and (16) in

we obtain

(1S; r?tx: (M, M)nw + > in**t,(M + iaL a) .0 r = 0

247

The terms nn(M,a) and np(M, - a) cancel in this formula so we

may replace XLo by )i='. The resulting formula is valid also if

M is not a weight. In this c?s€ 2r : 0. If , for a particulat d,

no M * ja, j > L, is a weight then nx+io: Q and the particular

sum !pfiN+i,(M * ia,a):0. If. M + ia, is a weight for some

j >- L, then no M - ka is a weight for k > | since the a-string

containin g M+ ja does not contain M. Hence the set {M+ka I e>1}

contains the complete a-string containing M + id. Then

\i:nx+ir(M + ia, a) : 0, since

f lp+ir(M + ia,a) : - / tw+iatsr((M + ia)S",a) '

Thus in all cases )i:fia+i,(M + ia, a):0 and (19) holds. The

argument shows also that

(20) i n**n,(M + ka, a) :0

holds for any ,","*r", U"ear funct ion Mon O. This implies that

D?:flx-r,(M - ka, - a) : nru(M, d) + }l:,'nx+x"(M + ka, a), so if

we substitute in (19) with Xt', we obtain

Tltx : (M, M)np +: nn(M, d) + ,7no*0,(M + ka, a) '

Setting 5 _ (1l2X>ln>oa) this gives the following formula: .

(2f1 TttN : (M, M * 26)nN + 2i nn+r,(M * ka,a) '

L1r'o

lf M:A the highest weight of 6 inll l , then nN:1 and /tx+*o:o

for a, > 0, & = 1. Hence (21) gives

r : (/1, A + 26) -- (/l + d, A + d) - (d' d)

and substitution in (21) gives Freudenthal's recursion forruula:

(22) ((A + d, tl + d) - (M + 6, M + B))nx - ztna+*'(M * ha, a) .

L=',,

that(18),

LIE ALGEBRAS

We shall now show that this gives an effective [rocedure forcalculating flr beginning with ltr : L. For this we rdquire a coupleof lemmas.

LpMu,c, 2. Let B -- (U2)2*>od. Then D(/zi) : 1, i * 1,2, . . .,1, ife;,ft h; is a set of canonical generators for 8. Also lf S + 1 is inW, then d - dS is a non-zero surn of distinct positiue ,roots.

Proof: If a is a positive root we know (Lemma 1) that aSi =a$r i :a -a(h ; )d , ; )O un less d , :d i , in wh ich case d ;S i : -d ; .Hence

dsi : G E")r, : (+,t_,) - + d, : 6 - d; .\ z i io / \, ,*,n

Also, (dSr, dr) : (d, aiSi) : (d, - a,r). Hence (d - ar, wr): (6, - a,r)and 2(d, ar) : (ai, a). Thus 6(hi) : 2(0, dil(db dt) : L,' i : t,2, . . ., l .Since any S e W maps roots into roots we evidently have dS:A - >p, where the summation is taken over the P - - aS > 0.If there are no such p, then aS > 0 for all a > 0. This impliesS: l, by Theorem 2, contrary to hypothesis.

Lpuu.r, 3. Let I be the highest weight of 6 in Tft. )Then

(23) ( M + d , M + d ) < ( ' l + d , l + d )

for any weight M + A of 6 in Tft.Proof: We shall prove that there exists a weight'M' > M such

that (Mt + d,Mt + d) > (M + 6,M + d). This will prove the resultby an evident ascending chain argument. Assume firqt there existsan i such that M(h) < 0, hi in the set of canonical generators.Then we take M' :'MS;: M - M(h;)a; > M and wg have

(M' + d, M' + d) - (M + d, M + 6)

- - zM(h')(M + 0, a;) * M(h)'(d;, d;)

- - ZM(h)l(d, d) ,

since M(h;) - 2(M, a;)l(ai, a;). Since (d, ai) > 0 this shows that( M ' + B , M ' + d ) > ( M + 6 , M + d ) . N e x t s u p p o s e I U I ( h ) > 0 , i -1 , 2 , . . . , 1 . I f n o M * a ; i s a r o o t a n d r + 0 s a t i S f i e s r h : M x ,then rei :0 for all i and r is canonical generator o! an e-extreme8-module. This must coincide with Dl, so M is the hJighest weight,contrary to hypothesis. Now let a; be one of thQ simple rootssuch that Mt - M * c; is a weight. Then M' > M'and

VIII. CHARACTERS OF THE IRREDUCIBLE MODULES

(M ' + B ,M '+ d ) - (M + B ,M + d ) - 2 (M+ d ,d i ) * (a ; ,d+) -

Since M(h;) > 0, (M, a;) 2 0. Also (6, a) > 0 and (ai, a;) ) 0, so

again we have (M' + d, M' + d) > (M + D, M + d).

It i. now clear that (22) can be used to determine the weights

and their multiplicities. Thus we begin with n,r: L. Suppose

that for a given fu[ - .4 - |,k;&, &i floil-n€gative integers with. at

least one hi > 0, we already know rtx' for every M' : A - y,k!;a;'

hl integers, 0 < k| < kr, M' + M. Then every term on the right

hand side of Q2) is known. Moreover, if (A + 0, I * d) :

(M + d,M + d), then, by Lemma3, M is not a weight and, n* -9.

Otherwise, the coefficient of ny in (22) is not zero so we can solve

for nx using the formula.

3. Weyl's character formulq'

In order to obtain a general formulation of Weyl's formula valid

for any field it is necessary to replace the exponentials which

appear in this formula by "formal" exponentials. This notion can

be- made precise by introducing the group algebra over the base

field of the group of integral linear functions on s. we recall

that an element M e b" is called integral if M(h;) is an integer

f .or i :L,Zr. . . r1, where e; , f t ,h; AtE canonical generators for 8 '

The set S'of integral linear functions is a group under addition'

and it is cleal that 3 is the direct sum of the cyclic groups gener-

ated by elements li of S such that t';(h) : dir. We now introduce

an algebra ?l over @ with basis {e(M)l M e 3} in 1 : 1 correspond-

ence with the elements of S in which the multiplication table is

e(M)e(M ' ) -e (M+M' )(24)

Then ?I is the group algebra over O of ,S and e(0) : 1 is the identity

element of ?1. The elements of fl. are the formal exponentials to

which we alluded before. We now define the character X of a

finite-dimensional module Ift to be the formal exponential

(25) x : \n re (M) ,u

where ny is the multiplicity of M e 3 as defined before: ltN:o

if M is not a weight and rtr:dimfir the dimensionality of the

weight space lnr if M is a weight. The summation in (25) is taken

over all M e S. This is a finite sum since flx * 0 for only a finite

250 LIE ALGEBRAS

number ot M.Let rr : e(f,;), ]o(h) - drj. Since any integral lineat function can

be written in one and only one way as M - Zmth lwhere the mi,are integers, the base element e(M): e(Zmif,;): e(mrlr)...e(mt)):e() , '1 r" 'e() t l * t : i { ' - . .xf t . We have cal led Mdominafr t i t M(hr)20f .o r i :L ,2 , . . . ,1 . Th is i s equ iva len t to rh20. Thd se t o f l inearcombinations of the e(M), M dominant, is a subalgetjra of ?l whichis the same as the set of linear combinations of the monomialstcYt...xTt, mi20. The set of these monomials obtalned from thesequences of integers (mr,...,mr) with lntt- 0 are l inearly inde-pendent. Hence the subalgebra we have indicated caq be identifiedwith the commutative polynomial algebra (0fx1, x2, .,. ., xtf in thealgebraically independent elements xr, . . ., xr. Every element of ?Ihas the form (x[ , . . .x i t \ - r f where the rr> 0 and f e Af ixr , . . . , . r r ) . I tis well known that a[w . . - r rr] is an integral domair[. It followsthat lI is a commutative integral domain.

with each element s in the weyl group w we associate thelinear mapping in ?I such that e(M)S - e(MS). Sincd

(e(M)e(M'))S: (e(M + M'))S - e((M + M)S)- e(MS + M'S) - e(MS)e(M'S) - e(IW)S^(M,)S ,

it is clear that s is an automorphism in ?I. The set of s in ?I isa group of automorphisms isomorphic to w. An eleilrent a € lI iscalled symmetric if as: a, s e W, and alternating if i as - (det S)a,S e W where det s is the determinant of the orthogohal transform-a t ionSin6 i . Thus de tS: r l and i f S :S, the Wey l re f lec t iondetermined by the root a then det S, - * 1. In partidular, det S; :-1 for Sr : Srr. Since the S; generate W, a is symmetfic if and onlyi f . aS. t :a , i :L ,2 , . . . ,1 , and a is a l te rna t ing i f and on i l y i f aSr : -af.or all i. The set of symmetric elements is a subalgebra; the setof alternating elements is a subspace. The product of two alternat-ing elements is symmetric and the product of an alternlating elementand a symmetric element is alternating. Since Wus : ltx forS e W, it follows that the character x:}nxe(M) ist a symmetricelement. We note next that the element

(26) Q - e(- d)Jl(e(a) - 1) : e(d)Il(l - e(- a)) ,

where a - (1/2)(2,rpt) the integral linear function defined in g 2, isan alternating element of ?I. Thus we have seen (prodf of Lemma 2)

VM. CHARACTERS OF THE IRREDUCIBLE MODULES

that dSi: I - ar so (- d)S; - - d + d,t and e(- d)Sr - e(- d)e(a)'

Also we know that S; permutes the positive roots * di ?nd sends

ai into - atr Hence

fl(r@) _ l)si :d > 0

and

QSo : ( n Crt'l - r)t<- ,,t)- r)e(- 6Y@;)a*o7

- e(- d)l[o(e(a) - lxl - e(a))a*o6

- -4 .

We set

(27) f :Qx .

This element is alternating. It turns out that we can obtain a

formula f.or f and this will give the desired formula for Z.

Let o: Xser(det S)S as linear operator in ?1. We have for any

T e w that oT : E"(det s)sT - (det T)-'x"(det sT)sT - (det T)oand similarly To - (det ?)o. If a is any element of ?I, then

(ao)T - (det T)(ao). Hence ao is an alternating element. lf^ a is

alternating to begin with, then aS - (det S)a and so ao : til)a where

ao is the order of the Weyl group. It follows that (llw)o is a pro-

jection operator of ?I onto the space of alternating elements. Hence

any alternating element has the form ad, Q. € lI, and consequently

such an element is a linear combination of the elements e(M)o, M

an integral linear function. Since Sa - -r o it is clear that e(M)o

can be replaced bV e(M$o and so we may express any alternating

element as a linear combination of elements e(M)o where M is

highest among its conjugates MS under the Weyl group. In parti-

cular, we may suppose that M > MSi : M - M(h;)u'i,, that is, we

may suppose that M(ht) Z 0 for i : 1,2,' ", l. Suppose M(h;,) : 0

for some i, or equivalently MSr = !l[. Then e(M)o - - e(M)Sto :

- e(MS;)o : - e(M)o and e(M)o - 0. We therefore conclude that

every alternating element is a linear combination of the elements

e(M)o where M(h) > 0, i : I ,2, " ' ,1.we now apply this argument to the element o defined in (26).

If we multiply out the product in (26) we see that I is a linear

/ \(Il,trtol -t))(e(- a) - r)

o*u5

252 LIE ALGEBRAS

combination of elements e(M) where fu[ - d - p and p is a sumof a subset of positive roots. Thus M: (ll2) o>oEcd Where eo : t 1and any conjugate MS of. M under the Weyl group again has theform d - p' : (Il?)2r>oeLa where p' is a sum of positive roots andet,: + t. If we apply the projection operator .(Llw)o:,to the expres-sion indicated f.or Q we obtain an expression f.or Q {s linear com-bination of elements e(M)o where M is of the form Bl- p, p a sumof positive roots. We have seen also that we may assume that Msatisfies M(h;) > 0, i : L,2, ...,1. These conditionb imply thatM : 6. Thus, if fu[ - d - p, a - Zkrai where the k; are non-negative integers, then 0 < M(h) : 6(h;) - p(h;): | - p(ht). Sincep e 3, p(ft) is an integer. Hence we must have p(h) < 0. On theother hand, 0 ( (p, o1 : I,hr(di, p) - Qlz)>k;(a;, ai)p(h) < 0. Hence(p, p): 0 and p : 0. Thus we have shown that O :'qe(6)o, T e O.Now e(6)o : )"(det S)e(dS). By Lemma 2, dS + d if S + 1. Hencee(6)o - e(6) + >:t e(M) where M:6 - 0, Q * 0 and a sum of positiveroots. Also it is clear from (26) that Q - e(d) + > * e(M). SinceQ - ae(d)o it follows that tt : l. We have therefofe proved thefollowing

Lonrnr.r 4. Let Q - e(- B)[I,>o@(a) - 1). Then

(28) Q - e(d)o :"$(det S)e(dS) . ,

We shall next introduce the vector space O* SrlI and we shalldefine certain linear mappings of this space and of the algebra ?I.The elements of O* 8?I have the form 2a@at p;le On, a; € V.The algebra composition in ?l provides a linear mapping of ?IQpUinto ?I such that a I b -, ab. This gives a lineaf mapping of(0*8?I )B2I :O*O(? I€ l? I ) in to O*@?r so tha t (p8a)Ob- .p&ab.It follows that if we set (Lp; I a)b : I,pi& a;b, then this productof Xpr I ai and D is single-valued and coincides with the image ofXp;8 a;8 b in 6* I ?I. It is clear that the product,()p; I a)b -

Xpr I arb turns 6* I ?l into a right ?I-module. We yecall that wehave the bilinear form (p, o): (hp, h,) on O* which defines the linearmapping of O* 816* into @ so that p I r - (p, c). If we combinethis with the linear mapping of ?I @ ?I into ?I we obtain the linearmapping of (6*8U)8(6*8?I) into a8U:?I suchl that (p8a)8(r 8 e) -+ (p,c)ab. This defines a O-bilinear mapping of b*'8 ?I suchthat the value (p I a, r I D) : (p, r)ab. Since (p, o) is symmetricand ?I is commutative this is a symmetric bilinear form. Also if

VIII. CHARACTERS OF THE IRREDUCIBLE MODULES 253

c e ? I , then ( (p8 a)c , r 8D) : (p&ac, r8D) : (p , r )acb and (p@a,

r8b)c : (p,r)abc and simi lar ly (p&a,( t8D)c) - (p8 a,r&b)c which

implies that (r, !), tc, y e 6* I ?1, is ?l-bilinear.

Next we define a linear mapping, the gradient, of ?I into 6* I ?l:

a -+ aG such that e(M)G : M I e(M) and a linear mapping, the

La\lacian, of ?I into [: a --, ay' such that e(M)y' - (M, M)e(M)' We

have

(e(M)e(M''o::rf*.^;f;#^M + M,): (M + M,)o,e(M)e(M,)

- M@e(MY(M') + M'8e(M')e(M)

- (e(M)G)e(M') + (e(Mt)G)e(M) .

The linearity then imPlies that

( 2 9 ) @ b ) G : ( a G ) b + ( b G ) a , a , b e W '

We have

(e(M)e(M'":,;l!n|rff ;'],,;:f.*r,Yi,"#,:,#;'::#,)e(M')

+ (M', M')e(M)e(M')

(e(MY)e(M') + 2(M & e(M), M' & e(M'))

+ (e(M')t)e(M)

- (e(M)t)e(M') + Z(e(M)G, e(M')G) + (e(M')t)e(M) -

This implies that

(30) @b)t - (at)b * Z(aG,bG) + (bt)a .

We now return to the formulas which we developed for the

multiplicity nx of the integral linear function M in a finite'dimen-

sional irreducible module of highest weight A, T the element of O

(rational number) determined by'the casimir operator. we consider

again (1.9):

Tltx : (M, M)nr + > in**i '(M + ia, a) .

We multiply both sides by e(M) and sum on M. This gives

@

rx : xl + 4 AZno*ntM + ia, a)e(M) ,

which we multiplY through bY

LIE ALGEBRAS

fl(e(a) - 1) : Il@@) - 1) il @? a) - 1) : * 8' ,a+0 d>0

bv (26). This gives

(31) ! rxQl+Q\Q'z

: Fe,4 fIe@) - r)/tx+t,(M + ja, a)(e(M if o) - e(M)) .

The coefficient of. e(M * aD in the right-hand side of 1 (31) is

f l(e(i l - D(2""*r,(M * ia, a) - f in"*o+ra(M + kl + t)4, a)){+-, \i':o r=:-o /

: II @(il r- I)na(M, a) .F*a

Hence (31) can be written in the form

(32) *rxQ'+@4Q':

Aff @@) - r)e(a)l,nn(M, a)e(M) l

- (> a&e(a) f l (e(9) - 1) , >M8n*e(M) \\dFo {*7 x /

: (=t Q'G,xG) - *Z((QG)Q,xG) .

Hence we have

rxQ' - @4Q' : 2((QG)Q, xG) : 2(QG, xG)Q

and canceling Q * 0 in the integral domain ?I, we obtain

r x Q - Q D Q - Z ( Q G , x G )- (xQ)/ - (xt)Q - (Q4x, bv ($0) .

If we set / : XQ, as before, we obtain

(33) rf : fl - (QOx .

Since O: X"erdet S(e(dS)) and (dS, d'S): (d, d) we have Qy' -- (d, d)8.Since r - (A + d, I + d) - (d, d) (eq. below (21)) thesd substitutionsconvert (33) to the following fundamental equation nor f

e > O

(34) f r ' :Q t+6 ,A+B) f .

The element / - rO is an alternating element ".rd

.orr*"quentlythis element is a linear combination of elementsj of the forme(M)o. Moreover, w€ can limit the M which are rfeeded here bylooking at the form of r and a. Thus we havel x -- I,n*e(M)

VIU. CHARACTERS OF THE IRREDUCIBLE MODULES

where we now consider the summation as taken just over the

weights M of the representation. Also we have seen that Q --

xsew(det s)e(ds). If we multiply we obtain xQ as a linear com'

bination of terms e(M+dS) where M is a weight: f:Znu+asa(M+ds)'M a w e i g h t , S e W . N o w

e ( M * D S ) r ' - ( M + 6 5 , M + 6 5 ) e ( M + d S ): (MS-' + d, Ms*' + d)e(M + DS)

so that e(M * dS) is in the characteristic space of the characteristicroot (MS-'+d, MS-'*d) of /. Since / belongs to the characteristicroot (l + d, I + d) f.or r' and characteristic spaces belonging to

distinct roots are linearly independent it follows that f is a linear

combination of e(M * dS) such that

(MS-' + d, MS-' + d) - (A + 6,1+ d) .

By Lemma 3, (MS-' * d, MS-'+ d) : (l + B, I + d) for the weight

Ms-'implies that MS-t : A. Hence we see that f is a linear com'

bination of the terms e(l + d)S. If we apply the projection operator(1.1w)o tolwe see that f - Ve(A * d)a : ?Es(dets)e(l + d')s. since

dS<D i f S +L , ( l+d ' )S < / l+ d i f S+1 . Hence the coe f f i c ien t

of e(1 + 6) in our expression for f is a. on the other hand, the

coefficient of this term in ?(Q is na: I. Hence rl :1 and we have

proved

weyl,s Theorem. Let wt be the irreducible module for 8 with

highest weight 1. Then the character Ta: \nne(M) of 8 in Tlt isgiaen by the formula

(35) r,r"E (det S)e(dS) :) det S(e(A + d)S) '

where O - (U2)Xa>odt a a root.This theorem means that the expression on the right is divisible

by 8 : Es (det s)e(ds) in ?t and the quotient is the character T, of the

representation.It is easy to see that this result gives Weyl's original formula

l, det S exP((l + d)S) (1,)

(36) X,1lh):

in the comprex case. *"r;t:-ployed his result to obtain by a

limiting process the dimensionality of TJt - 2n* - Zr(0). We proceed

256 LIE ALGEBRAS

to obtain the same result by a somewhat similar device.We introduce the algebra AQ) of formal power,series in an

indeterminate / with coefficients in O. We recall thalt the mappingof power series into their constant terms is a homonhorphism ( ofO(t) onto A. We can also define homomorphisms o{ tr into O<t>by employing exponent ia ls: expe:1* z +(2'z12!) + . . . which isdefined for any z e O<t> with zero constant term. We have therelation exp (e1 * zz) - (exp zr)(exp zz); hence, if ), A, Q e g*, thenexp (1, pX exp (p, p)t : €xp Q. * p, p)t. In particular, this holds for], -- M, p: M', integral linear functions on 6, which implies, inview of. (2t), that we have a homomorphism (o of ?I iinto @(/) suchthat e(M)C p : exp (M, p)t. Now consider 7-nC oC . Since Vo : ltrte(M)we have X,t|p:Zn*exp(M, p)t and, since the const{nt term of anexponential is 1, xnCK - 2n*: dim llt. We shall obtain theformula for dim!]t by applying CoC to (35), taking p: d - (1/2)>,a>&.

Let o : Euer(det S)S as before and let M, M' be ilntegral linearfunctions. Then

e(M)oCn,

a

Hence

(37) e(M)oCa - e(6)oCx - e(- d)C,. fI @@)(* - I)o > o

exp (- B, M)t fl (exR @, M)t - t)o > 0

since a - (1/2)X a>ca. Applying this to (3b) we obtain

Now

(3e)

(Lemma 4)I

: "!o(..oltr,

M)t - .*p+ (- a, irot) ,

F,(e"o* (a, M)t - exp

|<- a, Wt)

= f[ (a, M)tk (fnod /o-,') ,

VIU. CHARACTERS OF THE IRREDUCIBLE MODULES 257

where & is the number of positive roots. Hence if we divide both

sides of (38) by to and then apply the homomorphism ( which picks

out the constant term, we obtain

or

(40)

(dim ![t),f[o(a, d) : Eo(o,

6 + A) ,

fl(a, 6 + A)d im St :90

TTC;D-.

4. Some exa,tnples

We now indicate how dimIn - fTrroQl * d,a)/fl"ro(d,a) can be

calculated from the Dynkin diagram for 8. Let w;: L,2 or 3 be

the weight of the vertex ai ifi the diagram and let a,, be one of

the roots in the simple system such that w' = t' We replace the

scalar product (a, B) by (a, B)' : 2(a, 9)l@,, d,). Then we evidently

have dim st : II,>o(A * 6, a,)'lfl,ro(8, d)' . We write i - \m;f,;,tni 2 0, d -- 2ki4, h >- 0. Then since Ei)(/rt : 1 and D(hr') : 1'

j : L, : : . ,1, d: Xl i . Hence we require ( \Un; * 1) i ; , I ,kp ' ) t :

2;,t(mi * l)ft;(tr, c1)' and Li,ik{},;, a.)t . Now

(l;, di)' - ?(l;' a) - 2-()';' d) ' \o"

*'! - d;ttt)i(d,, d,)

- (at, a) (a,,, a',)

Hence (A + 6, a)' : 2(m; * L)w;ki and (d, a)' : I,w;ki and

T

\(m; * r)w;k;d imTJt - I I=*

t.rlt )wtk;

where the product is taken over all the sequences (&,, kz, '' ', kt),

ftr 2 0, such that Zhial is a root. We have seen that this set as

well as the w; can be determined from the Dynkin diagram.

G* Here u)r :3, I rz:1and the roots ?ta a,11 dz, dr* dz, dr*2d2,

ar * 3az, 2a, * 1ar. Then (41) gives

(42) dim IJt : hl(m,

* l)(mz + L)(3mr + mz + 4)

.(3mt * Zmz * S)(mt * mz + 2)(2m, * ms, + 3)l '

(41)

LIE ALGEBRAS

if the highest weight of tn is flhh * mzf,z. For I I - i' and1: lz we obtain, respectively, 14 and 7, which we frad obtainedpreviously.

B u l 2 2 . H e r e u t ) r : 7 1 ) z : " ' : r , t ) t - r : 2 , w t : l a n d t h e w e i g h t sare

( 4 3 ) d . i l d ; + r + . . . * a i , 1 < i < j < l - L ld r * d t + r + " ' * d t , l < i < l

d i * . . . * a t r r * 2 ( a i + . . . a a ) , 1 = i < ' i = 1 .

These contribute the following factors to the dlmensionalityformula:

(M) m i * m ; t * ' . . * m i * j - i * l

j - i + t , 1 < i < j ' < l - LI

t < i = l

- i +1

Z ( m i * m n * . . . * r n r ) * m t * 2 ( I - i ) * | l2 l - j - i + 1 '

r * m ) * Z ( m r *2 t - j - i + L

L e t t l : f , * , L < h = l - 1 , s o t h a t 7 n * : 1 , t n ; : Oproduct taken over the first set of factors in (44) is

f i 'd i r - i ,+2: f / \ .i : t i = h J - a + 1

- \ k /

'

i t < t< i

i , f t +n .

,<1 .

The

The second set of factors gives

+2 t -2 i+3r r - E

i i 2 t - 2 i+ t2 l + I

2 l - 2 k + t

and the last is

h

ili : r

, i 2 l - i - i +2 T r 2 t - i - i +3i=++ tZ I : J 4+2 , -#=* z t - r - i + r

(2t - k\ /2r\\ h / \ n ):T)

WrMultiplication of these results gives

(45) dim Ift - /21+ t\\ k ) '

Vru. CHARACTERS OF THE IRREDUCIBLE MODULES 259

We recall that this result was what was required to complete our

proof ($ 7.6) of the irreducibility of the module of &'vectors relative

to Br.

5. Applieations and further reeults

We shall now call a character of a finite-dimensional irreducible

representation of 8, a primitiue character. Such a character has

the form T,,r: e(1) * }nlrz(M), where the summation is taken over

M < I in the lexicographic ordering in 6f or in S. Since the

e(l) constitute a basis for the group algebra ?l of $ it is clear that

I,rr: Zz, implieS 11 : 1z and this implies that the associated repres-

eniationl are equivalent. Conversely, equivalence of finite-dimen'

sional irreducible representations implies equality' of the characters.

It is clear also from the expression we have indicated for a primitive

character that distinct primitive characters are linearly indep-

endent. If tJt is any finite-dimensional module for 8, IJt is a direct

sum of, say, mr irreducible modules with character x,rr, mz with

character Xh * X,rr, etC. Then the charaeter X of llt has the form

(46) X: / t l1Xt,* mzx,,rr + " ' 1 m*X,r*

Since this expression is unique we see that if Z and the primitive

characters are known then the /,/2; Cartr be determined. This gives

the isomorphism classes and multiplicities of the irreducible con-

stituents of [Jt. As a corollary, w€ see that these classes and

multiplicities are independent of the particular decomposition of Ift

into irreducible constituents.The characters can be used to determine the isomorphism classes

and multiplicities of the tensor product of two irreducible modules.

Suppose (rr, ..., tc*) is a basis for a module lJt such that xih :

A;(h)x; and (y,, " ',!,) is a basis for a module Tt such that yfi:

Mt@)yt. Then the lnn products n&ti form a basis for lIt@ft

and we have

(x;& t)h = (1; * M)(h)(x;8yi) .

Hence we have the following expressions for the characters of fi,

tt and lJt 8It: 2;e(1;), Lp(M), I,i,$QL * M) : l;,p(A)e(M).Thus we see that the character of llt @It is the product in ?I of

the characters of TJt and It. One obtains the structure of ![t @It

260 LIE ALGEBRAS

by writing the product of two characters as a sum bf primitivecharacters. In the case of the Lie algebra Ar there i$ a classicalformula for this called the Clebsch-Gordon formula. We shall givean extension of this which is due to R. Steinberg. This is basedon an explicit formula for the multiplicity nx of the weight M inthe iqreducible module with highest weight /1. This formula isdue to Kostant; the simple proof we shall give is due to Cartierand to Steinberg (independently).

We introduce first a partition function P(M) f.or M Q, S, which isanalogous to the (unordered) partition function of nurgber theory.lf. Me$, P(M) is the number of ways of writ ingMlas a sum ofpositive roots, that is, P(M) is the number of solutioi (hn, hp, . . .,

ho), of l,r>rkrd: fuf where the h, ate non-negative integers and

{a, B, "., p} is the set of positive roots. We have lP(0):1 and

P(M) : 0 unless M : Lm,a, m'inon-negative integer and al definedas before. Then e(M): ift . . . xft, xi : e(l). It is cdnvenient to

replace the group algebra ?I by the field fr of power iseries of theform (x?' -. - xl\-'f where / is an infinite series with coefrcientsin O in the elements rlr ... xlt, z; nor-r€gative integral. In fr wecan consider the "generating function" )a.E SP(M)e(M ) which isdefined since P(M) - 0 unless M : Zma r, //ti2 0. It i is clear thatwe have the identity

"I*P(lM)e(M):,4 $ + e(a) * e(2a)+' ' ') .

Since (L -e(a) ) - r :1*e(a) *e(2a)+ . " , we have the ident i t y

We re-write Weyl's formula (35) as

(Znne(u))) (det S)e(dS):"e(det S)a(l + d)S .

We replace M by - M in this and multiply the result through bye(d) to obtain

(48) 2n*(- M )) E (det S)e(d - dS)

By Lemma 4,

Q : X (det S)e(dS) - e(B) fI (1.- e(-a)) . ,

( 47 ) (> P (M)e (M l )ng -e (a ) ) - 1 .r u e $ / t ) 0

Hence

VIII. CHARACTERS OF THE IRREDUCIBLE MODULES

L (det S)e(d - dS) - II (1 - e(a)) .s e w d > 0

Hence if we multiply both sides of (a8) bV X'eg P(M)e(M), weobtain, by (47), that

\nxe(-M): ( L (det s)e(d - (A + D)s))(4",t )e@))(4e)

:X-%., s)p(M)e(hI+6-(, l+d)s).;:K

Comparison of coefficients of. e(- M) gives Kostant's formula for

the multiplicity nN in the irreducible module with highest weight

l, namely,

(50) nM:) (det S)P((l + d)S - (M + d)) .

We now consider the formula for X,trX,t, where 7.t;:'2nfi'e(M)isthe character of the irreducible module lltt with highest weight z/r.

We have seen that X,rr1,,tr: \m,il,t, where mn is the multiplicity in

!Il,8St, of the irreducible module with highest weight 1. Thesummation is taken over all the dominant l. If we multiplythrough by X"err (det S)e(dS) and apply Weyl's formula we get

/ '" l) I y, (det ?)e((1, + d)"))l l ' n f r ' e (M ' , , - /\ , t €S / \ rew

:|AouZr*(det S)e((l + d)S) .

The summation on the right-hand side can be taken for all I e 3if we define ln,tr:O for non-dominant A. Applying the formula (50)

f.or nfii we get

( / \ ' l

{ >( X (aets)P((1, + d)s - (M+ a))e(M) )ft&-g\ser / )

ffi @et T)e((/, + d)")):

A *^fi,(det S)e((l + d)S) .

Hence

r € s 8 , r € w

l€s sery

If we put (12 + d)? * M: ,,1+ d on the left we get

>- -E-Jdet ST)P ((A, * d)S + (1, * d)" - (A + z|))e[,] + a), 'es ,s,r€w

/ \

l € c i \ s e w /_ n € S(x +6)S: l+6

/ \> ( > (det S)ze(,,+d),s*r-6 le(l + a'1l € $ \ s e w /

,.te$ \seur /

Hence

LIE ALGEBRAS

X (det ST)P((A' + d)S * (A, + d)" - (1+ 2D))s , r€w

s€w

: lr tA + > (det S)zrz16rsi6 .8 € w8*r

It is easy to see that if I is(l + d)S - d is not dominant ift/f iu+Et,s,-d:0 if S + 1 and so we

dominant, then (l + 0)S and henceS + 1. Hence if I is dominant then

(51) lnA: X (a"t S?)P((zt, +

obtain the formula

d )s+ (A , *d )T - ( r+2d )8 , T E W

for the 'multiplicity of the module with highest weight i in!lt, 8 !I,.

Exerciees

The notations and conventions are as in Chapters VII andlVIII.1. Let p eSf. Give a direct proof that p lpSfor every S€Wif and

o n l y i f p ( h r ) > - 0 , i : 1 , 2 , . - . , 1 .2. (Seligman). Prove the uniqueness assertion in Theoreryr 2 by using the

fact that there exists a finite-dimensional irreducible modulerlJl with highestweight I satisfying: A(h,) are distinct and positive. Note that 1S : I ifnS : S, so that 2(A, a)l(at, ut) : 2(15, aaS)/(a;S, aaS) : 2(A, alS)l@r,S, aeS).This leads to l(hd : t l(h) if alS: a1.

VIII. CHARACTERS OF THE IRREDUCTBLE MODULES

3. Call a weight .,1 in 9Jt a frontier weight if for every root a * 0 either

A + a or I - a is not a weight. Show that if lJl is finite-dimensional irre-

ducible then any two frontier weights are conjugate under the Weyl group.

4. Let the base field be the field of real numbers. If a is a root let Po

be the hyperplane in S defined by a(h) = 0. A clwmber is defined to be a

connected component (maximal connected subset) of the complement ofgryop, in F. Show that every chamber C is a convex set. A set of roots

I is a d,ert.ning system for C if C is the set of elements ft, satisfying a(h) > 0

for all a € I. Defining systems which are minimal are called fundamental

systems. Show that these are iust simpte systems of roots determined by

the lexicographic orderings in S*.5. Show that the group algebra :X (of the group S of integral functions

on S) is a domain with unique factorization (into elements).

6. Prove that if Pe ?l is alternating, then P is divisible by Q = l5'(detS)e(DS).7. Let 4 be the automorphism of ! l such thate( l)n: s(-.1). Show that

if X is the character of a finite-dimensional module Sl then 1a is the character

of the contragredient module lJl*.

8. Let 9Jl be a finite-dimensional irreducible module whose character

satisfies Nn: t. Assume the base field algebraically closed. Show that the

image !n under the representation I in 9Jt is a subalgebra of an orthogonal

or a symplectic Lie algebra o{ linear transformations in !Jl.g. Let S be the split three-dimensional simple Lie algebra with canonical

basis e, .,1l, h. Show that the character of the (m + l)'dimensional irreducible

module 9Jl-+r for fr is c- * nn-r + .. . *r- tu where n: e(l) , t(h) = 1. Use

this to obtain the irreducible constituents of 9J1,,,11@ !Jl"nt.

l0 (Dynkin). Let !J? and Il be finite-dimensional irreducible modules for

g. prove that 9Jl I It is irreducible if and only if for every I in any simple

ideal of I either $ll : 0 or ltl : 0.

ll. Use Weyl's formula to show that the dimensionalities of the four

basic irreducible modules tor Ft are: 26, 52, 273, 1274. Use.Freudenthal's

formula to obtain the character of the 26-dimensional basic module'

12. Use 'Weyl's

formula to prove dim lJt = 2I if lJt has highest weight lr

for Bt.lB (Steinberg). Let lJt be the finite-dimensional irreducible module with

highest weight A. show that M is a weight of Dt if t - MS is a sum of

positive roots for every S e W.14 (Kostant). Prove the following recursion formula for the partition

function P(M):

P(M) : - I (det S)P(M - (d - dS)).8 € ws+r

263

CHAPTER IX

Automorphisms

In this chapter we shall study the groups of automorphisms of

semi-simple Lie algebras over an algebraically closed field of charac'

teristic 0. If e is an element of a Lie algebra B of characteristic0 such that ad e is nilpotent, then we know that exp (ad e) is an

automorphism. Products of automorphisms of this type will be

called invariant automorphisms. These constitute a subgroup Go(8)

of the group of automorphisms G(8) of 8. If r is any auto'

morphism, then r-t(exp ad z)r: €XP ad. z" and ad e" is nilpotent. It

is clear from this that Go is an invariant subgroup of G.The main problem we shall consider in this chapter is the de'

termination of the index of Go in G for 8 finite dimensional simple

over an algebraically closed field O of characteristic zero. We show

first that if 8 is finite-dimensional over O (not necessarily simple),then Go acts transitively on the set of Cartan subalgebras, that is,

if b, and 0z are Cartan subalgebras then there exists a o Q, Go suchthat Oi: Os. A conjugacy theorem of this type was first noted

by Cartan in the case of 8 semi-simple over the field of complexnumbers and it was applied by him to the study of the automor'phisms of these algebras. The extension and rigorous proof of

the conjugacy theorem is due to Chevalley. This result reduces

the study of the position of Go in G to the study of automorphismswhich map a Cartan subalgebra into itself.

If 8 is semi-simple, then the Weyl group plays an important role

in our considerations. We shall require also some explicit calcu-lations of invariant automorphisms due to Seligman. The finalresults we derive give the group of automorphisms for the Lie

algebras At, Bt, Ct, Dt, I > 4, G, and tr'r. It is noteworthy thatsimilar results can be obtained in the characteristic f case (see

Jacobson [8] and Seligman [4]). As usual, for the sake of simplicity we stick to the characteristic 0 case. In the next chapterwe shall extend our final results for non-exceptional simple Liealgebras over algebraically closed fields to algebras of this type

[265]

266 LIE ALGEBRAS

over arbitrary base fields (of characteristic 0).

I. Letnmas from algebraic geometry

Let sl be a finite-dimensional vector space with basis (ur,ttr,"',tt*)

over an infinite field A. Any r has a unique repfesentation as

r:ZE;u; and the Fi are the coordinates of r relative to the basis(u). Let f Qr .. ., f,*) be an element of the poltynomial ringAIl,,, .. ., A^l in the ln indeterminates )q with coefficients in O.Then .f (lr, . . ., f,^) and the basis (ze) define a mapping f of St intoO by the rule that the image f(x) - f (E): f (€, , " ' )€*) . ( In th ischapter we shall often use the notation /(r) rather than xf otxt .) We call f a Folynomial function on m. If (u'i, "', uk) is a

second basis, then it is readily seen that the function / can be

defined by another polynomial in O[i.r, .", ]^f with t'espect to (ul).

In this sense the notion of a polynomial function is independent ofthe choice of the basis for St. The set of polynomial functions isan algebra 0[!m] relative to the usual composition$ of functions.We recall that if / and g are functions on lJl with vallues in 0, then(f + g)(x) : f(x) + g(x), (af)(x) - af(r) for a in {0, (fil@ : f(x)s(x).The mapping f (1,, . - -, f,^) -,f e Ottrtl is a homomorphism ofo l J r , . . . , A - f i n t o @ [ f i J . S i n c e @ i s i n f i n i t e , f ( { r , ' . ' , ' € ^ ) : 0 f o r a l l

6c in O implies f (1r, .. ., ,l-) - 0. Hence the homonlorphism is anisomorphism. The isomorphism maps ,lr into the projection func-tion zr such that ni(x) : tn. Hence it is clear that the zi generate

the algebra O[St].Let Tt be a second finite-dimensional space ovet O iwith the basis

(ur, ur, '", an). A folynomial mapping P of Ut into Tt is a mappingof the fo rm x :ZTE*4ea ! : ) , i r l i a r where r i + b {E ' , " ' ,€^ ) ,

Pi(lr, . .., f,*) e Al)i, . . ., f,-7. This notion is indepbndent of thebasis. The set of polynomial mappings of St intd Tt is a vector

space under the usual addition and scalar multiplidation of map-pings. The resultant of a polynomial mapping P of Ul to Tt and

a polynomial mapping Q of Tt to S is a polynomial mapping POof ![t into S. The notion of a polynomial functioll is the specialcase of a polynomial mapping in which the image space is theone-dimensional space @. Hence if. P is a polynoniial mapping of!ft into Tt and f is a polynomial function on Tt, thefr Pf is a poly-

nomial frrnction on Sl. The mapping f - Pf is a fnapping of thealgebra @[It] of polynomial functions on tt into @tt!ml. We now

( 1 )

IX. AUTOMORPHISMS

look at the form of this mapping. Thus we have P(x) - y wherex : 2 8 & 6 ! - Z t l i u i a n d

tli : |i(Er, ' ' ', E^) ,

b i ( l r , . . . , f , - ) e Ol^b . . . , f ,^1. Also we have f(y) - f (n ' , " ' , ln) ,f (pr, -.., pn) Q AIpr, ..., Fn]l, pi indeterminates. Hence P/ maps r

into

fQ'G), br€), "', P"(E))

where bi€) = f {8, ,€r , . . . ,8*) . l f f , g e O[ l t ] , then

(f + s)(f,(t), ' ", P"(ED - f QJ.il "'s P"GD + s(b,(E), "', P"(E))af (p'(E), . .', p"GD : a(f (f ,(f), ' ' ', P"(€))(fg)(f'G)s '", P"GD : f @'G), "', b"(0)s(fr(f), ' ' ' , p"GD .

This shows that f --> P/ is a homomorphism o p of o[Tt] into otml.

Conversely, let a be an algebra homomorphism of @[Tt] into @[ill].

Consider the projection pi: X TJnt)n-rri. Suppose pi is the mapping

2E;u;- f i (Er,€r, ' ' ' ,E) and let P be the polynomial mapping of!m into Tl such that Z€ui-+Zttiai where vi:f i(E). Then Ppl

maps LErur into /r'(f) so that Ppi: p7. Since the pi generate O[It]

it follows that o coincides with the homomorphism a' determined

by P. Thus every homomorphism of @[yt] into OlUll (sending 1

into 1) is realized by a polynomial mapping of fi into Tt. It is

easy to see that if P and Q are two such polynomial mappings,then the homomorphism dp: oq if. and only if P: Q. Hence wehave a 1: 1 correspondence between the polynomial mappings of ![t

into It and the homomorphisms of O[It] into O[!ft].

Of particular importance for us is the . set of algebra homo'morphisms of @[It] into the base field @. This can be obtained in

a somewhat devious manner by identifying A with the algebra@tlnl of polynomial functions on fi : 0 and applying the foregoingresult. However, it is more straightforward to look at this di'rectly. We note first that if y e It, then the mapping or: .f -, f (y),

the specialization of .f tt !, is a homomorphism of OlUtl into O.Conversely, if o is any homomorphism of @[It] into 0, we let 7; :

p7, as before. Then it is immediate that o: dy wh€tl y - }niui.It is clear also that if lr * lz in Tt, then or, I 6vz.

Let f e OlVt) and let a : X a;u,i, € fi|. Then we can define alinear function d"f on Sl by

( 2 )

zffi LIE ALGEBRAS

( 3 ) (d"f)(YEu;): t (%\ Er .- f \o/ t /xr :a,

It is easy to see that d"f is independent of the basis used to definethis mapping. The linear mapping d,.f is called the diferential ol

f at a. We have the following properties

d"(f + 9): d"f a d"9

(4) d"(af) - a(d,f)

d"(fg) - f (a)(d"g) + g(a)(d"f) .

If / is an indeterminate and, f is extended to 11116r in the obviouswny, then we have the following relation in the algebra Oltl, whichis a consequence of Taylor's formula

f (a + tx) = f (a) + t(d"f)(r) (mod /z) .( 5 )

More generally, let P be a polynomial mapping x:}T|,ru;-t}Tfif)ui of St into It. Then we define a linear mapping doP, thediferenti.al of P at a by

(6) (d,P)(x):,8 $;W)^,_,,t0),, .Again, one can verify that this is independent of thetbases chosenin Ul and Il. Also one has the useful generalization of (5):

(7 ) P(a * tx) = P(a) + t(d"P)(x) (mod fz) .

A set of generators for the image space Vl(d"P) iis the set ofvectors

( 8 ) (d,P)(ut): + (K) ^.=,*,i .

Hence d"P is surjective if and only if the Jacobian rgatrix

( e ) ( ( % \ ) , i : L , . . . , f f i , j : ! , 2 , . . . , n\ v '/ \\ ard / t,*="*/

'

has rank z.

Lpuu.r 1. Assume A pe'rfect. Then if d"P is surjactiae for sonNea, the homomorphism oe is an isomorphism of OWtl irtto A[sXl].

Proof: Our hypothesis is that one of the z-rowed fninors of thematrix (dbtldl) is not zero. If. or is not an isomorphi$m, then thereexists a polynomial f(p,,. . ., p^) *0 such that f (pl€), pr(E),. - -, f*(E)) :

IX. AUTOMORPHISMS

0 for all fr in O. This implies that f (pr(,1), . . ., p"(l)): 0, that is,the polynomials fr(f,r, - . ., f,^), - . ., 2o(]r, . . ., f,^) are algebraicallydependent. If this is the case, then we may assume that the poly-

nomial f + 0 giving the dependence is of least degree. The relation

f (p,(l),. . ., y'"(t)) - 0 gives

(10)

This contradicts the hypothesis on the Jacobian matrix unless(0f l0p)(0,(l), ...,2"(i)) :0. Since the degree of / is minimal for

algebraic relations in the f '(l), ' ' ' , F^(]) we must have (0f i0p) - 0,j : 1, " ', n. This implies that f is a non'zero element of O-which

is absurd-if the characteristic is 0. If the characteristic is p+ 0,we obtain that f is a polynomial in p!, pl, "', $f;. Since @ is per-

fect this implies that f : 9p, g a polynomial in the p's. Theng(pr(l), . . . , P^(A)): 0 which again contradicts the minimality of the

degree of f.The main result we shall require for the conjugacy theorem for

Cartan subalgebras is the following

Tnnonpu 1. Let O be algebraically closed and let P be a foly'nornial mapping of silt into [t such that d"P i,s suriectiue for somea e fl|. If f i,s a non'zero polynomial function on !Il, then thereerists a non-zero polynomi,al function g on Tl such that if y is any

element of n safisfying g(y) + 0, then there erists an x in [n such

that f(x) * 0 and P(x) - y.

In geometric form this result has the following meaning: Given

an "open" set in Ul defined to be the set of elements r such that

f (r) + 0 for the non-zero polynomial f , then there exists an open setin !t defined by g(y) + A, g a non-zero polynomial which is com'pletely contained in the image under P of. the given open set in!m. (Suggestion: Draw a figure for this.) We shall see that Theo-rem 1 is an easy consequence of the following theorem on exten-sions of homomorphisms.

Tnsonpu 2. Int O be an algebraically closed field and let f bean ertension fi.eld of O, \ a subalgebra of f and 2I' an extensionalgebra of \I of the form ?I' : Vlur, Ltzt . ' , u,7, h e T. Let .f be anon-zero element of A'. Then there eri.sts a non-zero element g inT such that if o is any homomorphism of 1l into O such that g' + 0,

O:K:

27A LIE ALGEBRAS

then o has an extension homomorfhisrn t of ZIt iilo A such that

f ' +0 .Proof: Suppose first that r : l, so that ?[' : }LIul. Case I: z is

transcendental over the subfield 2 of f generated by ?1. We write

f : f o * f r u + " ' * f ^ t f , f r € Z l , f , + 0 . L e t g : f , a n { l e t a b e ahomomorphism of ?I into A such that g" + 0. Consider the poly-nomial f{ + f{ ^ + . .. + f:,^' in alll, .i an indeterminate. Since

.f: * 0, this polynomial has at most m roots in @ so we can choosec in O so that ),6fic' + 0. Let r be the homomorphism of ?I' :

?I[z] into @ such that 2 anu' -' l, aici. This is an extension of oand f

' + 0 as required. Case II: u is algebraic over E. Thecanonical homomorphism of ?I[i], I an indeterminate, onto ?I/ : ?Itul(identity on ?I, ^ -, u) has a non-zero kernel $. Since tf e V[ul, f isalgebraic over J also. Let bQ,), q(^) be non-zero polynomials in?l[,1] of least degree such that f(u) - 0 and A(f) : 0. ' Then thesepolynomials are also of least degree in lli.) such lhat p1u1 : g,

qU):0. Hence they are irreducible in tt,il. Let gr be the leadingcoefficient of p(^) and gz:4(0) and choosl g : 9$2. We shall showthat g has the required property for f . Thus suppose a is a homo-morphism of ?I into O satisfying g" : gigi + 0 and suppose r is anyextension of 6 to a homomorphism of ?I' -II[u] into A. Sinceg(f) :0 we shal l have g"( f ' ) - -0, which impl ies thal f '+ 0 s inceg"(0) + 0. Thus we need to show only that the hompmorphism 6can be extended to a homomorphism of ?I'. For this purpose letc be a root of P"(l) - 0 and consider the homomorphipm r' of ?I[,t]which coincides with a on ?I and maps .i into c. Lgt /t(i) be anyelement in the ideal $. Then the minimality of the {egree of..bQ)implies that there exists a non-negative integer & such that g\h(]')is divisible in ?Iti l by 2(,i). Since fr"(c):0, (gi)oh"(c) = 0 and sinceg{ + Q, h"(c) : g. Hence h(},)" - g and so $ is mapped into 0 byt' . It follows that r/ induces a homomorphism r of ?I' :2llu) =?lt,il/$ which is an extension of o.

Now assume the result holds for r - l. Let E = 2llu,l so that?I ' : Efur, . . . , t t , - t ) . Then thereexists an element hE, E such thatany homomorphism p of E such that hP + 0 has an extension r to? l ' s u c h t h a t f ' + 0 . B y t h e c a s e r : 1 ,

' t h e r e e x i s t s g e A s u c hthat any homomorphism d of ?I into 0 such that ,9" * 0 has anextension p to E - ?J[u,) such that hp + 0. Hence r is an extensionof o such that f' * 0 as required.

We now give the

IX. AUTOMORPHISMS

Proof of Theorem 7: The hypothesis on doP implies that or isan isomorphism of Olnl into O[!ft]. Let ?l : OlTtl"" s ?I/ : Olffil.If Er, - , ., r^ are the projection mappings of !m into @ we have?I' : ofnr, " ', fr^l: ?I[zrr, '", fr^). Since ?I' is an integral domainwe may suppose it imbedded in its field of quotients /'. Hencewe can apply Theorem 2 to ?l and ?lt. Let f be a non-zero elementof ?I/ - O[g?]. Then there exists a non'zero element g e @[It] suchthat if y is an element of Tt such that g(y) * 0 then the homo'morphism o: h"P -, h(y) of 2I : Al\t\"' into O, which satisfiesgoPn : g(y) + 0, can be extended to a homomorphism r of. OIfltl

into O satisfying f' + 0. We have seen that r has the form k -- h(x)

where r is an element of Sl. Then f (x) + 0 and for every h e sll,

h,nP : hnPn. This means that h(P(r)) : h(y). Hence P(x) -- y and

the theorem is proved.

Let B beclosed fieldof 8. Let

(11)

2. Coniugaea of Cartan subalgebraa

a finite-dimensional Lie algebra over an algebraicallyof characteristic 0 and let S be a Cartan subalgebra

8 :0 * I8 "

be the decomposition of 8 into root spaces corresponding to theroots 0, d, B, ... of 0 acting in 8. If lz e 0 and eu e g,,, thenthere exists an integer r such that e*(ad h - a(h)I)' : g. This isequivalent to the condition that a(h) is the only characteristic rootof the restriction of ad h to 8,r. The a are linear functions on 0.I f r e S p ( p - 0 o r p + 0 ) t h e n l x e r l - O o r p * a i s a r o o t a n d

lxe") e 8o**. In the latter case either llxerle'l : 0 or g * 2a is aroot and ffre,le,) € 8o*rr. If we continue in this way and we takeinto account the fact that there are only a finite number of distinctroots we see that r(ad er)k :0 for suffi,cientty high ft. This impliesthat ad e, is nilpotent for every ed e 8e, d + 0. It follows that ifnrr r Srrr " 'r €rk € 8r' dtr dzr " 'r dk non'zero roots, then

(12) ? : exp (ad e",) exp (ad edz) .. . exp (ad e"*)

is an invariant automorphism of 8.Now let (ht, hz, " ' , ht, €t+rt " ', 0o) be a basis for 8 such that

(hr, hr, ., ., h) is a basis for Q and the elements e;11, . . . , €n are inroot spaces 8r, a. * 0. Let ]r, . .., f,n be indeterminates, P -

271

LIE ALGEBRAS

o(f,r, -.., i*) and form the element I

/ t \(\];h; ) u*p (ad, )4+(t+') . . . exp (ad i"e")

( 1 3 ) \ i /

: *

b;( f , , , . . . , f ,o)h; * i ,p,( l , , . . . , f ,n)et

where the f; and fu are polynomials in the i's. These determinea polynomial mapping

(14) r, i Eihi * il,n,-r ! b;G)h;+ > f^nei

in 8.

The product a.p ... p of the non-zero roots is a non-zero poly-nomial function. It follows that there exist Ei e 6 such that if20 - \lfihi, then a(ho)B@o) ... p(ht) + 0. Then the characteristicroots of the restriction of ad ho to 8, * 8p + .. . * 8o are all differ-ent from 0 and so this restriction of ad &0 is non-singular.

We shall now calculate the differential dnoP of. P at ho. Forth is purpose we le t r :h*e , h :Z lErhr , e :2 i * r€p1, le t f be anindeterminate and we consider

P ( h a + 4 h + e ) )-- (ho + th) exp (ad tE*qp.t) exp (ad tEr.ze*z)

. . . exp (ad tE"e")

= (ho + th)(L * ad tE+gt+r). . . (t * ad tE*e,)t (mod /'): h0 + th + h0 adtE*p*r * .. . + h0 ad.tF^ea (mod f'z): ho * th * tErcrlhoer*rl + ... * tEofh,e,l , (mod t'z): ho + th * tlhoel (mod fr) .

If we compare this with (7) we we see that dnoP istthe mapping

( 1 6 ) h * e - + h * [ h o e l .

Since h-+h and e->lhoel are non'singular it follows that droP issurjective. We are therefore in a position to apply Theorem 1.Accordingly, we have the following result: If ,f is a polynomial

function * 0 on 8, then there exists a polynomial function g + 0on 8 such that if y e 8 and g(y) * 0, then there is an r in 8 suchthat P(r) - y and J@) + 0.

'We recall the definition of a regular element a of 8 as an ele'

IX. AUTOMORPHISMS 273

ment such that ad a has the minimum number l' of 0 characteristicroots. We recall also that if c is regular, then the set of vectorsZ belonging to the characteristic root 0 of ad a is a Cartan subal-gebra. It follows from this that if 0 is a Cartan subalgebra and0 contains a regular element a, then e is just the collection ofelements h e 8, such that h(ad a)' :0 for some integer r. We shallneed also the characterization given in g 3.1 of regular elements.For this we take the element u - }Ir)';h; * I,i*rlfit in 8r, P -

A(lr, - .., f,n) and we consider the characteristic polynomial

f "(A): det (i l - ad u): f ,n - r {A;)}"- t + . . . + (-L)"- ' ' ro- , , ( ) , r )A"

(17)

Then ro-1'(A1, ..., f,n) is a non-zero homogeneous polynomial of de-gree n - l' in the I's and if x - Zt;h; * \Eiet, then r-+ r,,-1,(r) :t,-y(Eu . . . , E) is a polynomial function on 8. The element r isregular in 8 if and only if rn-y(x) * 0. (It will be a consequenceof the theorem we are going to prove that l' : l.)

We now consider again the Cartan subalgebra S and the basis(hr, ' ' ' , h, , et+rt ' ' ' , €n) for 8. We apply Theorem I to the poly-nomial function .f : rilr,which is + 0 since tn-r, * 0 and op is anisomorphism. Accordingly we see that there is a non-zero poly-nomial function .q on 8 such that every y e 8 satisfying g(y) + 0has the form P(r) where .f (x) - r,_t,(P(x)) : ro_t,(y) + 0. Henceevery y such that .q(y) + 0 is regular and if x - I,,l:g ,h; * Li*rEiei,then y - P(r): (> E,h;)(expadEn&n) ... (exp adi*e*) - hn whereh:28;h and T is an invariant automorphism. Thus y is theimage of an elemerlt h e 0 under an invariant automorphism. Itfollows that h: yn-t is regular. It is now easy to prove the con-jugacy theorem for Cartan subalgebras.

Tsponnu 3. If Q, and bz are Cartan subalgebras of a finite-dimensional Lie algebra ouer an algebraically closed field of charac-teristic 0, then there exists an inuariant automorphism q such that0l : 0'.

Proof : There exists a non-zero polynomial functioo g; such thatif y is an element satisfying gi(y) + 0, then y - h'lt, h; a regularelement in 0; and h an invariant automorphism. Since g$z * 0we can choose y so that gr(y) + 0 and gz}) + 0. Then y : hl' :hzz, h; a regular element of Qr, I; ?n invariant automorphism. Thenhz: hl., T : rlf i tzt. Since hi is regular and is contained in ,br it

274 LIE ALGEBRAS

follows that 0e - Fl.Remarks. It is a consequence of the theorem that levery Cartan

subalgebra contains regular elements and that all Cartan sub'

algebras have the saine dimensionality / which is thd same as the

number // indicated above. It is easy to see also thalt if the nota-

tions are as before, then the regular elements of 8 belonging to Qare just the elements fto such that a(ho)P(h') "' p(h') + 0.

3. Non-isomorphism of the split aimple Lie al'gebras

We shall apply the conjugacy theorem for Cartafi subalgebras

first to settle a point which has been left open hit\erto, namely,

that the split simple Lie algebras which were listed in $$ 4.5-4.6

these algebras are given in the following

di,mensi,onalityt(t + 2)t(zt + L)t(zt + r)t(zt - r)

T45278

1332'48

are distinct in the sense of isomorphism. We recall that these

w e r e : A u l Z l , B t , l > 2 , C t , 1 2 3 , D t , l > 4 , G r , F r , ' , D a , E , a n d E .The dimensionalities oftable:

ty0eAtBtCt,DrG,FrEaEtEa

For the classical types and for Gr, Fn and Eu this was derived in

$ 4.6. 'W'e

have proved the existence of Et and .Ear in S 7.5. Thedimensionalities of these Lie algebras can be derived by determin-ing the positive roots directly from the Cartan matrices. We shallnot carry this out but we shall assume the result lfor these twoLie algebras.

To prove that no two of the Lie algebras we have listed areisomorphic it suffices to assume the base field algebfaically closed.This is clear since 8, = 8, implies 8,r zgrn for any extension Pof the base field. We therefore assume 0 algebraically closed.The subscript / in the designation Xt (e.g., At, E) fdr our Lie alge'bras is the dimensionality of a Cartan subalgebra. lThe conjugacytheorem shows that this is an invariant. Hence npcessary condi-

IX. AUTOMORPHISMS

tions for isomorphism of Xt and, Yy are I : l' and. dim Xr : dim Yr,.A glance at the list of dimensionalities shows that the only possible

isomorphisms which we may have are between Bt and Ct, I > 3and between Bo and Eu and between Cs and 86. The latter twohave been ruled out in an exercise (Exercise 7.6). It thereforeremaini to show that Br 7L Ct, I > 3.

Since Cr has an irreducible module in a 2l'dimensional space itwill suffice to show that if ![t is an irreducible module f.or Bt suchthat ! [ tBt*0, then dimgft >21 +1. We could establ ish th is, asin a similar discussion f.or G, ($ 7.6), by using the fact that theset of weights is invariant under the Weyl group. However, w€can now obtain the result more quickly by using Weyl's dimen-sionality formula. We observe first that (8.41) shows that if llt isan irreducible module of least dimension satisfying ![tBr * 0, then!ft is a basic module (cf. also Exercise 7.13). The dimensionalitieso f these are ( " f ' ) , k : I ,2 , . . . ,1 -L and 2 t (c f . (8 .45) and Exerc ise8.12). Since I > 3, these numbers exceed 2/.

This proves our assertion and completes the proof that Br andCr ?r€ not isomorphic if , = 3.

4. Automorphisme of semi-simple Lie algebrasoaer en algebraicalla clased, field

Let 8 be a finite-dimensional semi-simple Lie algebra over analgebraically closed field of characteristic 0, 0 a Cartan subalgebra,f r : {dr , . . . ,dt} z s imple system of roots relat ive to €), e i , f ; ,h; ,i: I,2, ' ..,1, d set of canonical generators for 8 determined by z.Thus the h form a basis for Q, 0r € 86n, ft, € 8,-"n and we have thefollowing relations:

(18)

[h ;h i l : 0

leJil - 6.iihi

le;hil: Ai;€;

Ifthl - - Arrf,

where (Ad is the Cartan matrix determined by z.Let r be an automorphism. Then 0" is a second Cartan sub-

algebra. Hence there exists an invariant automorphism a suchthat 0' : 0". Then the automorphism r' : 16-' maps 0 into itself.We now consider an automorphism r(:c') which maps the Cartansubalgebra 0 into itself. lf e, e I, we have le"h) - a(h)e". Hence

275

276 LIE ALGEBRAS

leLh'\ - a(h)eL. It follows that 8l : 8p where B is a root' In thiswayweobtain a mapping d--rp of thesetof roots. Since e, 'e 8Bwe have leLh") - p(h")el. Hence we see that '

(19) a(h) -* P(h')

Let rx d,enote the transpose in Q* of the restriction qf r to'Q. Bydefinition, if E e,b*, then t"(h):f(h'). Then (19), implies that

0,* : d ot g - d,",-'. Hence we have the followingl

PnoposrrroN 1. Let t be an automorphi,sm of I such that $' - p

for a Cartan subalgebra 0 o/ S. Then if a is any ro'ot of b in 8,

(20 ) 8 | :86 ( r+1 - l

where r* is the transpose in b* of the restriction of c; to Q.

We note next that if fr : {dr, .. ., a,r} and Fi - d\' i ')-r, then z' :

{Fr, .- ., B,} is a simple system of roots. Thus ever}r root has the

form + E k&i where the &i are non-negative integerd. If we apply("*)-t we see that every root also has the form *. Z kih This

guarantees that z' is a simple system. We prove next

PnoposttloN 2. Let n and n' be si,mple systems of roots. Then

there exists an inuariant automorphism o such that Q" - $ and ota*t-t -

Tt' .

Proof : Le t r : {d r ,dz t ' ' ,d r } , r ' : {F r , " ' ,0 t } . Then we know(Theorems 8.1, 8.2) that nt : TcSar,,S"r, '" $oir for suitable Weyl

reflections S,,. It is therefore clear that it suffice$ to prove the

result for r'- zS"a. For this purpose we introduce the invariant

automorphisms expad Efi and expad €et, E e A. In lour calculation

we shall use the formulas for the irreducible representations of

the three-dimensional split simple Lie algebra given in (36) of $ 3.8.

Wenotethat the matrices of the restrictions of ad/6, adei to 8; -

Aei * Ahi * Ofi, using the basis (er, hi,lh;f;l) are, respectively,

Hence for exp adEfi, expad Ee; in 8,i we have the matrices

r))' *''-'(+ j, s)(2r) "or -, (l s

(22) A{E): (i i ' ';\, B,(E': ( ̂ 'r i\o o t l \2f ' -zE

IX. AUTOMORPHISMS

It follows that the matrix of the restriction to 8; of

(23) o;(E) = expad Ef;expadE-'erexpadti

is

f,;)(24) A;(E)8,(E-')a,(t) -- ( 3 j,

\28_, o

In part icular, h i i (o - -hr Next lethebsat isfy au(h)-_ 0. Then

lherl: 0 : lhfil and consequently lf ttE) : h. Since 0 is the directsum of Ah and the subspace of vectors /l such that a;(h) : 0 itis clear that Q';'g'= b. Moreover, the restriction of ar(F) to 0 isthe reflection determined by h;: h --> h - lZ(h, h;)l&t, h;)lh; - h -

[2(h, h,)l(h,1, h,)lhat h - l2a,(h)loti(h,t))h,r lf p e $J*, then

o(n - 21u,(h), /,",) : p(.h) - 'o\l"r), a{h) : p(h) - ?(l'o'), a{h) .-

\ d { h " ) ' "

/ d { h ' ) \ d i , d ; )

This shows that the transpose inverse of the restriction to .b ofor(f) is the Weyl reflection Soi in 0*. Hence the invariant auto-morphism al(f) satisfies zrc;{€)*)

- ' - vSot: vt , as required.Proposition 2 and the considerations preceding it show that if r

is an automorphism such that 0' : b, then there exists an in'variant automorphism 6 such that b":0 and if r ' : ra-', theno((r')*)-' : r for the simple system ft. We simplify our notationagain by writing r for r' . Then we have a permutation i -' i' of.i - - I ,2, . . . , / such that o\ t ' ) - r * d i , . Also we have le ' rh i l - Zei ,

t f :h i l - -2 f : , te i f f l : h i . S ince e l e8 , , , , , f i e8 , - " , , wehave e i :

It i€i,, f i : v;f;,. Then l4l -- p;viht,. Since le;,hr, ' l :2er and [eihi):Zei so that plv; leyh;, \ :Zp&r, , we have vi : yt ; 'and hi : h. i , . Since

leih ' i l : Ai$ ' r , le; ,h i , ) : Ai ; i : r , . Hence we have

( 2 5 ) A n , j , - - A o j , i , j : 1 , 2 , " ' , 1 .

The subgroup of the symmetric group on 1, 2, - . . ,l of. the permuta-tions i -, i' satisfying (25) will be called the group of automorphismsof the Cartan matrix (A). If we recall the definition of the Dynkindiagram of the Cartan matrix, it is clear that if 4,, dzt .. ., d7 ztythe vertices of the diagram, then any element i -t i' of the groupof automorphisms of (At) defines an automorphism of the Dynkindiagram, that is, a 1 : I mapping d; -+ e,;, such that (dr., at) :(ar,,ar) and for any i, j, the number of lines connecting d4 to di

LIE ALGEBRAS

is the same as that connecting dr and, ai,(cf. $4.il. The argu-ment used to show that the Dynkin diagram determines the Cartanmatrix shows also that the converse holds, that is, if di-'+ 4r, isan automorphism of the Dynkin diagram, then i -- i' is an auto'morphism of the Cartan matrix.

Now suppose that r is an automorphism such that 81,t: &rt,i: !,2, ...,1. Then we have the identity mapping f -i, ' : i ' inthe foregoing argument. This shows that ei : F&r f: - po'fo,hi : hr. Hence r acts as the identity in 0. Convetsely, Prop. 1

shows that if the restriction of r to 0 is the identity,'then gL: 8,

for every d so ef, : pier f', -- trtt f r For these autortrorphisms we

have the following

PnoroslrtoN 3. If c as an automorhhism such that h' -- h foreaery h of a Cartan subalgebra O of 8, then r ls an inaariantautomorphism.

Proof: We have seen that ef, -- p&; and, f{ - piL.fi. Let o;(6) bethe invariant automorphism defined by (23) and let

(26) ou(E) - ot(E)oi(I)

We have seen that hli(E' - -hi, hai(Et - h if at(h)that h-irit : h for all h. The matrix relative tothe restriction of. ot(E) to 8; is the product of theby the matrix obtained

(n)Hence we have

(28)

from this by taking 6 : 1.

d iag{ t2 , l , t - ' } .

e?i'et : E'er: Enooe, .

- 0. It follows(ei, h;,lhJtl) ofmatrix in (2+1The result is

We wish to calculate next e?t'E' for j + i. We have 1fr;1: 0 and

lfihr): -A,iifi. Hence fi generates an irreducible inodule for 8i.There are four possibil i t ies -A;t:0, 1,2 or 3. If. A;t: -2 themodule is equivalent to 8; and the argument just used shows that

fe;Gt - g-eufi and this implies that e?i''' : (-F)tti€i. lf Aii - Q,

[,fr8r] : 0 and this implies that elict - et: (-E)t;le;. Next letAu : -L. Then the matrices of f;, e; acting in the module gener-ated by f5 are, respectively

(n) lo 1\ /0 0\\o o) ' \-t o) '

It follows that the matrix of a;(f) is

(30) (_? , iland that of ar;(f) is diag {-8, -5-'}. Hence we have f}'G' : -E.fi

znd e'i iGr : -E-tet: (-E)oiiei. Finally, let A;i: -3. Then the

representing matrices forf; and e; are, according to to (36) of $ 3.8:

l 0 1 0 \ l 0 0 0 01

(31) fl o 1 ol. [ -^s o, I l l\ - - l \o o o r l ' Io -4 o o / -\ 0 0 0 0 / \ 0 0 _ 3 0 t

matrix of. oa({) as defined

t 0 0 0 t ' l 6 tI o o -rE ol.I o zE- 'o o l\ - 6 F - ' 0 0 0 l

IX. AUTOMORPHISMS

It follows that the

(s2)

in (23) is

Consequently, the matrix of ornE) in the module generated by ftis diag {-t" , -8, -E-' , -F-t}. Hence f f'u' : -E'f i and solitet -

?E)-'ei -- (-E)^njei. Thus in all cases we have

(33) seit€t - (-E)^niei

( i : i o r j + i ) . N o w s e t

(34) Q = Q(E' ,FI , . . . , F,) : co(-€r)on(-Fr) " ' t ' t { -E) .

Then (33) implies that

(35) e l - g l ' , tE f i . - . € f ie i , i : 1 ,2 , - - . ,1 .

We recall that the matrix (Ad is non-singular and its determinantis clearly an integer d. Hence we have an integral matrix (Brr')

such that (Ad(Bi : dI. Let i be fixed and set Eo: (p'fd)"i*,

h : l , " ' , 1 . T h e n

o0 (E 1..... € 1t _ 7p2rrt

d1a i tA I i+. " + B i. a t i e i : errl \r, e i : / qe j(36)

e'

g? {Er . . . , e t t : e " r r l o ) " i rA | i+ " '+B i t ^ , t rn : e , i f i + j .

It is clear that a suitable product of the invariant automorphismsdefined here for j : I,2, . . . ,l coincides with the given auto'morphism r in its action on the ei,f;, h;. Since these are generatorsit is clear that r is an invariant automorphism.

It is now easy to see that the index of the invariant subgroup

280 LIE ALGEBRAS

Go in G does not exceed the order of the group of adtomorphismsof the Cartan matrix. In fact, we can display a sublroup K of. Gwhich is isomorphic to the group of automorphisms qf the Cartanmatrix such that every r e G is congruent modulo Go, to a te e K.Let P: i-ri' be an automorphism of the Cartan matrix. Then wehave the relations

lh, i , ,h i , l -0, feo, f i , l : Bi lhy

fet'hi ' l : Aitet' , [f; 'hi ' f : - Ainfn' ,

Hence the isomorphism theorem for split semi-simplel Lie algebras(Theorem 4.3) implies that there exists a (unique) dutomorphismtp of 8 such that eI": er, .f{':f;,. The set ofr these auto-morphisms is evidently a subgroup Kof G isomorphictto the groupof automorphisms of the Cartan matrix.

Now let r be any automorphism of 8. We have seen that r iscongruent modulo Go to an automorphism r, such that 6r" : €).Also we know that rr is congruent modulo Go to an automorphismrz such that Q"z: Q and rc(t;t-r : n for the simple sy$tem of rootsT : {d r ,dz t - . . ,d t } re la t i ve to 0 . We have 8L1: &rn , , ' , i : 1 ,2 , . . . ,1and i -r i' is an automorphism P of. the Cartan matri*. Let r.r" b€the corresponding element of K. Then 6 : tztFt satisfies 8Zr: 8r*ga-or. 9-rn, i : 1,2, . . .,1. Hence o is an invariant automorphism,by Prop. 3. Thus r is congruent modulo Go to rr.

It can be shown that no element of K is in Go, urhich meansthat G is the semi-direct product of. K and Go. This is equivalentto showing that the index of Go in G is the order of the group ofautomorphisms of the Cartan matrix. A proof of this will beindicated in the exercises. We shall now restrict our attention to8 simple. The result stated will follow for the ll-ie algebrasAr, Bt, Ct, Dt, I + 4, Gz and Fn from the explicit determination ofthe groups of automorphisms for these Lie algebrps which weshall give in the next section. The result will alsq be clear forEt and. Ee.

We now examine the groups of automorphisms of the connectedDynkin diagrams. We recall that these are the ones which corre-spond to the simple Lie algebras 8. The types are tAt, I 2_ 7, Br,1 2 2 , C , l > 3 , D t , 1 2 4 , G r , F r , E a , E z , E a . I f w e l o o k a t t h e s ediagrams as given in $4.5 we seethat for Ar,Bt,Cr,Gr,Ft ,E, andEe the group of automorphisms of the diagram is lthe identity.

IX. AUTOMORPHISMS

1 1 1 1Fof ,4r, I > 2: a-o. . . s-e- we have in

dr dz dt-r dt

identity mapping the automorphism dt-, dt+rt.

1

281

addition to the

For

I I I - z o d tD r : a - ; . . . 0 1d1 d2 dt_z \9 o,_,

I

we have the identity automorphism and the mapping dr'-+ dii < I - 2, dFr -'+ dtt dt -+ dt-t which is an automorphism. Theseare the only automorphisms if / > 5. For /: 4 the diagram

Dr i o-o1'" o'

d1 dz \o dg

has the group of automorphisms which permute dt, ds, dl and leaved2 fixed. This is isomorphic to the symmetric group on threeelements. For

liae

Ea, 1-1_!1-l_:d1 d2 ds, dq, d5

the group of automorphisms consists of the identity mapping andthe mapping a6--+dat d;-) a6-;, i 35.

It is now clear that we have the following

Tuponpu 4. Let 8, be fi.nite-dimensional simple ouer an alge-brai,cally closed field of characteristi,c 0, G the group of auto-morphisms of 8, Go the inuari,ant subgroup of inuariant automor-phisms. Then G : Go unless 8 is of one of the following types:At, I > t, Dt or E6. In all of these ca,ses except Da, the inder[G : GoJ < 2 and for D', [G : Gu] 5 6.

Remarh. The group G is an algebraic linear group (cf. Chevalleytzl). It is easy to see that Go is the algebraic component of theidentity element of G. If @ is the field of complex numbers, thenG is a topological group and Go is the connected component of 1in G.

5. Explieit determination of the automorphismsfor the simple Lie algebras

Let 8 be a semi-simple subalgebra of @r,, 0 the algebra of linear

LIE ALGEBRAS

transformations of a finite-dimensional vector space tover o. Letz fu an element of B such that ad Z is nilpotent. since the alge-bra of linear transformations adg t3 is semi-simple it tfollows fromTheorem 3.17 that there exists an element H e I such that [zH] -z. This implies (Lemma 4, $ 2.5) that z is nilpotent. we havead Z - Zn - Zz (Zn: X - XZ, Zz: X -- ZX). Hence

exp ad Z : Exp (2" - Z") : exp Zn exp (.- Z) ,

since [Z*2"]: 0. Then :

(37) exp ad 7 - (exg Z)nGxp (- Z))"- (exp Z)nbxp Z);' .

If we set A : exp z, then the automorphism exp ad z of. 8 is themapping

(38) x -+ A-txA .

we now consider the simple Lie algebras of types iAt, Bucu Dt,Gz, and Ft.

At. Here 8 is the Lie algebra of linear transformattions of tracezero in an (/ + l)-dimensional vector space. we can identify gwith the Lie algebra of matrices of trace 0 in Ot+r. lt A is a non-singular matrix, then X-+ A-|XA is an automorphistn of 8. Sincethe only matrices which commute with all matrices bf trace 0 arethe scalar matrices it is clear that the automorphism$ X-- A-'xA,X -. B-'XB are identical if and only if B : pA, p e A. Besides theautomorphisms X--+ A-IXA we have the automorphfism X-- -X,of 8 and, more generally, we have the set of atrtomorphismsX ---+ - A-'X'A. Suppose the automorphism X -, -X' coincideswith one of the automorphisms X --, A-'XA. Then we have

(39) A- 'XA: -X'

for all X of. trace 0. This implies that

-X - A' X'(A')-r - - A, A-'XA(A')' |

so that A(A')-' commutes with all X. It follows tthat A' : pAihence At : *.A. The condition (39) can be rewrittenr as A-'X'A :-X. The set of X's satisfying this condition is either the sym-plectic Lie algebra or the orthogonal Lie algebra. Accordingly thedimensionality is either (l + t)(l + 2)tZ (odd / only) or I(l + t)lZ (/ evenor odd). Since the dimensionality of the space of (l * l) x (/ + t)

IX. AUTOMORPHISMS

matrices of trace 0 is 12 + 21, we must have either l' + 2l :( / + l X / + 2 ) l Z o r l ' * 2 1 : l ( l + l ) 1 2 . T h e o n l v s o l u t i o n i s l : Lin which case /' * 2l : (/ + lX/ + 2)lZ. Thus we see that X-'-X'coincides with an automorphism of the form X- A-'4A only if

f ; "it,j;t;.,n..

we have -x' : A-'xA for A: (j, l) "'u

The result obtained at the beginning of this section shows that

every invariant automorphism of ,8 has the form X-, A-|XA where

A is a product of exponentials of nilpotent matrices. For the Lie

algebra A, this is the complete automorphism group. For At,

I > l, we have the automorphism X -t -X' which is not invariant.

Hence lG: Gol - 2 and the automorphisms are the mappings

X -' A-|XA and X '- - A-'X'A.

Tnponpu 5. The grouf of automorphisms of the Lie algebra of

2x2 matricesof trace 0 is the set of ma\bings x-- A-'xA. The

g:roup of automorphisms of the Lie atgebra of n x n matrices of

iroru 0, n > 2, is the set of mafpings X-+ A-tXA and X-' - A-'X' A'

(The base f.eld a is algebraically closed of characteristic Q.)

Bt, Ct, Dt. These algebras are the sets of matrices X satisfying

S-tX'S - -X where S: 1 for Bt and Dt and S' : -S for Ct'

The siie of the matrices is 2t for Ct and Dr and 2l + 1 for Br'

W e t a k e t > 2 f o r B t , / > 3 f o r C r a n d l > 4 f ' o r D t . L e t O b e a

matrix such that

(40) O'SO : PS

where p + O is in @. Then O is non-singular and if X e 8, then

Y: O-'XO satisfies

S-t y,S : S-t(O-tXO),S = S-rO, X,(O,)- IS

(41) : -S- 'O'SXS-'(O')- 'S: -pO- 'X(p- 'O)

: -O- 'XO : -Y .

Hence Y e 8 and consequently X -r Y -- O-'XO is an automorphism

of 8. Since the base field is algebraically closed we can replace

O by p-LO : O, and we obtain Or'XO, - Y, OiSO' : S. If we

write O f.or Or, then we see that for Br and D,, O is an orthogonal

matrix (S: 1) and for C,, O is a symplectic matrix. It is easy to

verify that the enveloping associative algebra of 8 is the complete

matrix algebra. We leave it to the reader to prove this. It follows

284 LIE ALGEBRAS

that the only matrices which commute with all the dlements of 8are the scalars. Hence if Or'XOr: O;'XO, for all X e 8, thenO z - - o O y p € A .

Let Z be an element of 8 such that ad Z is nilpoterft. Then wehave seen that Z is nilpotent and the automorphismt exp ad Z hasthe form X-+ A-'XA where A:expZ. The nilpotqnce of Z im-plies that exp Z is unimodular (Exercise 5.4). Also, We have

S-'(exp Z)'S - S-'(exp Z')S- exp S-'Z'S: €xF en - (exp Z)-' .

Hence A : €xp Z is orthogonal for Bt and Dr and symplectic forCt. For ,Br and Ct every automorphism is an invariant automor-phism. Hence in these cases the automorphisms of 8 have theform X-'O-'XO where O is unimodular and satisdes O'SO: S.For Bt this states that O is a proper orthogonal dnatrix (corre-sponding to a rotation). For Cr it is known that if Ol is symplectic(O'SO : S) then O is necessarily unimodular (see Artfin [1], p. 139,or Exercise 12 below). Hence this condition can be dropped. Nowconsider Dt. In this case there exist orthogonal matrices O ofdeterminant -1 and we cannot have O : pO, where O, is a properorthogonal matrix. Thus it O : QOr, then p: +I and in eithercase det O : det pOt : det Ot : 1. This contradictiort implies thatthe automorphism X--'O-|XO, where O is improper brthogonal, isnot invariant and we see that G = Go. If I > 4, then the indexof Go in G is 5 2; hence this index is 2 and every hutomorphismof. Dt has the form X-rO-'XO where O is orthogonal. We there-fore have the following

Tnsonnu 6. Let O be algebraically closed of characteristic 0 andlet 8, be the Lie algebra of skew matrices or the "symplectic" Liealgebra of rnatrices X such that S-tX'S - -X where S': -S.

Assume the number of roLt)s n 2 5 in the odd-di,mensional skewcase, n ):6 in the symplectic case and n Z l0 in the eaen-dimen-sional skew case. Then the groups of automorphism, of 8, in theshew cases consist of the mappings X-->O-|XO wherg O is ortho-gonal. In the odd-di.mensional case one can add the condition thatO is brofuer. In the symflectic case the group of automorphisms isthe set of mappings X-tO-'XO where O'SO - S.

The case of. D, is not covered by this theorem. It can be shownthat in this case the group GlGo is isomorphic to [he symmetric

I


group on three letters and the group G can be determined. This

will be indicated in some exercises below.We consider next the Lie algebra Gz. Here we use the representa'

tion of 8 as the algebra of derivations of a Cayley algebra 6 ($ 4.6).

Now, in general, if O is a non-associative algebra, D a derivation

of G, A an automorphism of 6, then A-'DA is a derivation. It

follows that the mapping X--+ A-'XA determined by an automor-phism of 6 is an automorphism of the derivation algebra 8 - E(6).

In the case of O, the Cayley algebra over an algebraically closed

field of characteristic 0, we know that every automorphism of B

is invariant and so it is a product of mappings of the form

X-' A-'XA where A: axP Z, Z in 8 and Z a nilpotent linear

transformation in 0. Since Z e 8, Z is a derivation of 0. 'It follows

that A : €xP Z is an automorphism of 6. We therefore see that

every automorphism of 8 has the form X -, A-'XA where A is an

automorphism in 6.

The same reasoning applies to the Lie algebra Ft. Here we

represent 8 as the derivation algebra of the exceptional Jordan alge'

bra Mi and we obtain the result that the group of automorphisms

of 8 is the set of mappings X -- A-'XA where A is an automor-phism of. Mi.

Tnponpu 7. Let 8, be the Lie algebra of deriuations of the Cayley

a lgebraE 'oro f the lo rdana lgebraMfouerana lgebra ica l l yc losed

rt.eM of characteri,stic 0. Then the automorphisrns of t haug the'form X--+ A-,XA where A is an automorphism of 6' or of Mf .

Exercises

The base field in all of these exercises will be of characteristic 0 and all

spaces are finite-dimensional.l. Show that any non-singular linear transformation A can be written in

the form AoAc where Aw is unipotent, that is, Au:l + N, N nilpotent,

and At is semi-simple and A, and At are polynomials in 14. Prove that if

A: BtBo and Bo and Bs commute where Bo iS unipotent and 8c is semi'

simple then Au : Bw and Ar - BE. (Hintz Use Theorem 3.16.)

2, Let r be an automorphism of a non-associative algebra ?[ and let r =

rrrrr 3s ir 1. Prove that rr and cu are automorphisms. (Hint: Assume the

base field is algebraically closed and use Exercise 2.5. Extend the field to

obtain the result in the general case.)3. Let I be a finite-dimensional simple Lie algebra over an algebraically

286 LIE ALGEBRAS

closed field and let Go be the group of invariant automorphisnis of 8. Provethat 8 is irreducible relative to Go.

4. Let G be the split Cayley algebra over an algebraically closed field,Go the subspace of elements of trace 0 in G: a * d : 0. We ftrave seen thatthe derivation algebra t(:6r; acts irreducibly in 0o ($7.6). Use this resultto prove that Go is irreducible relative to the group of automorphisms G of G.

5. Same as 4. with G replaced bv lff (cf. g 4.6).Exercises 6 through 9 are designed to prove that if I is semi-simple over

an algebraically closed field, then G/Go is isomorphic to the lgroup of auto-morphisms of the Cartan matrix or Dynkin diagram determintd by a Cartansubalgebra 0 of 8. In all of these exercises 8, 0, G, Gd', etc., are asindicated in the text.

6. I f a€8 and Eo:[zl lzal:} l , then dimGo ] l , the dimensional i ty of aCartan subalgebra of 8. Slcetch of proof: Note that dirn Gc : dim 8 -rank (ad o). Show that if o is a regular element, that Go : S a Cartan sub-a lgebra so d imGa: l . l f . (u , . . . ,x tn) is a bas is for 8 over @,and ( { r , . . . , fn)are indeterminates, then c : 2 €rut is a .regular element of 8p, P : @(tt) sorank(adr) :n-1 . I f 6 : \a lur the spec ia l izat ion h :Ft shows thatrank (ad a) < n - I. Hence dim Go 2 l.

7. If r is an automorphism in I let I, be the set of fixed points ofr: D" : D. Prove that if r is invariant, then dim I" 21. Strcetch of prmfz\il'e have r = €XF Zr exp Zz - . . exp Z, where Zt : ad zt is nilpotent and wehave toshowthat rank ( r -1) < n-L Let f i , . . . , f " be indeterminates andlet P be the field of formal power series in the 6c, that is, the quotientfield of the algebra Q < tu. . ., f" ) of formal power series in the fo withcoefficients in @. Then r(f) = exgtrZrexglzZz... expt Z, ii an invariantautomorphism of 8r. The matrix of r({) relative to the basisr (uu . . . , un) ofI has entries which are polynomials in the {a and the specialization fe: 1gives the matrix of r relative to this basis. Hence if dim lrrer 2 l, a special-ization argurnent will show that dim I" > | (semi-simplicity anrd the rank areunchanged in passing from I to 8p). Now the exponential formula can beusedto show that { f ) = expErZr . . .expt rZr :expZ where Z:ad?, ze 8p(Exercise s.LZ). Then I'rer f Gz so dim frrel Z I by 6.

8. Let r be an invariant automorphism such that 0" = 6 And ,n(e*)-t - r,for a simple system of roots t. Then h, : h for every h e 0. Sketch ofproof : (r'*)-t induces a permutation of the set of roots which we can writeas a product of cycles (9r, . . . , B)(Tu .. . , Ts) . . . . Since n(er)-r : n, (r*)-tleaves the set of positive roots invariant; hence the 8c in a cgcle (Bu ...,9r)are either all positive or all negative. 8pr * ... + 8B, is invhriant under r.If ap, is a base element for 8pn, then we have el, : vtaB2t oEr:),rza1s, . . . , e'Fr:,treFt so that the characteristic polynomial of the restrlction of r to8 f t + . . . + 8 B " i s 1 7 - v r . . . v 7 . I f v r . . . y r * L t h e n t - 1 i s i n o t a f a c t o r o fthis and consequently (8p, + . .. * 8Br) fl I, = 0. If this holds for every


cyc le , t hen . I , c 0 s i nce I = S@(8p r@ " ' @8p , )O(8v rO " ' O8 r , ) + " "

Since dim.[, ] t by 7. it will follow that .It - S and h" = h for all h e 0.

Now le tobe theau tomorph i sm inSsuch t ha t l r t =h , h€b and Qo ; =p f t 1 *0 ,

fi : pi'ft (et, ft,h, canonical generators)' We have shown that o is invariant'

If a is a positive root then o : I kst and tZ = p!'pt' "' u!',' and eo-a :

u lk rhokr . . . t rkae-r . The automorphism r '= or sat is f ies the same condi t ions. r ' ' , t ' I

as r and e ' p ' r= r ' r eq . , " ' , a -P r= ' t r aB1 whe re ' l " ' u ! - v l " u l p i t " 'P i l i f t he

Btare a l l ios i t ive 'and Fr* ' " * B,= I is iar*0 s t non 'negat ive in tegers ,

a n d v r . . . r l : v r . . . v l F i " . . . h t ' t i f t h e 8 r a r e a l l n e g a t i v e a n d B r * ; . . * 8 r =

- Lirrnr. SinEe e, * 0 for sotne i we can choose the p's so that u{ " ' ti + 1

forlvery cycle. Then the argument used before shows that h: ' =h, h€8'

H e n c e l L r = h , h e S .9. Prove that G/Go is isomorphic to the group of automorphisms of the

Cartan matrix.10. Let I = (Di-r, I + 1 : 2r and let r be an automorphism of' the form

f,-r - A-LX'A in 8. Show that dim I, zr and that there exist r such that

dim ft : r.ll. Let I be semi-simple over an algebraically closed field and let I/ be

the subgroup of Go of 7 such that S4 g F, K the subgroup of Go of I such

t h a t h r = h f o r a l l f t . € s . S h o w t h a t K i s a n i n v a r i a n t s u b g r o u p o f l l a n dthat I{/K = W. (This gives a conceptual description of the Weyl group')

12. Use the proof of Theorem 6 to prove that if O is a symplectic matrix

with entries in a field of characteristic 0, then det O : l'

1 3 . L e t O b e a s p l i t c a y l e y a l g e b r a a n d l e t d : a I - Q ' o i f a = a l * o o ,

o o € G o . S e t N ( o ) : q , d : d a a n d ( o , b ) : l l N t o + b ) - N t o ) - f V t U ) l ' V e r i f y

that (o, b) is a non-degenerate symmetric bilinear form of maximal witt

index. Prove that N(ac, b) : N(o, bZ) and N(co' b) : N1o' 6b)' Hence show

tha t i f c€Go , t hen ca ( t+ r c ) , c t ' l r +c r ) and p , : l ( c r ' +cn ) a re i n t he

orthogonal Lie algebra (Dr) of the space G relative to the form (o' b)' Show

that every element of this Lie algebra has the form Bt + I [8.,Fa11 where

c, c i , d t € Go.l{. Use the alternative law (eq. (4.79)) to prove the following identity in G

c(ab) * (ab)c : (ca)b + a\bc)

or Z(ab)Rc - \a,cr,)b * a(bcn). Use this to prove the principle of local triality:

For every linear transformation A in G which is skew relative to (o, b) there

exist a unique pair (8, c) of skew l inear transformations such that

(ab)A : 1o.B)b * otbC) .

15. Show that the mappings A - B and A -* C determined in 14 are auto'

morphisms of the orthogonal Lie algebra. Prove that every automorphism

of this Lie algebra is of the form X -. O- tXO where O is orthogonal or is

the product of one of the automorphisms defined in 14. by an automorphism

X - O - |XO.

288 LIE ALGEBRAS

16. Let 8r and 8s be two subalgebras of Dr isomorphic tb Ds. provethat there exists an automorphism of. Dt mapping 5J1 onto fJ2.

17. Show that the automorphism in At, I ) l, such that fu - ft, ft - et(cf. p.127) is not invariant. Show that for Du I24, this automorphism isinvariant if and only if I is even.

18. (Steinberg). If 6 is a Cartan subalgebra let Go(b) be group of auto-morphisms generated by au exp ad e where e belongs to a rdot space of grelative to 6 (cf. Chevalley [7]). Let &rr be a second Cartan Subalgebra andlet Go(6r) be the corresponding group of automorphisms. Show that thereexists ? € Go(S), ?r € Go(br) such that S? : S?t. Then S : i0ltr-t. Thisimpl ies that Go(0) :4r7r 'Go(f i r )TrT-, so that

Go(6) : t i-tGo(p)n : TlLtGo($)71 : Go(Or) . l19. Prove that Go(6) : Go, the group of invariant automorpfrisms.

CHAPTER X

Simple Lie Algebras overan ArbitrarY Field

In this chapter we study the problem of classifying the finite-

dimensional simple Lie algebras over an arbitrary field of charac-

teristic 0. The known methods for handling this involve reductions

to the problem treated in Chapter IV of classifying the simple Lie

algebras over an algebraically closed field of characteristic 0. One

first defines a certain extension field /' called the centroid of the

simple Lie algebra which has the property that 8 can be considered

as an algebra over I' and (8 over /-)o : J7-s.r8 is simple for every

extension field I of. T. It is natural to replace the given base

field @ by the field /'. In this way the classification problem is

reduced to the special case of classifying the Lie algebras I such

that 8o is simple for every extension field g of the base field. If

8 is of this type and J? is the algebraic closure, then the possi-

bilities for ,8o are known (A1, Bt, Cr, etc.) and one now has the

problem of determining all the 8' such that 8o is one of the known

simple Lie algebras over the algebraically closed field .?.

This problem can be transferred to the analogous one in which

the algebraically closed field I is replaced by a finite'dimensional

Galois extension P of. T and 8r is one of the split simple Lie alge-

bras over P. This is equivalent to the problem of determining

the finite groups of automorphisms of 8p considered as an algebra

over A which are semi-linear transformations in )3p over P.

We shall consider this problem in detail for 8r one of the classi-

cal types At - D except Dr. Our results will not give complete

classifications even in these cases but will amount to a reduction

of the problem to fairly standard questions on associative algebras.

For certain base fields (e.g., the field of real numbers) complete

solutions of these problems are known, so in these cases the

classification problem can be solved.The classification problem for the field of real numbers is quite

old. A complete solution was given by Cartan in 1914. Simplifica-tions of the treatment are due to Lardy and to Gantmacher. For

[28e]

290 LIE ALGEBRAS

an arbitrary base field of characteristic 0 the results for the classi-cal types are due to Landherr and to the present aluthor, for Gzto the author and for F. to Tomber. It is worth notilng that mostof the results carry over also to the characteristid p + 0 case.This has been shown by the author. References to the literaturecan be found in the bibliography.

7. Multiplication algebra and, centroid of anon- awoeiative algebra

Let ?I be an arbitrary non-associative algebra over la field o (cf.S 1.1). lf. a e [, the right (left) multiplication aa (a")iis the linearmapping tr --+ tra (r -+ ax). we define the multiplication atgebra T(Dt)to be the enveloping algebra of all the an and at, s e A. ThusE(?r) is the algebra (associative with l) generated hy the ar,

"rrd,an. If ?I is a Lie algebra, then !(?I) is the enveloping algebra ofthe Lie algebra ad ?r. we define the centroid r(?I) of ?r to be thecentralizer of E(?I) in the algebra 0(?r) of all linear transformationsin ?I. Thus the elements of /- : /'(2I) are the linear transformations7 such that [r, A]:0 for all A e !(?I). Evidently,'y e T if andonly if lvanJ: 0 : [rar.] f.or all a e 21, and these conditions can bewritten in the form

( 1 ) ( a b ) r : @ f i b - a ( b r ) a , b e ? I

Lnuur 1. If A' : T, then I- i,s commutatiue.Proof : Le t y ,6e I ' , a ,beTI . Then (ab) r t - ( (a r \b )d- (a : )e | )

and (ab)r8 - (a(brDd : (a|)(bi. If we interchange T and d weobtain (ab)dr : @6)(bi - @)(bS). Hence (ab)(rd - dr) : 0. Since?It : ?[, any element c of ?l has the form c - D a;hr It followsthat c(rd - dr) : 0 for all c and f" is commutative. I

A non-associative algebra ?I is simple if tr has (two-sided)ideals + 0, + [ and ?l' + 0. since ?12 is an ideal it follows that?l' : 2I for ?r simple. Hence the lemma implies that /' is commuta-tive.

The ideals of a non'associative algebra ?r are just the subspaceswhich are invariant relative to the right and left multiplications.These are the same as the subspaces which are invariant relativeto the multiplication algebra E = t(?l). It follows that ?r is simpleif and only if ! is an irreducible algebra of linear tralnsformations.lf. r e 2I, then the smallest E-invariant subspace contdining r is rE.

rloitr

X. SIMPLE LIE ALGEBRAS OVER AN ARBITRARY FIELD 29I

Hence if. x+0 and U is simple, then rT.:?J. The converse is

easily seen also: if ?I'z * 0 and rT - ?I for every x + 0, then tr is

simple. We recall the well-known lemma of Schur: If ! is an

irreducible algebra of linear transformations, then the'centralizer

of A is a division algebra (for proof see, for example, Jacobson

[2], vol. lI, p. 27L). In the special case of ! - E(?l) for ?l simple

this and Lemma 1 give

Tnponou L. The centroid f of a simple non'associatiue algebra is

a field.Since the centroid I' is a field we can consider ?l as a (left)

vector space over l 'by setting Ta:&T, & €?I, 7 € l ' . Then con-

dition (1) can be re-written as

( 1') r@b) - (ya)b - a(rb) ,

which is just the condition that ?I as vector space over /- be a non-

associative algebra over /' relative to the product ab defined in 2t

over (0.

A non-associative algebra ?I will be called central if its centroid

/' coincides with the base field. If ?I is simple with centroid /',

then ?l is central simple over l.; for we have

Tnponnru 2. Let \ be a sirnple non-asnciatiue algebra ouer a rteldO and let I'(= O) be the centroid. Consider ?l as algebra ouer I' by

d.efi,ning Ta: aT, a € ?I, 7 € l'. ThenTI is simfle and central ouer I'

and. the multiptication algebra of V, oaer T is the sarne set of trans'

formations as the multiplication algebra of il ouer A.

Proof: Since f =- A it is clear that a /'-ideal of ?I is a O-ideal so

?I is f-simple^: Similarly, the centroid f of ?I over f is contained

in /'; hence F : T and ?I is central. Let t denote the multipli-

cation algebra of ?l over l'. Then it is clear that i -- f\. the set

of /'-linear combinations of the elements of !. Now let Eo be the

subset of E of elements A such that rA e t for al| r in /'. It is

clear that to is a @'subalgebra of !. lf a, x € U, then r@x) - a(Tx) :

Qa)r which means that Tar,: atT : (ra)r.. Hence at e' To and simi-

larly, an e To. Thus ts contains all the left and the right multipli-

cations and consequently to : a. W'e therefore have t : /'! : !'

We consider next the question of extension of the base field of

a non-associative simple algebra. The fundamental result in this

connection is the following

292 LIE ALGEBRAS

THponpnr 3. If \ is a non-associatiue central simple a'l,gebra oaer oand P is any extension field of A, then ?Ie is central sdmble ouer P.Next let w be an arbitrary non-associatiae algebra ouerla, let y' be asubfi,eld (ouer A) of the centroid and suppose the r'-algehra T 8z?l fssiatple. Then il is simple ouer O and y' - l.

Proof: For the proof of the first assertion we shall need a well-known density theorem on irreducible algebras of lirlear transfor-mations. (See, for example, Jacobson [2], vol. II, p. ZT2.) A specialcase of this restrlt states that if t is a non-zero irredubible algebraof linear transformations in a vector space ?l ovef a and thecentralizer of ! in @(U) is the set @ of scalar multiplijcations, thent is a dense algebra of linear transformations in ?I ever O. Thismeans that if {xr, xr, ..., )tn} is an ordered finite set of linearlyindependent elements of ?I and !r, !2, .. . t ln ?ro n lrbitrary ele-ments of ?I, then there exists a T e T. such that k.iT : y.;, i -

I,2, ..-,n. The density theorem is applicable to the rhultiplicationalgebra t of a central simple non-associative algebrar Now let Pbe an extension field of a and consider the extension algebra ?Ip,for which we choose a basis {urle e I} consisting r of elementsu, e 2I. Let x and y be any elements of ?Ip with x + O. Then wecan write r : ZiFttci, y : Zinrx, where {xr, xr, . .., xn} is a suitablesubset of the basis {u"} and the f's and 7's are in p. We mayassume also that E, + 0. The extension to 2I" of anf element oft is in the multiplication algebra t of ?lr ov€r P. i Hence theree x i s t s a n A ; e i s u c h t h a t x l A i : x ; a n d ) c i A t - $ i i f i * I , i :1 ,2 , " . ,n . Then t rA ; : E f i r and A:ZT: rV$tLA i € t an6 sa t is f iesxA -- t. Thus xt e y and since y is arbitrary, rt. : ilr. This im-plies that ?Iie is simple by the criterion we noted before. Let Cbe a linear transformation in ?Ip such that LCAI : 0 flor all a e t.Let tio be one of the base elements and apply the previous con-siderat ions to r :uo: x l t y: ud,c. Let AreE sat isfy )c1A1 : 1 '1,r ; A 1 : 0 , i + I . T h e n u , C : x 1 C : x r A r C : r r C A r : g A r : T J C . :vJfla. Since Zr depends on uta, we write 4r: Qo, and so we haveurC : Quu4ot 0, Q P. Next we note that if u, and uB 6re any twobase elements, then there exists 2 BoF e t such that urBaF: uF.This is clear because of the density of t. Then ppup: uBC :utBrBC : uoCBap: gocrrBry: pu6p. Hence Qa : 0F: p and C isthe scalar multiplication by the element p. This shows that thecentroid is P, so ?Ir is central over P. This completes the proofof the first assertion.

X. SIMPLE LIE ALGEBRAS OVER AN ARBITRARY FIELD 293

Now suppose ?I is a non-associative algebra ovet A such that

f &o?l is simple over y', I a subfield containing A of. the centroid

/- of ?I. We can consider ?I as algebra over / (6a : a6, a e il, d e /)

and we show first that ?l is simple over y'. Thus let E be a l'

ideal in ?I. Then the subset of elements of the form X r;8 bi,

Tt e T, D, e E is a l-ideaL in /" 8, ?I. The properties of tensor

products over fields imply that if E + 0, ?I, then the ideal indicated

is a proper non-zero r'-ideal in T &oII' Hence E: 0 or E - ?I'

Also ?I' + 0 since (r8r?I)'z+ 0. This shows that ?I is simple over

l. The proof of T'heorem 2 shows that the multiplication algebra

E of ?l over 0 is the same set as the multiplication algebra of ?l

over y'. Since we have xT.- ?I for all r*0 it is now clear that

?l is simple over 0 also and /' is a field. Consider ?I again as

algebra over l. Then one checks that the mapping

2rr8 ao- . } r rar(= \a t ; )( 2 )

T;Q T, or€V is a l'algebra homomorphism of I8r?I onto ?I'

Suppose f =l and let 7€f , $ lso that 1, Tare r ' - independent.

T h e n a r e l a t i o n t & a r + 1 8 a 2 : 0 , a t € ' 2 L , i m p l i e s t h a t t h e a t : o 'C h o o s e a 1 : a * 0 1 a z : - T a . T h e n r & a r + 1 8 a 2 + 0 a n d ' t h e

image of this element under (2) is Ta - 7a:0. Hence if. f + r',

then (2) has a non-zero kernel as well as a non'zero image. This

contradicts the assumption that /'8, U is simple over y'. Hence

f : y' and the proof is comPlete.It is immediate that a dense algebra E of linear transformations

in a finite-dimensional vector space is the complete algebra g(U).

Thus if (xr, . .., xn) is a basis and A is any linear transformation,

then t con ta ins ?such tha tx tT : tc tA . Hence A:TeT" I f ? l

is non-associative simple with centroid /-, then we have seen that

the multiplication algebra ! is a dense algebra of linear transfor-

mations in ?I considered as a vector space over f. Hence if ?l is

finite-dimensional over 0 and consequently over /., then ! is the

complete algebra of linear transformations in ?I over l-. We there-

fore have the following

THponpru 4. Let ?1 be a fi,nite'dimensional simple algebra with

centroid T and. multiptication algebra 7.. Then T" is the complete

set of linear transformations in ?I considered as a aector space

ouer T.Let the dimensionality [?l : O] of ?I ovet O be n and let lT : Ol:

294 LIE ALGEBRAS

r, Ia: rJ - nx- Then it is well known that n - rm,and we nowsee that

Again let ?I be arbitrary and let a -, at be an isomorphism of ?tonto a second non-associative algebra ?I-. Then d is a 1 :1 linearmapping of ?I onto 2I- and (ab)e : a0b0, which implies that( 3 )

Thus( 3')

a70 : o(at), , b*0 : 0(bo)n .

o-tazo - (ao)" , o-rb*o : (be)* .

This implies that the mapping x-+0-tx0 is an isdmorphism ofthe multiplication algebra !(u) onto !([). It is clear ilso thatT -) T0 : 0-'T0 is an isomorphism of the centroid r([) onto r(D.lf aeDl we have (ar)o ._qaee-try: a0T0. In particular, if ?l issimple and ?I is considered as a vector space over 1., then we have

(ra)t : Toao

Next let D be a derivation in ?I. Then D is linear I and (ab)D _(aD)b * a(bD), which gives

( 4 )

( 5 )

( 6 )

Ia", Dl: (aD), , fb*, Dl: (bD)*This implies that the inner derivation x-+ [x, Dl in o(ll) mapst(?I) into itself. _ consequently, this induces a derivation T'-+ Ti, =lrDl in r(tr). By definition of Tu we have (ar)D_(aD)r*a7d sothat for simple ?l we have

ja)D: T"a + r@D) .we can state the results we have just noted in the following con-venient form.

Tsponpu 5. Let ?r and E be centrat simble non-associatiae alge-bras ouer a rt,eM T and tet A b_e a subfietd. of f . Then any iso-morphisrn 0 of \ ouer a onto E oaer o is a semi-liinear trinsfor-mation of a ouer T onto E oaer l. Any deriaation,D in T ouero satisfies (ra)D: Tda + r@D), T e T, a ea where d i,s a deriaa-tion in T as a fi,eld oaer O.

we add one further remark to this discussion. suppgsE (ur,...,ilo)is a basis for ?I over /- and that u;tr: : Z T;irur, Iiir G /', gives thelrultiplication table. A I : 1 semi-rinear transformatiOn d of ?I onto?I- maps the ur in to the basis u! r , i :1 , . . . , " t i iL ,o*r - r . * ais a @-isomorphism we have utol: Zrtiouf; where d acting ol Tit*

X. SIMPLE LIE ALGEBRAS OVER AN ARBITRARY FIELD 2%

is the automorphism in /' associated with the semi-linear mapping

0 of. It. In many cases which are of interest one can choose a basis

so that the rtir e @, that is, ?l : ?Iop where ?lo is a non'associative

algebra over o. Then we have ulul : 2 rrtouto which implies that

?I and I are isomorphic as algebras over /-.

2. Isomorphiern of extension algebras

Let tr be a non-associative algebra over O and let P be a finite-

dimensional Galois extension field of A, G : {1, s, . . . , u} the Galois

group of. P over O. In this section we shall obtain a survey of

the isomorphism classes of non-associative algebras E such that

8.. : ?Ip. We recalt that if ?l is a vector space over @ then I can

be identified with a subset of ?I". This subset is a O-subspace of

?Ip which generates ?Ip as space over P and has the property that

any set of elements of ?I which is @-independent is necessarily

P-independent. Moreover, these properties are characteristic. If

(Qr, Qz, . . ., Q*) is a basis for P over @, then any element of ?Ir has

a unique representation in the form 2 p;)c;, tr € ?I' If s e G, then

s defines a semi-linear transformation U, in ?I" by the rule

(7 ) (\ oux)U, : 2 pln .

It is easy to check that (J" is independent of the choice of the

basis in P over O, that the automorphism in P associated with U'

is s (that is, (tr) U": E (xU"), E e P, x e Ur) and

( 8 ) U " U I : U * , U t : t -

Hence the U, form a group isomorphic to G. If we take pt - 1,

then the elements of ?I have the form Prxr, xr e ?I and it is clear

from (7) that these elements are fixed points for every U", s e G.

It is easy to show directly that ?I is just the set of fixed points

relative to the U". This will follow also from the following basic

lemma:

Lpuu.c, 2. Let tl be a aector space ouer a fi.eld P which is a

f.nite-d.imensional Galoi,s extension of a fi.eld O. Suppose that foreach s in the Galois group G of P oaer O there is associated a

semi-li,near transformation U" in ?I with associated autornorphism

s in P such that (Jr: L, (J,(lt: U*. Let 2I be the set oL fixed

\oints_relatiue to the (J,, s e G. Then V is a O'subspace of ?I such

that 2I - ?Il".

LIE ALGEBRAS

Prgof: One verifies directly that lI is a @-subspdce of [. Ifrc e ?I and p e P, then 1,66p"(x(J,) e ?I since (A"p'(x(J,))U, :2"p't(xU,(Jr) : Z,p"'x(J,, - )"p'(x(I"). Let (pr, Qz, . . ., An) be a basisf.or P over o. Then it is well known that the n x n matrix whoserows are (pi, pi, .. ., pL), s € G, is non-singular. It follows fromthis that r is a P-linear combination of the n elements 2,pi@(t,)which belong to ?I. Thus the p-space m spanned, by ?I is fr.Next let frr, ' '', fr, be elements of 2I which are @lindependent.Assume there exist E; € P, not all 0, such that Zi|iqi : 0. Thenwe may take r minimal for such relations and *" Ftruy supposet' : 1. Evidently we have r ) L and we may assumd that Er 6, o.Then 0 : (Xif 6)(J, : lfi]xi and we can choose s iin G so thatEi + gr. we can therefore obtain a relation 0 - xlF.r h - zi€lxr :l,i-r(Er - Ei)x' and this is non-trivial and shorter than the relation}ilixi :0. This contradiction shows that the rr.'s are p-inde-pendent. Hencefr :?1. .

l f x:ZEf i ; , €t € P, 11 e ?I , then r(J, :Z€' txt . f fhus the (J,are the transformations we constructed before for trn. we returnto the situation considered before in which rve were given [l : lI.and we defined the u" bv (z). we saw that ll s g, the @-subspaceof fixed elements relative to the t/". On the other hAnd, *. hu.r.fl: ?r" - Ep. Let {e,) be a basis for ?l over o alrrd, let b e E.Then 6 : X E$t Ere . P , x , ie {e , } . S ince ther randD ia re in E th isrelation implies that these elements are o-dependent and since thex; zte O-independent we have D: X d.tx;, d,;e (0. Hence De?I andE : 2l so that lt is just the set of fixed elements relative to theU,. Our results establish the following

Tnnonou 6. Let fr be a aector space oaer p, a fi.nite-dimensionalGalois ertension of the fietd o. Jut {u,ls€G} be a:(finite) set ofsemi'linear transformations in ?r ouer p such that:i (1) (Jr: L,u,u1 : u", (2) the automorphism in P associated with u, is s. Let2I be the A-space of fixed elements relati.ue to the U,. ,,Then fr : U"and the correspondence {t/,}-?I is a bijection af thq set of finitesets of semi-linear transformations satisfying (r) and (p) and the setof A-subspaces 2I of Lsailsfying fr: 2I..

Now assume that ?r is a non-associative algebra over o. Thenit is easy to check that the u" defined by (T) are alrtomorphismsof ?I - ?Ip considered as an algebra over A. Convprsely, if thegroup { U"} of semi-linear transformations in fr is given and every


U, is an automorphism of fr over O then the set ?I of fixed ele-

ments is an algebra over A. Hence the correspondence of Theorem

6 itduces a bijection of the set of groups {U"}, U, za automorphismof ?I over @ and the set {?I} of O'subalgebras ?l such that ?Ir -fr.

Let ?I and E be two of the t['algebras such that 2I,e : Ep and

suppose A is an isomorphism of ?I onto E. Let {%} be the group

of semi-linear transformations associated with A, {U,} that with2I. The isomorphism A has a unique extension to an automorphism

A of. fr over P. The mapping A-'(J"A is a semi-linear transfor'

mation in fr with associated automorphism s in P. Also A-'U;A:

1 and (A-'U,A)(A-'U.A): A-'U"A. The set of f ixed points rela'

t ive to the A-t(J,A is the space E. Hence A-'U,A: V", s e G.

Conversely, let {U,}, {V"} be groups of semi'linear mappings as

in Theorem 6 such that the (J,, V, are automorphisms of ?I as non-

associative algebra over O and assume there exists a P'auto'

morphism A of. fr such that

( 9 ) V , :A - 'U "A , sec .

Then 2IA: E for the associated non-associative algebras 2I and E.

Hence ?I and E are isomorphic as algebras over O.We therefore have the following

THnonsM 7. Let fr be a non-associatiae algebra ouer a fi'nitedimensi.onal Galois extension P of a fi.eld O. Then the corre'spondence of Theorem 6 between sets of semi'linear transformati,ons

lfJ,j in E satisfyins $) and (2) and O'subspaces V such that 2Ie :

E induces a bijection of the set of {U") such that eaery U" is aO-algebra isomorphism and. the W which are A'subalgehras of 8.

The corresponding O-subalgebras_are isomorphic if and only if thereerists an autornorphism A of V oaer P such that (9) holds for theassociated groups.

If E is a non-associative algebra over O such that Ep : ?Ie, thenE can be identified with its image in fr : ?I". In this way we see

that Theorem 7 gives a survey of the isomorphism classes of alge-bras E such that Ep =ei, via certain similarity classes of groups

of automorphisms of fr over A. This is the type of result wewished to establish.

In the sequel we shall be concerned with finite'dimensional non-associative algebras over a field O of. characteristic 0 and we shallbe concerned with the question of equality Vo : Eo for two such

2W3 LIE ALGEBRAS

algebras, g being the algebraic closure of the base field o. Let(er, " ',€*), (fr, -..,.f^) be bases for ?I over o and E over o re-spectively. Then the equality ?Ip : Eo implies thal fi: Z p;iei,p;te Q, i, i --1, " ',ff i . Let J be the subfierd over oigenerated bythe p;1. Since €v€r}' p11 is algebraic over o and thebe are only afinite number of these, , is a finite-dimensional extension of t0.since J? is algebraic and of characteristic zero, J isl contained ina subfield P of' J? which is a finite-dirnensional Galois extension ofO. Since the prie P it is clear that ZTpfrtrZ1pei and the con-verse inequality holds since et : Z p,;tf i, ei) : @i)-r. Also it isclear that the f; and the e; both form linearly independent setsover P. Hence \Pe;-?Ip and ) p.fo: Er, so ?Io:,Eo for ;7 thealgebraic closure of the base field A implies ?[p = Er for p asuitable finite-dimensional Galois extension of. 0.

3. Simple Lie algebral of types

we now take up the main problem of this chapter: the classifica-tion of the finite-dimensional simple Lie algebras over any field, oof characteristic 0. If 8 is such an argebra and /. is its centroid,then /' is a finite-dimensional field extension of @ arrd g is finite-dimensional central simple over r. converscly, if g is finite-dirnensional central simple over any finite-dimensional extensionfield /- of a, then 8 is finite-dimensional simple overia. If gr and8z

"r€ two isomorphic simple Lie algebras over @, thdn the respec-

tive centroids f, and Tz "te

isomorphic so both algebras *"y U"considered as central simple over the same field | -TrzTz.Moreover, Theo'em 5 shows that we have a semi-linear transfor-mation d of ,8r over /. onto 8z over /. such that d is a O-isomorphismof 8r onto 82 as non-associative algebras over o. These resultsreduce the classification problem for simple Lie algpbras over Oto the following problems: (l) the classification of the finite-dimen-sional field extensions /. of o, (z) the classification of the finite-dimensional central simple Lie algebras over the f in (l), and (3)determination of conditions for the existence of a semii-linear trans-formation d of two central simple Lie algebras ovef /- such thatd is a tD-algebra. isomorphism. To make this concrete we considerthe important special case: o the field of real numbers. Here anyalgebraic extension field f of o is either @ itself or is isomorphicto the field of complex numbers. If. f is the field of complex

A-D t

X. SIMPLE LIE ALGEBRAS OVER AN ARBITRARY FIELD 2W

numbers, then /- is algebraically closed and we know the classifica-

tion of the finite-dimensional simple Lie algebras over l-. We

recall that the algebras in this classification (Ar, Br etc.) all have

the form 80",80 an algebra over O (Theorem 4.2). It follows from

the remark following Theorem 5 that the algebras in our list (which

are not l--isomorphic) are not isomorphic over @. Thus, to complete

the classification over the reals it remains to classify the central

simple algebras over this field.

Now suppose 8 is a finite-dimensional central simple Lie algebra

over /' which is any field of characteristic 0. If g is the algebraic

closure of /a, then 8o is simple. Conversely, if 8 is finite-dimensional

over /'and 8o is siniple, then 8 is evidently simple so its centroid f'

is a finite-dimensionat field extension of l'. This can be considered

as a subfield of g. Then 8", iS simple since (8r')o: 8o is simple'

Hence 8 is centrat by Theorem 3. Thus 8 is central simple over

l' if and only if ,8o is simPle over .o.

Since g is algebraically closed of characteristic 0 we know the

possibilities for 8o . They are the Lie algebras 4r, I > L, Bu I 2 2,

C,, l2 3, D1, I > 4, Gr, Fr, Eu, Er, E in the Kil l ing Cartan l ist ' If

8o is the Lie algebra X in this list then we shall say that 8 is of

tyfe x. usually the subscript / will be dropped and we shall

speak simply of g of. type A, type B, etc. For each type X we

shall choose a fixed Lie algebra 8o of this type. For example, we

can take 8o to be the split Lie algebra over /' of type X. Then

our problem is to classify the Lie algebras 8 such that 8o: 8oo'

For a particular 8 there exists a finite-dimensional Galois extension

P such that 8p : 8oe and we have seen that to determine the 8

which satisfy this condition for a particular P, then we have to

look at the automorphisms of 80p over /'. We shall study the

cases of 8o of types, At, Bt, Ct, Dt, I > 4. In this section we shall

give some constructions of Lie algebras of types A to D. In the

next section we give the conditions for isomorphism of these Lie

dlgebras and in $ 5 we shall prove that every Lie algebra of type

At, Bt, Ct, Dt, I > 4 can be obtained in the manner given here.

The starting point of our constructions is the fundamental

Wedderburn structure theorem on simple associative algebras: Any

finite-dimensional simple associative algebra U is isomorphic to an

algebra @ of all the linear transformations of a finite'dimensional

.ruito, space St over a finite'dimensional division algebta r'. An

equivalent formulation is that ?I : y'n the algebra of n x z matrices

3OO LIE ALGEBRAS

over a finite-dimensional division algebra l. (For I a proof see:Jacobson, structure of Rings, p. gg, or Artin, Nesbitt, and rhrall,Rings with Minimum condition, p.gz.) If the base field t- is alge.braically closed then the only finite-dimensional division algebraover f is f itself , so in this case w: Tn for some zi If the basefield is the field of real numbers, then it is known that there arejust three possibilities for y': y' : T, / - T(i) the field of complexnumbers or y' the division algebra of quaternions. i (Theorem ofFrobenius, cf. Dickson [1], p. 62, or pontrjagin I1l, t p. 1ZS.) Thecenter o of a finite-dimensional simpre associative algsbra is a fieldand the centroid consists of the mappings cr: cn, c e G. This canbe identified with the center. If ?I is central (G - f)l and g is thealgebraic closure of T, then ?Io is finite-dimensional simple over g.Hence ?Io = 9n for some n. Since [?I : f] : [Uo : gl = lg*: e) : 1xzthis shows that the dimensionality of any fi.nite-dimerhsional centralsimple associatiue algebra is a square.

Let 2I be a finite'dimensional central simple assocfative algebraover /-. Consider the derived algebra g _ ?Ii of the Lie altebrawb If g is the algebraic closure of l', then go: (fli)o : 1ul;i =g'nu since ?Io * 9o. On the other hand, we know that if n _ I * l,I > L, then gLr, is the Lie algebra of (/ + 1) x (/ * r.) matrices oftrace 0 and this is the simple Lie algebra At over g. Since go = gLtit follows that 8 - ?IL is a central simple Lie algebrb of type Ar.

Next let ?I be a finite-dimensional simple associative algebra withan inuolution J. By definition, / is an anti-automorpfrism of periodtwo in ?I. Hence -"/ is an automorphism of ?Ir. Tfie set g(U, /)of /-skew elements (a" : -a) is the subset of fixed blements rela-tive to the automorphism -/. Hence this is a subhlgebra of ?rz.The anti'automorphism / induces an automorphism i in the cenrer6 of E which is either the identity or is of period two. In thefirst case./ is of first kind and in the second / is an involution ofsecond kind.

we assume first that / is of second kind. More precisely, w€assume that G : P: r(il a quadratic extension of tlie base field fan_d that q" : -q. Let 0(2I, D be the space of /-symrrfetric elements( o ' - o ) . A n y a e ? I h a s t h e f o r m a _ b * c , b _ L A l l + a ' ) i O

" " ac : L @ - a ' 1 e @ . I f D e b , q b e 6 a n d i f c e 6 t h e n E c e g . H e n c ethe mapping r -' qx is a 1 : 1 linear mapping of 6i onto 6 so wehave the dimensionality relation: [€ : f] - [6: TI. Since 6 n 6 _ 0a n d ? I - 6 + O , ? I - 6 S g a n d [ ? I : r ] _ 2 [ g : r ] . T h u s [ ? I : p ] _

X. SIMPLE LIE ALGEBRAS OVER AN ARBITRARY FIELD 3OI

+tlt, Tl:[6:f]. We recalt also that [?I:P] is a square z', so

ig , f l :n2 . Le t (a r , . . . ,anz) be abas is fo r6over / ' . Then every

.t"tn.nt of ?I has the form \iza,ra; + [i'|n(qar) : \ p;ail P;:

d; * F;Q, d;, F; e T. It follows that the a; form a basis for ?I (or ?Iz)

over P. This implies that 6p - 2Ir over P. Let I be the algebraic

closure of T which we may assume to be an extension of the field

P. Then 6o: (6p)o : (U, over P)o -- Qnr. since (?I over P)a = 9"'

Let 8 be the derived algebra @' : e(A, n' . Then 8a = QLz so

[ 8 p : g ] : n 2 - 1 a n d [ g ( U , h ' : f ] : n ' - I . l f n - l + t , 1 2 1 , 8 o

is the simple Lie algebra At. Hence 8 is central simple of type

A t .We have now given two constructions of Lie algebras of type A'

We summarize our results in the following

Tnponpru 8. Izt 2I be a finite-d,imensional central simple associatiae

algebra ouer T, 2I + f . Then the deriaed algebra DIL is a central

sllmpu Lie atgebra of type At, I > r. Int 2I be a fi.nite-dimensional

simple associ,itiae algebra with center P a quadratic extension of the

Oasi fieU T and, tuppott ?I possesses an inaolution J of second kind.

Supiose also that Il + P. I-et g(U,,D be the Lie algebra of shew

etiments of W. Then the deriaed, algebra @(V, I)' is a central simfle

Lie algebra of tYPe At, I > l.

Before pro.""ding to the discussion of the Lie algebras 9(?I, /),

/ of first kind we quote some well'known results on involutions in

algebras of linear transformations (cf. Jacobson [3], pp. 80-83).

Let ![t be a finite-dimensional vector space over a division algebra

I and let G be the algebra of linear transformations in Sl over y''

Then it is known that G has an involution A -' Ar if and only if

/ has an anti-automorphism d-'-- d of -period one or two. If the

period is one, then d : d and ffir: drd, implies that l- is _com'mutative. Conversely if y' is commutative, then one can take d = d'

lf. d, --+d is given, then one can define a non-degenerate he-rmitian

or skew hermitian form (x, y) in lIt ovet y' relative to d -'+ d. Such

a form is defined bY the conditions

(10) (r, + xz, !) -- (trr, y) -t (xz, !) , (x, y' * yt) : (x'y') * (r' y')

(dr, y) : d,(x, y) , (x, dY) - (x, Y)d

for x, )Ct, JCz, !, !r, lz e IIl, d e /,

(11) (r,:y) : (f,x) or (x, y) - -T, x) ,

302 LIE ALGEBRAS

according as the form is hermitian or skew-hermitian, and non-degeneracy means that (r, z) : 0 for all r implies z .+0. If tt = d,then we obtain a symmetric or skew bilinear form. lf',(ut,ctz,. . .,14o)is a bas is fo r ! f t over y ' , then (x ,y ) :ZE{ , ; fo r x*ZE; r4 ; , ! :Znrui defines a non-degenerate hermitian form and (r, y) : ) E;prr;is a non'degenerate skew hermitian form if. V - - p + 0. rf. tl-= d,then non'degenerate skew bilinear forms exist for sl if and onlyif !m is of even dimensionality over /'. Let (x,y)r be any non-degenerate hermitian or skew hermitian form associated wittr d -- ttin l. lf. A € 6 we let Ar denote the adjointof. ,A,rel4tive to (r,y),that is, Ar is the linear transformation in !ft such that (r4, y) _(r , tA') for x,-yeSt. Then i t is easi ly seen that 'A- l" ' i -s aninvolution in @. Moreover, one has the fundamentalr theorem thatevery involution t of.0 is obtained in this way.

In particular, suppose y' : /- is an algebraically closed field. Sincey' : T, the only anti-automorphism of. / as algebra over /' is theidentity mapping. Hence (x, y) is either a non-degenerate sym-metric bilinear form or is a non-degenerate skew bilinea, form.The latter can occur only in the even-dimensional case. The Liealgebra g(@,

"I) determined by the form is the Lier algebra Bt if.the form is symmetric and dim ![t :21 * I. The Lie algebra g(@,,/)is cr if the form is skew and dim m - 2l and it i. D, if the formis symmetric and dim Sl :2,.

Now let ?I be a finite-dimensional central simple associative alge-bra over /" which has an involution / of first kind. If g is thealgebraic closure of f , then ?Io z !)o. The extension ! of. J to alinear transformation in ?Io is an involution in ?Io and 6(?I o, D _@(21, I)o. Since 2Io = Qn and / is an involution in g, the resultabove shows that 6(?Io,./) is one of the Lie algebraS Bt, Ct or Dt.We assume now that l >- 2, I2 3 or l > 4 according as 6(?Ie, /) isBt, ct or Du Then the algebras @(?Io, J) are simlle and we

'see

that (?I, "/) is central simple over A of. types Bu Ct or D1. We

recall that n is the dimensionality of the space lll considered be-fore and that n' : fQo: Qf - [?I : r]. For Br we hdve n - zl * rand [g(?I, I): I']: [O(?Io, D: A : t(Zl + 1). For Ct, n : 2l andI@: fl: K2l + l) and for Dt, n : Zl and [@: f] : l(Zt - 1). Wecan now state the following

Tnponpu 9. Let ?I be a finite-dimensional central sirnple associatiuealgebra ouer I of dimensionality nt and suppose 2r hus an inuolution


J of fi.rst ki,nd. Let q(}I, D be the Lie algebra of J'skew elernents of

fu. If n : 2l * L, then [g(u, I): rl : l(21 * r) and 6 es central

simple of tyfe Bt for t > 2. If n :21, then [@ : /'] : l(21 * t) or

tef-D. Inthefornrcr case a3surne, = 3. Then @ is central simple

of type ct. If n -2t and t6: rl : l(21 - L) then we assulne I > 4.

Then @ is central simPle of tYPe D-

4. Conditions for isomorPhiem

Let 8, and lJz be finite-dimensional central simple Lie algebras

over the field /- of characteristic 0- Suppose O is a subfield of f

and that 8r and 8z are isomorphic as algebras over A:8r:o8r.

Then we know that a @'isomorphism 0 of ,8, onto 8, is a semi-

linear transformation of 8, over /' onto 8z over /'. We denote the

associated automorphism in l' by d also so that we have (ra)' :

T t o t i f . T e r , a e g r . L e t ( a t , . , . , a * ) b e a b a s i s f o r S , o v e r / ' w i t h

the multiplication table la;ai7: Z Tiixa*; Then @1, ' ' ', ot^) is a

basis for 8z over l- and laqtall : X rlioa?,. Let I be the algebraic

closure of. T. Then it is a well'known result of Galois theory that

the automorphism 0 in T has an extension to an automorphism d

in 9. The ar, form a basis for 8ro over g and the a0' form a basis

for 8zo over g. It follows that the mapping \a&r+ ) ortna'r, to; € 9,

is a @-isomorphism of 8rp onto 8zo. Thus $qo =o$zo. On the other

hand, 8ro is simple over the algebraically closed field g' Hence

it has a basis over J? whose multiplication coefficients are in the

prime field and so are in O. This implies (remark following Theo'

r e m S ) t h a t 8 r o 4 o g z o . W e h a v e t h e r e f o r e p r o v e d t h e f o l l o w i n g

Lpuu.o, 3. I-et \ and' gz be finite'dimensional central simple Lie

algebras ouer a. rt,eU. f of characteristic zero- Izt 0 be a subfi'eld ol

T and' 9 the algebraic closure of r' Then 8' 3o 8' implies 8ro *p8zo'

This result implies that the only @-isomorphisms which can exist

for the Lie algebras of Theorems 8 and 9 are those between the

algebras defined in Theorem 8 and between algebras of the same

type (8,, Ct, D) of Theorem 9. We shall call the Lie algebras of

tite torm ?I! of Theorem 8 Lie algebras of tyfe Ar, those of the

form g(?I, I)' , I of second kind, Lie algebras of type An' For the

latter class we assume / ) 1 if the type is Ar. This amounts to

a s s u m i n g t h a t t ? I : P l : n ' > 4 ' W e s h a l l s u p p o s e a l s o f r o m n o won that t > 4 for the algebras of type Dr. We consider next the

304 LIE ALGEBRAS

enveloping associative algebras of the Lie algebras of Theorems8, 9.

LsuMA, 4. Let o, T, e be as in Lemrna s. (r) Let tr oe a finite-dimensional central simple associatiae algebra ouer I of Eimensionalityn" > I. Then the enueloping associatiue algebra of g 12IL ouer A is?I. (2) Let 2I be a finite'dimensional simple associatiub algebra withcenter P a quadratic extension of T and with an inuolution I o/second kind. Assume [?I: P] : n, > 4. Then the enueloping associa-tiue algebra of 8 : 6(?I, I)' oaer o is rr. (3) I*t rr bq finite-dimen-sional central simple ouer T with an inuorution ! of first kind and[?I: rJ : n' > l. Then the enueloping associati.ue algebra of g --

Proof: we note first that in all cases the enveloping associativealgebra of 8 over @ is the same as that of g over its centroid /..T h u s s i n c e S i s a v e c t o r s p a c e o v e r / . , y l e g i f 1 6 / . a n d / e g .Hence the two enveloping associative algebras indiQated coincidewith the set of sums of products 1,,1, .. . lt, lr e g. This remarkshows that we may as well assume the base field b - T and weshall now do this. In the cases 1 and 3 we introduce tire algebra?Io: !)n, fl ) 1, and we consider 8o which is a subalgebru of !2*".We know that 8o is the Lie algebra of matrices dl, 8,, C, or Dtas defined in g 4.6. In all cases an elementary dirdct calculationwith the bases given in g 4.6 shows that the g-subalgdbra generatedby 8o is Qn, that is, (8o)* - lJn where the * denotes fhe envelopingassociative algebra. If 8* denotes the enveloping aspociativ. utg.-bra over a of 8 (in ?I), then it clear that the o-subspa ce of. e*spanned by 8* is (t3o)* : gn. Hence 8* contains a Uasis for gnover 9. Since the elements of this basis are conitained in ?I iifollows that they constitute a basis for ?I. Thus we have g* : lI.The argument just used cannot be applied readily, to the case Z(Lie algebras of type Au) since in this case 2lo:,gnD!)n. Wetherefore proceed in a somewhat different mannbr. Let p _0(q) where qt : -q, as before. Then q e 6(til,jD and gt -g.(U, n + qg@, I) : g(?I, ,/)*. Hence it suffices to show that8 * = 6 ( ? I , ) . S i n c e 0 - T a n d , B - @ , w e h a v e L 6 : t \ ) : n z a n d[8: @] : n' - l. Hence, it suffices to show that g* contains anelement a e @, e@'. Assume the contrary that g* n g : 6,. Letbi, i:1,2,3, be skew elements. Then it is immediate that {brbz6r} =brbzbs + bsbzbt is skew. If the bt e @', then {6,Dr6r} € @' : g* n-6.


We now consider the algebra (?I over P)o: 9,. We have seen

that this algebra has a basis consisting of elements of 6 and thatg'*r. has a basis of elements in 8 : 6'. The multilinear character

of. {brbzbs} now implies that {b,brbr) e g'u for all br € 9',r,- If we

take bi: b this implies that Ds € 9'*r, if b e g'"r;. Thus we must

have tr D' : 0 for all b satisfying tr b : 0. This is impossible since

n t 2. For example, we can take b : Zer, - €zz - €ss so that tr b :

0, tr b' :6 + 0. This contradiction shows that 8* = O and 8* : ?I.

We are now ready to prove our main isomorphism theorems. In

all of these A, I' and I are as in the foregoing discussion.

Tnponsu 10. Let 2I and E be fi.nite'dimensional central simfle

associatiue algebras oaer I' and. let 0 be an isomorphism of ?Jl ou€r

O onto E!. ou€r 0. Assume [?I: f] : nz ) | and [E: /-] > 1. Then

if n :2, 0 can be extended in' one and only one way to an iso-

m o r p h i s m o f ? J o u e r o o n t o E o u e r o a n d i f n > 2 t h e n 0 c a n b e

extended in one and only one way to either an isomorphism or the

negatiue of an anti-isomorphism of W ouer CI onto E ouer O.

Proof: We have 2lo= Q*andEna g^. By Lemma 3, !J'6 =oE!,oi

hence Q'*, = 9'*r. and m : n. Thus we may assume that ?Ia : 9n:

Eo so that Q* has a basis (ar, . . . , ant) such that the at form a basis

for ?I over f. We may assulrle also that (ar, " ' , anz-t) is a basis

for W', over /'. Now d is a semi-linear mapping of. 2I'r. over I' onto

Si over f whose automorphism in I' we denote by 0. Then

@1, . ' - , atr- , ) is a basis for El over / ' . I f 0 is extended to the

automorphism 0 in 9, then Ll'- 'r2tar->rtno! is an automorphism0' in 9'*r, over @ which is a semi-linear transformation with as-

sociated automorphism 0 in 9. Let (e i ) , i , i : I , " ' , f l , be the

usual matrix units f.or Qo. Then the mapping \ at;i€;i -r !, ofti€ii,

o;; € g, is an automorphism 0" of the associative algebra Q* over 0

whose associated automorphism in A is 0. d" induces an auto'morphism 0" in the Lie algebra 9'*r. and since 0' and d" have the

same associated automorphism in Q, T : (0")-td' is an automorphismin Q',t over 9. By Theorem 9.5 this has the form X -' M-'XM if.

n:2 and i t e i ther has this form or the form f , -+ -M- 'X'M i f '

n > 2. Since the mapping X-+ M-'XM is an automorphism of !)"

and X -, M-'X'M is an anti-automorphism of Qn, 7l can be realized

by an automorphism of. !l* over Q if. n - 2 and either by an auto-

morphism or by the negative of an anti-automorphism of. Qo over9 f.or n > 2. Then 0' : 0"T can be extended to an automorphism

C of 8" over o if n - 2, or to an automorphism ( og the negativeof an anti-automorphism C of g, over A if n > Z. Since d, Is themapping 2 a&r+ ) ,tal it is clear that 0tcoincides with the given0 on LrL. Since the enveloping associative algebra over o of ?Il is?I and that of al is E it follows that c maps r[ onto E and conse-quently 0 can be extended to an isomorphism or the negative ofan anti'isomorphism of ?I over o and E over @. sinc0 ?Il ienerates?I over @ it is clear that the extension is unique.

This result shows that if ?Il and El are isomorphib as algebrasover o, then ?I and E are either isomorphic or anti.tisomorphic aso'algebras. The converse is clear since any isombrphism of ?Ionto E induces an isomorphism of ?Ii into Ei and if d is an anti-isomorphism of ?r onto E then -d induces an isomorphism of ?iionto El. our result also gives a description of the group of auto-morphisms of ?Il. If n - 2 or if ?I has no anti-automprphism, thenthe group of automorphisms of III over O can be itdentified withthe group of automorphisms of 2I over a. u n)z and ?I has ananti'automorphism /, then it is easy to prove by a feld extensionargument that the automorphism a -, -at in ?Il is not of the forma - eo, 0 an automorphiism of ?I. It follows that :the group ofautomorphisms of ?I over o is isomorphic to a subgfroup or ina"*two of the group of automorphisms of ?Il over 0.

If we take o : l, then it is a known result of the associativetheory that every automorphism of ?I over o is inner. This couldalso be deducedi from the form of the automorphisms of. ll,*" u,

"field extension argument. Then it follows thai the automorphismsot ?I1 over A are of the form x -, m-,trln or of the form x --, -m-r)c"mwhere J is a fixed anti-automorphism in ?I.

Tnponpu 11. lzt ?Ii, i:1,2, be a finite-dimensional simple as-yfoliy! algebra ouer I with center a quadratic fietd p; - r(i) and'itti*t

t?I':';;n?>4 and let a be a subfietd of r. Then any o-isoworphism 06(2[', J)t onto 6(?Ir, J)' can be extended in one and onry one waya 0-isomorphism of 21, onto 2Iz. The Lie algebra @(?[,, Jr), is-isomorphic to any Lie algebra EL of type Ar.

LIE ALGEBRAS

Proof: We may choose a basis (ar, . .., e,,z)(ar, . .. t anz-r) is a basis for 6(?11, /,)/ over I.If .? is the algebraic closure of. T (chosenthen (?Ir over Pr)o: !)n, so 6(2t,, fr)o: gnr"

for ?It over P, so thatand fof ?I{z over Pr.

to contdin P, and Pr),and 6(?I,, f)' : O'^r".

t -ofto

not


Similarty, w€ have 6(Ur, h)b = Q'*rr. Assume there exists a a'

isomorphism of @(fl',,I')' onto Q(9Ir, h)'. Then we know thatg'*r"=oQt*rz so nr: t lz: n andwe may assume that ?Ir and ?Iz are

@-subalgebras of g,n such that any Pr-basis for ?Ii is a basis for 9".

Let 0 be a @-isomorphism of 8r : 6(U,, /r)t onto 8z : @(21r, Jr)'.Then d is semi'linear in 8, over /' onto 8z over l' with associated

automorphism 0 in T. If. (au ' ' ' , anz-r) is a basis for 8r ov€r /",

then (al, . .., at*r-r) is a basis for 8, over l' and both of these arebases for Qt*z over g. If the automorphism d in /" is extended

to an automorphism d in g, then the mapping 2l'-'tror-'-2co1la0;is an automorphism in Q'*t over A whose associated automorphism

in g is 0. The proof of Theorem 10 shows that this can be

extended to an automorphism or the negative of an anti-auto-

morphism of the associative algebra gn over o. Since the envelop-

ing associative algebra of @(?[, ])' over @ is 2Ir it follows that the

isomorphism d on 8r over O can be extended to an isomorphism or

the negative of an anti-isomorphism of U, ovet O onto ?Ie over O.

If the second possibility holds let ( denote the anti-isomorphism.

Then /'C is an isomorphism of ?1, onto ?Iz and if a e 6(?I,, /')' then

a0 : -a€ -- qrr( since arl : -a for a e @(VL,I)| . Hence /t( is an

isomorphism of ?Ir which coincides with d on 6(?I b ft)' . Thus in

every case we can extend d to an isomorphism of ?It onto ?Iz. This

extension is unique since ?Ir is the enveloping associative algebra

of O(U,, ,I,)'. This proves our first assertion. Next let 8r : @(?I,,./,)'

and suppose we have a O-isomorphism d of 8r onto 8z : Et where

A is central simple associative over l'. The argument just used

for ?1, and ?Iz shows that 0 can be extended to an isomorphism d

of ?I, onto E. Since the centroids of the Lie algebras 6(?I',.I,)'

and Ei consist of the multiplications (x-t yr) in these Lie algebras

by the elements T e f it fotlows that 0 maps l" into itself. Hence

Pr: l@r) is mapped into a subfield of E which properly contains

f . On the other hand, Pt is the center of ?I, so this is mapped

into the center f of E. This contradiction proves that 8r: @(Sr, "/r)'cannot be isomorphic to El.

We consider next the Lie algebras of types B, C and D which

we defined in Theorem 9. The main isomorphism result on these

is the following

THoonsu 12. I*t Vi, i,: L,2, be a finite-dirnensional central simlle

associatiae algebra oaer T such that Vi has an inuolution t; of f.rst

LIE ALGEBRAS

hind. If [\: r] : n?, then [O(?I,, J): I,]and [@QI;, J): T]: I;(2h * t) or t;(21; - t)

: l;(Zh * L) V nn: Zl,i, * Lif ni:21;.

"In the respec-

h23, l r ) t4 . Then i f Ag(U,, l) ont7 @(?lr, !) canO-isomorphism of V1 onto

tiae cases indicated we assulne that l;2 Z,is a subfield of f , any O-isomorphism 0 ofbe extended in one and only one way to au,2.

Proof: If g is the algebraic closure of O, then W";s v gn6 andCI(2[r, l)o is the..lil.algebra Bt, ctn or Dlnaccording zs ?tt : zrt * r,rti : 2lt and [6(?I;, J) :/-] : tt(2t; *

-t) o, nr : Zliand [E(?I ;, f;) :/-] :

l;,(21; - 7). It follows that if g(?I,, Jr) =r@l(Wr, lr) then nr!' 1z: ltand we may suppose that ?Iro : go - plzo and g(?[r, Jt), - @(Vr, Ir)ois either the Lie algebra of skew symmetric matrices in p" (ivpesB or D) or the Lie algebra of matrices satisfying @-'A,e:'_A,I a skew symmetric matrix with entries in trre prirne field (typec ) . L e t ( a t , . . . , a * ) ( m : l ( z t + l ) o r t ( z l r ) ) b e a b a s i s f o r@(u', /r) over /. and hence for 6a over o. If d is a o-isomorphismof 6(?I', /r) onto @(V2,-J2), 0 is semi-linear with associated auto-morphism 0 in f and (aer, .. ., ol) is a basis for 6(?Iz, /2) over /. andfor @o over 9- If d is extended to the automorphism d in o thenZaia;+ ! ,ulal, otr € g, is an automorphism 0' in 6o ov€r o withd as its associated automorphism in e. If (e;r.) is a usual matrixbasis for g* over g then X at;i€ii-r ) aeiieli is an aut5morphism d//of 9o over J? whose associated automorphism in g is id. Moreover,since I has entries in A, d,, maps 6o into itself and so it inducesan automorphism 0" in 6o over @. It follows that 0t70rr1-r is anautomorphism of @o over a. In all cases this rhas the formx-+ 114-txM and so it can be extended to an inner automorphismof !2n over g. It follows that 0' can be extended to an automor-phism ( of 9, over Q. Since the restriction of ( rto 6(1I,,,I,) isthe given d and since the enveroping associative alge$ra

"r gifr,, /,1is .Ui,

c maps ?I, isomorphically on rlz and coiti.ides with' 0' on6(u',

"/'). Thus 0 can be extended to a o-isomorphism of ?I, onto?I2. Since O(2t,, /,) generates llr this extension is unique.

5. Completeness theoreml

be a finite-dimensional central simple Lie aflgebra of typer. Then there exists a finite-dimensional Galois extensionof. I such that 8p = P1,", n 2 2. Thus we m4y suppose that

Let 8A overfield PS i s a l'-subalgebra of Plz such that the p-space spanned by g is


P|, and elements of 8 which are /'-independent are P-independent.We have seen also that for each s in the Galois group G of. P over

x there corresponds an automorphism lJ, of Pl,,t oYet I' such that(px)U, - p'(x(J"). If (ar, " ', Q'oz-r) is a basis {or 8 over /-, hence

ior P!,, over P, then (J, is the mapping >l'-'P;a;--+2pia;. We

have (J, - I, (J,(Jt: U* and 8 which is the set of elernents \76ait

T;e T, is the set of fixed elements for the U,, seG. We have

seen in the last section that U" has an extension U' in the enve-

loping associative algebr? P* of. Plt, such that U' is an automor-phism of Po over l' if n:2 and U, is either an automorphism of

Po over T or the negative of an anti-automorphism of Po over T

i f n)2. Every y€, P* is a sum of products xf iz" ' )G; x;€ P'nr, ,

and if p e P and (J" is an automorphism of Po, then (prr "' x,)U,:(prrU,)(rzu,) ' " (x,U,) - p"(xr(J,) "' (x,U,) : p'((r, "' x,)U,).' If U"

is the negative of an anti-automorphism, then (px, ''' x,)U,:

(-L)'(r,U") . '- (rzlJ,)(pxJJ"): (-1)'p'(x,U")" '(rrU,): p'((x' " ' x,)U").

Hence in either case U" is semi'linear with automorphism s in P.

since (J,(Jt and (I* have the same effect on the set of generators

Plz of P" we have (J"(Jr: (J"t. Similarly Ut:1 is valid in Pn.

lf. (J, and (Jt are the negatives of anti-automorphisms then Uss-r :

(J,[1;' is an automorphism. It follows immediately from this that

the subset H of. elements s e G such that U" is an automorphism

of Pn over l' is a subgroup of index one or two in G.

Case I. H: G. The subset of Po of. fixed elements under the

U, is a l--subalgebra U of P* such that }le: P* (Theorem 7).

Hence ?I is finite-dimensional central simple over l-. Evidently

8 g U a n d s o 8 ' : 8 = V L . O n t h e o t h e r h a n d , t S : r l : n 2 - 1 a n d

LVL:r l :n ' -1. Hence 8:?I l is a Lie algebra of type ,4r 8sdefined on p. 303.

Case II. H + G. Then I/ has index two in G. We know alsothat n > 2 in this case. The subset of. P of elements F such that

E" : E, s e F/, is a quadratic sub4eld l'(q) over l.. It is clear from

the form of the U, that x : 212-'p;ar satisfies *(f , : 1, s e F/, if

and only if all the fie/'(q). Thus 8rror, the set of f(q)-linear

combinations of the a;, is the set of elements of. Pl," which arefixed for all the U,, s €, H, and I/ is the Galois group of P overf@). It follows from case I that Srror:Ul where ?I is centralsimple over f(q) and is the enveloping associative algebra of 8rror.Now let / € G, 4 H. Since f/ is of index two in G, an element

Zi'- 'p;a; of 8",0r is in 8 if and only if (2p;ai)(L:2i;ai. Now

310 LIE ALGEBRAS ,

ut: -/ where / is an anti-automorphism of p" over 1.. since(2 pP)Ur: 2 F;a; for pi e f@) and p the conjugate of p under theautomorphism + L of. f(q) over I', J- -(Jt maps B;101 into itself.Hence / induces an anti-automorphism in the enveloping algebra?I of 8.,0,. If pe T(q), -ye?I, then (py), - p!". Herrcelirr[u.",the automorphism p + p in I'(q), so / is of second kind. sincea u t : a f o r a G , 8 , a t : - a a n d a e g ( ? I , / ) . T h u s g L g ( u , / ) a n d8' : 8 s @(?r, /)'. comparison of dimensionalities oirer /- showsthat 8 - 6(2r, r)'. Hence 8 is a simple Lie algebra of type Ar.

We have therefore proved the following

Tnponprrr 13. Any central simfle Lie algebra of type Ar. is i,so-rnorphic either to a Lie algebra V,n, 2I a fi.nite-dimensional centralsimple associatiue algebra or to an algebra @(a, I), whete 2r is f.nite-dimensional simple associatiue with an inaolution I of qecond kind.

we consider next the Lie algebras of types B, c and D in thefollowing

Tnuonsu 14. Izt 8, be a centrar simple Lie algebra of type 81,I > 2, Ct, 12 3 or Du 14 5. Then g is isomorhhic to a Lie- ilgebrag(u, /) where { ,s a fi.nite-dimensional central simfle assoiatiuealgebra, J an inuolution of fi.rst kind. in 21.

Proof: There exists a finite-dimensional Galois extension p of Tsuch that 8e is the Lie algebra @(.pn, J) of /-skew ntatrices in powhere/is the involution x-x': the transpose of x[n p,,, or theinvolution x--Q-'x'Q where e': -e and the entriies of e arein the pr i rne f ie ld . A lso we have n>E i f n -21 *L , nz6 i f .n:21 and the involution is x-re-'x'e, and z > 10 in the re-maining case. For each s in the Galois group G of p over /. wehave the automorphism (1, of. 8.p over f : ZTpra,; + .X plar where(ar, . .., a*) is a basis for 8 over f and for Bp oyer p. g isthe subset of 8p of elements which are fixed relative to theu,. The conditions on n insure that (J" can be extended to anautomorphism u" of the enveloping associative algehra p* of. ge(Theorem 12). The extension (/, is semi-linear in po wilth associatedautomorphism s, U, - 1 and (J,U: (J* hold in p". Hence thesubset of P* of elements which are fixed relative to the (J", s € G,is a subalgebra { of. P^ such that ?Ip - po. Hence ?I is finite-dimensional central simple over l-. lf. X € 6.e, then Xr : _X andXU, e 6r. Hence XrfI, - -X(J, _ (X[J)r. Thus f(Jt: U,/ holds


in 6ip. Since P" is the enveloping associative algebra of @p:@(Pn, D it follows that fU": U"J in Pn also. This implies that Imaps ?I into itself and hence / induces an involution in ?I over l-

which is of first kind since l. is the center of U. lf. a e I then

&t : -a so a e @(W, I) and 8 s @({, /). On the other hand,g(?I, hp = g(P*, I) : Br. Hence 8 : 6({, /). This completes theproof.

6. A claser look at the isomorphism conditions

We have seen in $ 4, that if Jr, i : L,2, is an involution (either

kind) in a finite-dimensional simple associative algebra 2L ov€f O,

then 6({ r, Jr)' =o@(2I2, /')/ implies that ?Ir and ?Iz are isomorphic.

It therefore suffices to consider one algebra ?I - ?I' - ?Iz and con-

sider the condition for isomorphism of g(?I, /)/ and g(?I, K)' where

I and K are involutions in ?I. We have seen that any isomorphism

d of 6(?I, /)/ onto g(?I, K)/ can be realized by an automorphism

of ?I. If. ae g(?I, J)t then ato : -a0: aoK. Thus l0 - dK holds

in 6(tr, /)'. Since the enveloping A-algebras of 6(fl, /)' and g(21, K)l

are ?I we have I0:0K in ?I or K:0-'J0. We shall call the in-

volutions / and K of II cogredient if. there exists an automorphismd of D such that K:0-'f0. We have seen that cogredience is a

necessary condition for isomorphism of @(V, n' and g(?I, K)'.

Conversely, if. J and K are cogredient and K: 0' '/d where d is

an automorphism then d maps g(?I, /) onto g(U, K) and @(2I, I)'onto 6(?I, K)'. Hence d induces an isomorphism of the Lie algebrag(?I, /)/ onto g(?I, K)'. Thus g(?I, J)' =rg(U, K)' if. and only if /and K are cogredient. We see also that the group of automor-phisms of the Lie algebra 6(?I, I)' can be identified with the sub-group of the group of automorphisms d of the algebra ?I such that0J: J0.

Now let ?I : @ the algebra of linear transformations in the finite-dimensional vector space !ft over the finite dimensional divisionalgebra y'. Let a--+d be an involution in / and let (x,y) be ahermitian or skew hermitian form corresponding to this involution.Then the mapping A-, A*, A e 8,, A* the adjoint of A, is aninvolution in @. We recall that A* is the unique linear transfor-mation such that (xA,l):(x,lA*) and we have (y,x) - eTrJ)where e : 1 or -1 according as the form is hermitian or skewhermitian. Suppose d--d' is a second involution in I and (r,t),

3T2 LIE ALGEBRAS ,

a second hermitian or skew hermitian form correspodding to thisso we have (t, x)r: er(tr,yX where er : tl. Suppose tbe involutiondetermined by this form is the same as that given by (x, y). Thusi f . (x,yA*):(xA,y) then (x,yA*)r : (xA,y)r . Let u and o be arbi-trary vectors in $t. Then r->(x,u)a is a linear tran$formation inul and one checks that its adjoint relative to (x, y) is x -+ e(r, a)u.Hence we have

((r, u)a, !)r: (r, e(y, a)u),(r, u)(u, !)r: e(r, u)r(y, u)l

Since x, !, 14, a are arbitrary this shows that (x, y)r:)(x, t)p, p * 0in /. lf. a e y', (x, at), : (x, !)rd' gives (x, y)dp _ (*, y)pa,; henced' : p-tdp. Also we have (y, x)p: er((f,, y)p), * eet(r, !), :erp-tp1fi1l: eqp-tp(y, x)p. Hence p: e*,D. Conver$ely, let p beany element of r' satisfying F : + p.+ 0. Then a direct verificationshows that a-rd' : p-tdp is an involution inl and (k,y)r=(r,y)pis hermitian or skew hermitian relative to this involution . lf. (x, y)is skew hermitian and p : -p, then (r,y)p is hermitiin. Hence ifI contains a skew element + 0, then a skew hermitiihn form canbe replaced by a hermitian one which gives the sanre involutionA-.A* in G. If. / contains no such elements theni p: p for allp e / and this implies that / is a field. Hence we may restrictour attention to hermitian forms and to alternate forrns (/ a field).Two such forms (x, t), and (.r, y) give the same involution in @ ifand only if. (x,J)r : (r,y)p, F: p if a+a is the involution of (r,J).

It is known that any automorphism of G has the forrn A -+ s-rASwhere s is a semi-linear transformation in lut over y';(Jacobson [3],p. 45). If d is the automorphism in y' associated with s, then onechecks that'(r,y)r: (rs,Js)o-t is hermitian or alterirate with in-volution a--7a0ye-'. Moreover, if. A € G then

(xAS,yS)o-' - (rSS-'AS, yg)e-t* (rS, rS(S-t,451*;e-t

. : (rS, y(S(S-'eS)*S-r;5;e-1

Thus (rA, t)r: (fr,y(S(S-r.4S)*S-'), and the involutlon in @ de-termined by (r,J), is A-'S(S-'AS)*S-'. Thus if we, call A- A*,l, and A -, S-'AS, d, then the new involution is K - 0f0-t which iscogredient to J. Elecause of this relation it is natural to extendthe usual notion of equivalence of forms in the following manner:

X. SIMPLE LIE ALGEBRAS OVER AN ARBITRARY FIELD 3I3

Two hermitian forms (r, y) and (r, y), are said to be S'equivalent

if there exists a 1: 1 semi-linear transformation S with associated

isomorphism d such that (r, J), : (rS, yS)0-'.

It is well known that any two non-degenerate alternate forms

are equivalent in the ordinary sense. Hence the involutions in O

determined by any two such forms are cogredient. Moreover, these

are not cogredient to any involution determined by a hermitian

form. Our results imply also that the non-degenerate hermitian

forms (x, y) and (r, y)r define cogredient involutions in 0 if and only

if. (r,!)t: (rS,JS)o-'p where S is semi-linear with automorphism

0 and p is symmetric relative to a'-rd'=@)t-'

7. Central simple real Lie algebras

We shalt now apply our results and known results on associative

algebras to classify the central simple Lie algebras of types A-D(except D.) over the field O of rcal numbers. By Frobenius' theo-

rem, the finite-dimensional division algebras ovet A are: O; the

complex field P - AQ), i' : -l; the quaternion division algebra r'

with basis L, i, i , & such that

(12) i ' : i ' : k 2 : - L ,j k : - h j : i ,

I has the standard involution a: d + pi + ri + Bk -> d :

d - Pi - rj - 8k. Since the automorphisms in / are all inner,every involution in y' is either standard or it has the form a + q-'dq

where 4: -q. The dimensionality of the space of skew elementsunder the standard involution is three; under a-+q-tAq the dimen-sionality is one.

'We denote the automorphism ;t 1 in P over O bV

p + F .By the Wedderburn theorem, the finite-dimensional simple as-

sociative algebras over O are the full matrix algebras Oo, P" andy'o. These can be identified with the algebras @(O,n), @(P,n) and@(/,n) of linear transformations in lUl over O, Pand y' respectively.The algebras @, =E(O,n) and y'n?@(l,n) are central and haveonly inner automorphisms. The center of @(P, n) = P* is P. Inaddition to inner automorphisms, E(P, n) has the outer automor-phisms X- S-tXS where S is a semi-linear transformation withassociated automorphism p+V in P.

All our algebras have involutions and 6(P, n) has involutions of

i j - - j i = = k ,

h i : - i k - i .

314 LIE ALGEBRAS

second kind. The involutions of o(@, z) have the fbrm x-- x*where x* is the adjoint of, X relative to a non-degenerafe symmetricor skew bilinear form in ![l over o. Involutions ddtermined byskew forms are not cogredient to any determined by ta symmetricform. Any two non-degenerate skew bilinear forms a;e equivalentso these give a single cogredience class of involutions. If (r, y)and (r, !)r ate non'degenerate the criterion of the last $ection showsthat the involution determined by (r, y) is cogredieht to that of(r, y), if and only if. (x, y) is equivalent to a multiple of (r, y)r.Since (r, y) is equivalent to any positive multiple of (x,ly) it followsthat the involution given by (r,y) is cogredient to that of (r, t),if and only if. (x, y) is equivalent to !(r, y)r. If k, g is non-degenerate symmetric it is well known that there exists a basis(ur, ur, . . . , un) for Ul such that

(u;, u+): 1 , L< i<p( u t , u ) - - 1 , p 1 j < n( u ; , u ) : 0 , i + j .

The numbr P is an invariant by Sylvester's theorem. Accordingly,we obtaiir' [nlz) * 1 cogredience classes of involutions Correspondingto the values b:0,1, ... ,[nlL). A simpre calculation using thecanonical basis (13) shows that the dimensionality orrer o oi th"space of skew elements determined by (r, !) is n(n - L)12. Thisimplies that the Lie algebra 6(G, I) of these elementts is of typeB or D according as z is odd or even. The Lie algebra determinedby a skew bilinear form (x, y) is of type C.'we

consider next the involutions of second kind in @(p, n). Suchan involution is the adjoint mapping relative to a ndn-degeneratehermitian form (x, y) in fi over p. A basis (ur, r4r, . ,,., uo) can bechosen so th4t (13) holds. If (r, y) is S-equivalent tol(x,y)r in thesense thit there exists a semi-linear transformation S With automor-phism p+0 in Psuch that (x,y) - (rS,yS),, then we have (u;S,ug)r-1 , 1 S i = p , ( u $ , u $ ) , : - I , p < j 3 n , ( u i S , u $ ) r : 0 i i + j . T h e n(r,y) and' (r,y)t are equivalent in the usual sense. since (r,y) isequivalent to y(x,l), r real and positive, it follows that (r, y) and(r, r)t determine cogredient involutions if and only if (r, y) isequivalent to t(r, J)r. This and Sylvester,s theorem for hermitianforms implies again that the number of non-cogredient involutionsof second kind in @(P, n) is lnlTl * L.

(13)


The discussion we have just given carries over verbatim to

hermitian forms in St over I for which the involution in r' is the

standard one. A basis satisfying (13) can be chosen (cf. Jacobson

[2], vol. II, p. 159). The number of cogredience classes of involu-

tions given by the standard hermitian forms is lnlLl + l. If we

use a canonical basis for which (13) holds it is easy to calculate

that the dimensionality of @(@(1, n), t) determined by the associated

involution ,/ is n(Zn + l). Since lv,4 P2, @(1, n), 7 Prn' Since

@(E(l,n), I)" has dimensionality n(Zn* 1) over P it follows that

@ is central simple of tyPe C".It remains to consider the hermitian forms (x, y) in fi over y'

whose involutions in y' are of the form a--q-'dq,4: -Q. If we

replace (x, y) by (r, y)q-' we obtain a skew hermitian form relative

to the standard involution. We prefer to treat these.

LsMMa 5. If q, and. qz are non-zero skew i,n / (relatiue to the

stand.ard inaolution), then there exi,sts a non'zero a in A such that

Q z : A | t A .

Proof: We note first that if. b, and bz are elements of y' not in

@ which have equal traces and norms, then there exists an isomor-

phism of. AQ) onto O(Dz) mapping b, into Dz. This can be extended

to an inner automorphism of. / (Jacobson, Structure of Rings, p.

L62). It follows that br and bz are similar, that is, bz: cbrc-t for

some c in r'' In particular, if N(qz) : N(4')' then since the traces

T(qr): 0 : T(qr), there exists a c such that qz: cQ$-t : N(c)-'eqtc.

We note also that since the norm of any non-zero element is posi'

tive, 41 is necessarily similar to a suitable positive multiple of qz.

we now see that it suffices to prove the lemma for Qz: TQr' r > 0'

Then we consider the quadratic field O(q') - A(qr). We have 7 :

Mc) for some c in this field. Hence Qz: N(c)Qr: eQ$ as required.

The usual method of obtaining a diagonal matrix for a hermitian

or skew hermitian form now gives the following

Lputur 6. If (x, y) is a non'degenerate skew

form relatiue to the standard inuolution in y',

basis (ur, ur, . . ' , ctn) for fit such that the matrix

(14) diag {q, Q, "' , Q}

where q is any selected non'zero skew element of /.

This result implies that there is just one cogredience class of

involutions in G(r', z) given by skew hermitian forms in fi over l.

hermitian bilinearthen there exists a((u;, ui)) :

316 LIE ALGEBRAS

If we use (14) we can calculate the dimensionality for the Lie alge-bra defined by the involution. This is n(Zn- 1) which implies thatthe Lie algebra is central simple of type D*.

we can now list a set of representatives for the isomorphismclasses of central simple Lie algebras of types A-D over

-a, asfollows.

-J E

Type Ar. OLz, n ) \ r't*", n Z L .Type Au. Suppose n > I and let

(ts) se:dias{f;,-ff i1}.Let 6(P, n,p) denote the Lie algebra of. rnatrices xe p^such thatS; 'X 'S ' - -X . Then the L ie a lgebras @(p,n ,p) fo r ,b :0 ,1 , . . . ,Inl2l constitute our list. :

Tybe B- Let.@(o,n,p) be the Lie algebra of mafrices xeoosuch that SrlX'Sr: -X. Then our list is the set of

tl-i"

"fl.Lrr.@(O,n ,p) w i th n odd, n ) :5 .TyDe c. Let 6(/,n,p) be the Lie algebra of matrices xe r*

satisfying s;lx'se: -x and let 6(o,2n,e) the Liie algebra ofmatrices in Azo such that e-rx,e- _X where e is ui| skewsymmetric matrix. Then the l ist is: 6(/, n, p), n2 B, b : 0, 1.., [ntZland, 6(O,2n, Q), n Z 3.

Tyfe D. Let .@(r', n, Q) be the Lie algebra of matrices in y'osatisfying Q-'x'Q - -x where a is u rk"* hermitian matrix.The list is; @(y', n, e), n ) S, and @(O,2n, So), n 2 S, and p _0 , 1 , " ' , n .

Exereises

1. Determine the groups of automorphisms of the simple real Lie algebrasof types A-D, except D*

2' Determine an invariant non-degenerate symmetric bilinear form forevery central simple real Lie algebra of type A-D. use thib to enumeratethe compact Lie algebras in the list (cf. g 4.2).

3. show that the Lie algebras s(r',4, e) and s(@, g, sr) in the notation of$7 are isomorphic, (This shows that Theorem 12 is not valid for Lie alge-bras of typr' Du since y'r and @a are not isomorphic.) |

4. A (generalized) quaternion algebra over an arbitrary field @ is definedto be an algebra with basis L,i, i,/c such that I is the identity and themultiplication table for i, i, k is

i 2 : a I , j r = g l , k z = - a 1 l * 0i j = - j i : k , j k = - k j - - p i , k i : _ i k = _ a i .

X. SIMPLE LIE ALGEBRAS OVER AN ARBITRARY FIELD 3T7

Show that an algebra y' is a quaternion algebra over @ if and only if y'o = 9z

for Q the algebraic closure of @.

d. A non-associative algebra G over an arbitrary field @ is called a Cayleg

algebra if G has an identity 1, G contains a quaternion subalgebra y' con'

taining 1 and every element of G can be writen in one and only one way in

the form a * bu, o, b in y' and' a an element of G such that

a(bu): (ba)u , (bu)a: Qa)u , (au)(kt) : 16a ,

where p is a non-zero element of @. Show that G is alternative (cf. $ 4.6)

and that the mapping & = o + bu+ d - ba is an involution in G. Prove that

cE : Ms)L = in where N(o) e @ and satisfies N(nil = N(r)N(g). Let (n, g) :

ltN(r + il - N(c) - N(g)I. Show that (r, gr) is a non-degenerate svmmetric

bilinear form and that two Cayley algebras are isomorphic if and only if

their forms (n,U) are equivalent. Prove that a non-associative algebra 6 is

a cayley algebra if and only if Go is the split caylby algebra of '$ 4.6.

6. Use Theorem 9.7 and Exercise 5 to prove that two Cayley algebras

over a field of characteristic 0 are isomorphic if and only if their derivation

algebras are isomorPhic.7. Prove that a central simple Lie algebra over a field of characteristic

0 is of type G if and only if it is isomorphic to the derivation algebra of a

Cayley algebra.

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ARTIN, E.

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Blocr, R.

tll New Simple Lie algebras of prime characteristic. Trans. Am. Math.

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Izt On Lie algebras of classical type. Proc. Am. Math. Soc. 11 (1960), pp.

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Bonnr,, A.

tl] Topology of Lie groups and characteristic classes. Bull. Am. Math.

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Bonpl A., and Cnpva'lr,oY, C.

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6 (1953),

INDEX

Abelian Lie algebra, 10 Characteristic polynomial, 60Adjoint mapping, 9 Characters, 239, 249Ado's theorem, 202 primitive character, 259Algebra Cohomology groups, 93-95, 174-785

alternative algebra, 143 Compact form, 147associative algebra, 3 Complete Lie algebra, 11associative algebra of linear trans- Complete reducibility, 46,75-83, 96-

formations, 5 103constants of multiplication of an Composition series, 48

algebra, 2convention on algebra terminology, Derivation, 7, 73-75, 79

15 anti-derivation, 179of differential operators, 175 inner derivation, 9Jordan algebra, 144 (sr, sz)-derivation, 177Lie algebra, 3 Derivation algebra, 8Lie algebra of linear transforma- Derived series, 23

tions, 6 Direct sum, 18non-associative algebra, 2 Dynkin diagram, 128

Almost algebraic Lie algebras, 98Automorphisms of Au Bt,, Ct,, Dt, Gz Engel's theorems, 36

Fu 28I-285 Enveloping algebra, 32Exceptional simple Lie algebras, 142

Basic irreducible modules, p. 226 Extension of base field, 26-28Extension of a Lie algebra, 18, 88

Campbell-Hausdorff formula, 175 Exterior algebra, 178Canonical generators, 126Cartan matrix, 121 Factor set, 89Cartan's criteria, 68, 69 Filtered algebra, 164Cartan subalgebra, 57 Fitting decomposition, 37, 39, 57

conjugacy of Cartan subalgebras, Free associative algebra, 167273 Free Lie algebra, 167

splitting Cartan subalgebra, 108 Freudenthal's formula, 247Casimir operator, 78 Friedrich's theorem, 170Cayley algebra, L42, 317Center, 10 Goldie-Ore theorem, 165Central non-associative algebra, 29L Graded algebra, 163Centralizer, 28

Centroid, 291 Holomorph, 18Chamber, 263

t32el

330 INDEX

Ideal, 10, 31Indecomposability (for a set of linear

transformations), 46Invariant form. 69Irreducibility (for a set of linear

transformations), 46absolute irrepucibility, 223

Irreducible modules for At, Bt andGz, 226-235

Iwasawa's theorem. 204

Jacobi identity, 3

Killing form, 69Kostant's formula, 261

Levi 's theorem, 91Lie element, 168Lie's theorems, 50, 52Linear property, 146Lower central series, 23

Malcev-Harish-Chandra theorem, 92Mqdule, 14

cyclic module, 272contragredient module, 22a-extreme cyclic module, 272tensor product of modules, 21

Morozov's lemma, 98Mostow-Taft theorem, 105Multiplication algebra, 290

Nil radical, 26Nilpotent Lie algebra, 25Normalizer, 28, 57

Orthogonal Lie algebra, 7

Poincar6 - Birkhoff - Witt theorem,159

Polynomial mapping, 266differential of a polynomial

mapping, 268Primary components, 37 -43

Qnaternion algebra, 316

Radical

of associative algebfa, 26of Lie algebra, 24

Rank, 58Real simple Lie algebras, 313-316Reductive Lie algebraf 104, 105Regular element, 58Representation

adjoint representation, 16for associative algebra, 14for Lie algebra, 16 iregular representatipn, 16of split three dimehsional simple

Lie algebra, 83-186tensor product of rtepresentations,

2lRestricted Lie algebra

istic p, 187Abelian restricted Lie

194Roots, 64

of character-

algebra, 192-

Schenkman's derivation towertheorem, 56

Semi-simple and nilpotent componentsof a linear transformation, 98

Semi-simple associatirle algebra, 26Semi-simple Lie alge\ra, 24

structure theorem {or semi-simpleLie algebras, 7[

Simple system of roofs, 120indecomposable simple system, 127

Solvable Lie algebra, 24Specht-Wever theoreni, 169Spin representation, ?29Split Lie algebra of linear transform-

ations, 42, 50Split semi-simple Lie, algebra, 108

existence theorem nor, 220isomorphism theorein tor 127, 221

INDEX 331

Split simple Lie algebras At, Bt. Ct,Dt ,Gz,F+,80, 135-146; Ez andEa, 228

Split three-dimensional simple Liealgebra, 14

Splittable Lie algebra (in the senseof Malcev), 98

Steinberg's fotmula, 262Subinvariant subalgebra, 29Symplectic Lie algebra, 7

Trace form, 69Triality, 287Triangular Lie algebra, 7

u-algebra of a restricted Lie algebra,r92

Unipotent linear transformation, 285

Universal enveloping algebra, 152Upper central series, 29

Weakly closed subset of an associa'

tive algebra, 3lWeight, 43

weight space, 43, 61Weyl group, lI9, 240-243Weyl's character formula, 255Weyl's dimensionality formula, 257Whitehead's lemmas 77, 89Witt's formula, 194Witt's theorem, 168

A CATALOGUE OF SELECTED DOVER BOOKSIN Att FIELDS OF INTEREST

A CATALOGUE OF SELECTED DOVERBOOKS IN ALL FIELDS OF INTEREST

CELESTIAL OBJECTS FoR coMMoN TELEscopEs, T. w. webb. Themost used book in amateur astronomy: inestimable "it

}L, il;til ;;identifring- ngTlv 1:m0 celestiar-obj"_"ir. Edi;e;6;","d uy rtriie*zt w.Mayall. 72 iltustrations. Tora, *r#ll:;,m;S?1" ;;;;;, :";;; ;;

HISTORICAL STUDIES IN THE LANGUAGE OF CHEM$iNY, M. P.crosland. The important part lan-gu1gp t "r

pLy"Jl"-td i;-;"b;;; ;ichemistrv from the svmbolism oialJhemy- 6 ih" "dopuo" "f l;;;l;;u;nomenclature in 1892. ". . . wholeheartudly ,"*-*u"i"d,;jd;il;:-G

illustrations. 4l6pp. of text. SEla x 8y+. ASiOZ_S;-ii.-ib.00

BURNHAM'S CELESTIAL HANDBOOK, Robert Burnham, Jr. Thorourh.readable guide to the.stars beyond org so\ :yrt"* E--fi;tir" ilil;;i;fully illushated. Breakdown is alphabeticar by constellation, Ard;;;a;to !e!us in Vol. l; chamaeleon to orio" p yoi. z; *J pa; to=u;;;J;in vol. 3. Hundreds of illustrations. Totar of about 200opp. 6v6 ;5r;:

23567-& 2sS68-8, 29679-0 pa., Three.uol iJ ii6.85

THEORY OF WING SECTIONS: INCLUDING A SUMMART OF AIR-F-oI! DATA, Ira H. Abbott and A. E. von Doenhoff. concise.rLrrr/ Lr^Ltr, rra rr. ADDott anc A. rg. von Doenhofi. concise qompilation

:f :"!l!:1i9. aero$ryamic characteristics- of modern NeSa *irig';;;;:plus description of theory. s5Qpp. of tabtes. 6$;p. ;-,t'; bi;:

"""""-'

6058&8 Pa. $7.00DE nE METALLICA,.Georgius Agricola. Translated by Herbert,c. Hooverand Lou H. Hoover. The famous fr*rr", translation of greatest rtreatise ontechnological chemistry, engineering, g*logy, ;iil;""i

""rty--J"l"itimes (1556). All 28e original *od",rlr. o56pp.r3h:rt;;.;;

l

THE ORIGIN oF coNTINENTs AND gqEANS, Alfred wefener. oneof the most influential, most controv'ersial books io-s"ierr"e, fhu

"l"rriostatement for continental drift. Full 1966 translation of Wegene/, -ilI

( f929) version. 64 illustraUons. Z46pp. S% x gtrh. 6fZ0g-Z pa. 0Zg0

THE PRINCIPLES oF psycHol.ocy, w-illiam James. Famous longcourse complete, unabridged. Stream of thought, time perceptiotr" -"-ory]experimental methods; _great work decades ahead of its time. jStill ;"iie;useful; read in many classes. g4 ffgures. Total of l3glpp. S%, x g\h-.--'

20381-6, 2O}BZ-4 pa., Two-vol.'set $19.00

CATILLOGUE OF DOVER BOOKS

TONE POEMS, SERIES II: TILL EULENSPIEGELS LUSTIGESTREICHE, ALSO SPRACH ZANATHUSTRA, AND EIN HELDEN-LEBEN, Richard Strauss. Three important orchestral works, including veryDopular Till Eubnspbgets Morrg Pmnlu, reproduced in full score fromirieinal edifioru. Shrdy scpre. 3l5pp. 9% x LZY+. (Available in U.S. only)

23755-9 pa $7.50

TONE POEMS, SERIES I: DON JUAN, TOD UND VERKLARUNGAND DON QUIXOTE, Richard Strauss. Three of the most often per-forrred and re-corded works in entire orchesbal repertoire, reproduced infull score from original editions. Study score. 286pp. 9% x LtY+. (Avail-able in U.S. only) 2,glil-O Pa. S7.5O

11 LATE STRING QUARTETS, Franz foseph Haydn. The form whichHaydn deftned and "brought to perfection." (Gtooe's). 1l strin_g quartetsin complete score, his last and his best. The ffrst in a projected series ofthe complete Haydn string quartets. Reliable modern.Eulenberg

"ditioq'otherwis! difiiculi to obtain. 320pp. 8% x tlYl. (Available in U.S. only)2375V2 Pa. $0.95

FOURTH, FIFTH AND SDMH SYMPHONIES IN FULL SCORE, PCTETIlyitch Tchaikovsky. Complete orchestral soores of Symphon{ No. 1 TF Minor, Op. 36; 3ymphony No. 5 in E Minor, Op. 6{; S;'mn-hofr No' 0in B Minoa 'fatheuque," bp. 74. Bretikopf & Hartel eds. Study sg!:4S0pp.9% x lLYt. 23861-X Pa' $10'95

THE MARRIAGE OF FIGARO: COMPLETE SCORE, Wolfgang A.Mozart. Finest comic opera ever written. Full score, not to be confirsedwith piano renderings. Peters edition. Study score. a8pp. 9! x .\lY^e;(Availabb in U.S. o"ly) 23751-6 Pa. $11.95

.IMAGE' ON THE ART AND EVOLUTION OF THE FILM, EditEd bYMarshall Deutelbaum. Pioneering book brings together for ffnt time 38groundbreaking articles on early silent fflms from Imoge and 260 illutra-tions newly shot from rare prints in the collection of the InternationalMusetun of Photography. A landmark work. Index. 256pp. 8% x 11.

29777-X Pa. $8.95

AROUND-THE-WORLD COOKY BOOK, Lois Lintner SumpUon andMarguerite Linbrer Ashbrook. 373 cooky and frosting recipes from 28cpuntries ( America, Austria, China, Russia, Italy, etc. ) include Viennesekisses, rice wafers, London strips, lady ftngen, hony, sugar spice, mapleoookies, etc. Clear instnrctions. All tested. 38 drawings. l82pp. 5% x 8.

2f,8024 Pa. $2.50

THE ART NOUVEAU STYI.E, edited by Roberta Weddell. 579 rarephotographs, not available elsewhere, of works in iewelry, metdwork, glass,@ramics, textiles, architecture and furniture bv f75 artists-Mucha, Seguy,Lalique, Tiffany, Gaudin, Hotrlwein, Saarinen, and many others. 288pp.8,||6 x lLYt. 235rS7 Pa. i0.95

CATALOGUE OF DOVEN BOOKS

THE COMPLETE BOOK OF DOLL MAKING AND COLLECTINC,catherine christopher. Inshuctions,_ patterns for dozens of ;;d- f;';;gdglt on up to elaborate, historicaily accurate tgut"r. Mould ii"";; ;clothing, make doll houses, etc. Also collecting "information.

ll""y 'iri*-

trations. 288pp. 6 x 9. 2ZOffi-A p"1 CnSoTHE DAGUERREOTYPE IN AMERICA, Beaumont Newhall. wonderfulportraits, 1850's townscapes, landscapes; full text plus rOa plotogrdtr.The basic book. Enlarged 1976 edition. 27hpp. gV; ; lLve.-'

-

2.$n-7 Pa. 97.95CRAFTSMAN HoMEs, Gustav stickley. 2g6 architectural drawines. floorplans, and photographs illustrate 40 different [i"a"

-oT ..Mil;i"';rt'y;,

homes from The craftsman ( 190l-16), voice of e-"ri""n style "i

,ir";liltand organic harmony. Thorough coverage of craftsma' idea in texi anipicture, now collector's item. 2}4pp. gd x f f . 9f,f/gl_6 p". $OOOPEWTER-WORKING: INSTRUCTIONS AND PROJECTS, Brrrl N. os-born. & Gordon o. wilber. Introduction to pewter-wotr.i"i roi-

"*"t"*craftsman. History and characteristics 9f pe*iei; i*rr,-*"13.;t, ;;;;-step instructions. photos, line drawing!, diagrams.' Total -;i

i6;.7r/e x IoVa^. 23Z8Gg pa.-$i:db

THE GREAT cHIpAGo _{IRE, edited bv David Lowe. r0 drabratic, eye-witness accounts of the lgzr disaster, includirg """

oi trru "ftj;;th

;;rebuilding -plus 70 contem_potrty prroiogr"pl', -""a*illustratidns

of they5qorthouse,_Palm^e_r House,- Gieat den'tral b"pot, etc. Introductionby David Lowe. 87pp. 8yn x Lt. ZS77t-0 pa. g4.00SILHOUETTES: A PICTORIAL ARCHIVE OF VARIED ILLUSTRA.lI9ws' 4ited bv carol Belanger Graftog. ou"r-ooir-lihouettes from thelsth to 20th centuries.include iroffles

""J ruti ffg;; ;-a;;; ild ;;;;,children, birds and animars, q.6rrpr

""a ,*""r,-;";;,-rhips, an arphabet.Dozens of uses for commerciir ariists

""J;J[i;;J":' L!!bgrdq6-; iiyi.23751-8 Pa. $4.00

S-\I-YA!I - 1,419 COPYRIGHT-FREE TLLUSTRATIoNS or MAM-MALS, BIRDS, FISH, INsEcrs, ETC., edited-ttJi* it"rt"r. clear woocrengravings present, in extremely lifelike po-ses, orr"i t,000 species-ot iii--mals. one of the most^extensive copyrighi-fr"" pi"to.ili ro*""books of itskind. Captions. Index. 284pp. g x rb. 'zg76s_4-i;:

I?.b;INDIAN DESIGNS FROM ANCIENT EcuADoR, Frederick w. shafier.282.original-designs by pre-columbian Indians ot n",i"a"i-is,iir:lsoo-,i5. i.Designs-include people, mammars, birds, reptiles, trtt,-prl"L, il;L.;;:yetrig designs. use as is or alter for advertling,'te*tiie;l;;ih;;;;i, ";;.Introduction. 95pp. BVe, x tLVe. is76a:d-i;.

-h.;i;

SZIGETI ON THE.VIOLIN, foseph Szigeti. Genial, looselyr structuredtour _ by premier violinist, featuring a pleasant mixture of ieminis"";;;insr-shts into great music and musicianq innumerable tips foJ p;il;;violinists. 38s musical passages. 256pp. s% x gye. zdzog-x

-f;. $5:tE

CATALOGUE OF DOVER BOOKS

GEOMETRY, RELATIVITY AND THE FOURTH DIN{ENSION, RUdOIf

Rucker. Exposition of fourth dimension, mealrs -of visualization, conceo-ts

"i tlf"ir*ti ar Flatland characters continue adventures. Popular, -easily

iollowed yut """,rt"te,

profound. l4l illustrations. l33pp. 1!"^*_8\.^ _-23400-2 Pa. $2.75

THE ORIGIN OF LIFE, A. I. Oparin. Ntodern classic in biochemistry, the

ffrst rigorous examination of possi6le evolution of life from nitrocarbon com-

porrndi. Non-technical, easily followed. Total of 295pp' 5t1" x 8\h,'60213-3 Pa. $4.00

PLANETS, STARS AND GALAXIES, A. E. Fanning. C,omprehensive in-troductOry survey: the sun, sOlar system, stars, galaxieS, Universe, cosmOlogy;quasars, iadio siars, etc. 24pp. of photographs. 189pp. SYa x 8rh. ( Avail-

able it U.S. only) 21080-2 Pa. $3.00

THE THIRTEEN BOOKS OF EUCLID'S ELEMENTS, translated withintroduction and commentary by Sir Thomas L. Heath. Deftnitive edition.Textual and linguistic notes, mathematical analysis, 2500 years of criticalcommentary. Do not conftrse with abridged school editions. Total of l4l4pp.1Ve x 8th. 60088-2, 60089-0, 60090-4 Pa., Three-vol. set $f8.50

DIALOGUES CONCERNING TWO NEW SCIENCES, GAIiIEO GAIiICi.Encompassing 30 years of experiment and thought, these dialogues_ dealwith giometrie demonstrations of fracture of solid bodies, cohesion, lever-age, speed of light and sound, pendulums, falling bodies, accelerated , mo'tion, etc. 300pp. \Ve x 81h. 60099-8 Pa. $4.00

Prices subiea b change withwt notice.

Available at your book dealer or write for free catalogue to Dept.- GI, Dover

Publicatioru, Inc., 180 Varick St., N.Y., N.Y. f0014. Dover publishes more

than 175 books each year on science, elementary and advanced mathemaHcs,

biology, music, art, literary history, social sciences and other areas'

[lie.algebras].nathan.jacobson

Documents

uniaersal

lie algebras

universal

fitting null

universal

uniuersal

lie algebras

simple lie