IC/68/23

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

ON THE REPRESENTATIONSOF LIE ALGEBRAS IN LINEAR SPACES

C. D. PALEV

1968PIAZZA OBERDAN

TRIESTE

IC/68/23

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ON THE REPRESENTATIONS OF LIE ALGEBRAS

IN LINEAR SPACES *

C D . PALEV**

TRIESTE

April 1968

* To be submitted to "Journal of Mathematical Physics".

** On leave of absence from the Institute of Physics, Bulgarian Academy of Sciences, Sofia, Bulgaria.

ABSTRACT '

A general rnethod for finding representations of an arbitrary

Lie algebra is presented. The method is based on the fact that every

abstract Lie algebra is homomorphic to an algebra of bilinear com-

binations of formal creation and annihilation operators. SU(2) and

SU(1,1) are treated as examples.

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ON THE REPRESENTATIONS OF LIE ALGEBRAS

IN LINEAR SPACES

I. INTRODUCTION

In a series of recent papers [1-6] .creation and annihilation

operators satisfying Bose-like commutation relations have

been successfully used in order to find some classes of represent-

ations of groups which are important for physical applications. This

construction depends on the possibility of writing the generators of

the group as polynomials of creation and annihilation operators. If

this is the case, then a representation can be realized in Fock space

generated by creation operators from a single "vacuum" state or,

more generally, in the functional Fock space of entire analytical

functions of n complex variables [4, 7],

The aim of this paper is to present a general method for

constructing representations of Lie algebras in a form convenient for

physical applications. To do this we first show that for every abstract

Lie algebra "H- there exists a homomorphic mapping of ^ into the Lie

algebra of bilinear polynomials of creation and annihilation operators

b. , a. , i, j = 1 , . . . ,n, which satisfy Bose-like commutation relations

[a., b.] = 6,. with n > n1 ; n1 depends on A . If the algebra gy is,

for example, semisimple the homomorphism becomes an isomorphism.

Then we prove that representations of every Lie algebra can be real-

ized in a generalized Fock space, which is a direct product of some

representation space of the algebra VV (for instance, the trivial one)

and the space of polynomials of the creation operators in arbitrary

complex power.

We emphasise that in the representation theory of, for example,

SU(1,1), SL(n, c), the method given in this paper is commonly used,

although in some modified version [4] . Hence the examples given

in the last section contain no new result.

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II. ISOMORPHISM BETWEEN LIE ALGEBRAS AND BILINEAR

COMBINATIONS OF CREATION AND ANNIHILATION

OPERATORS

In this section we first introduce the terminology and prove some

auxiliary theorems. We then prove that every abstract Lie algebra for

which there exists at least one faithful matrix representation is iso-

morphic to an algebra spanned by bilinear combinations of creation

and annihilation operators.

Let B be an arbitrary Lie algebra (over the field of real or

complex numbers) and let X be its universal enveloping algebra, i, e.,

the algebra spanned by polynomials of the basis elements b..,. . . , b.

of B , Then a linear representation of B can be realized in X ,

i. e . , for every b1# b e B and V x e X

( b i V b 2 b i ) x = l b i V x • (1)

Let the space of an algebra <r split into a direct sum of sub-

spaces "Jt and B and let "H- be a subalgebra and B be an ideal of

the algebra cA- . Denote the universal enveloping algebra of B by X

and let L be a linear space carrying some representation ctta °f

the algebra rt . With every a ec^ we associate a linear operator

F defined everywhere on the space G = X $3 L , according to the3.

formulae

and

Fb(g) = bx

Fh(g) =

where h e ' K , b e B and we r e p r e s e n t g by g - x-J-, x e X and

i e L . Then the ope ra to r s F , a e < i define a l inea r r e p r e s e n t -

ation F of cH in G . This can be proved by s t ra ight forwardt

calculat ions .

We now introduce the a lgebra ic enti t ies a and b . ( i , j - l # . . . , n

satisfying the commutat ion re la t ions [a . ,b . ] = 6.. . As these commut-.] 6..

.. 3..

ation relations resemble those for the creation and annihilation

operators used in quantum field theory, we shall call b. and a.

creation and annihilation operators'*. In what follows we denote bya *

B a row (b1(.. .,b ) and by A a column (• ) .i n afl

Lemma. For every two matrices M and N of order n the follow-

ing relations are valid:

(a) B(oM + /3N)A = orBMA + £BNA(3)

(b) [BMA, BNA] = B[M, N]A ,

where BMA means matrix multiplication, i, e. , \ b. M.. a. .

i>3

The commutator on the left-hand side of (b) denotes a commutator

between bilinear combinations of creation and annihilation operators,

whereas on the right-hand side it is the usual commutator between

the matrices M and N .

The eq, (a) is obvious. For (b), calculating directly, we have

[BMA,BNA] = [b.M..a.,b N a ] = M.. N [b. a., b a ] =1 rj ] ' p pq q lj pq l j p q

M..N (6. b. a. - 6. b a.) = b. [MN],. a. = B[MN]AIJ pq JP i J iq P j i 13 J

which proves the Lemma.

Theorem 1. Let ^{, be an arbitrary Lie algebra and C-j be a finite

n-dimensional matrix representation of 7C- . Denote by VJ* the set

of all elements V^ of the form V^ = BC,A , V h c H .n n h

Q(a) The set VJ* is a Lie algebra.

(b) The mapping h —> g{h) = V is a homomorphism g of cK'C

onto V ^ . If the representation C^ is faithful,the algebras

and V-, are isomorphic.

* V e do not use the notation a* for b because we do not need to define an involution in the re-

presentation spaces where a and b are linear operators.

(2To prove (a) we note that VV certainly is a linear space.

From this and using the above Lemma we have

t a l V + °2Vh, ' V = ["lB Ch, A + " 2B C h , A ' B C h A l

C + a 2 C )A.BChA] = BI^ C +«,C .LA 1 &

- «1 B [ C h / C h l A + « * * C h ' Ch1 A = " 1 [ VV V + "2[Vh •Vh1 '

where a., ffo are numbers.

It is trivial to show also that [V, ,V, ] + [VL ,V, ] = 0 and thath l h2 h2 h i

Cthe Jacobi identity holds. Hence Vrfp is an algebra. The existence of

the homomorphism g mapping 3> onto V, is also a direct con-

sequence of the Lemma, If the representation C p of "K- is faithful

then the kernel Z of the homomorphism g contains only the element

h = 0 . Indeed, from g(Z) = Vz = BC^A it follows that the equation

g(Z)= 0 holds only for C™ = 0 and hence for Z = 0 . Therefore,

g (0) = 0 and the algebras "% and V , are isomorphic [8] . WeCcall the algebra V« the representation-generating algebra of the

algebra ^ .

Let tv be a semisimple Lie algebra with generators e. and

let [e. e.] = C, . e , then the mapping f(e.) = C gives an iso-1 J Kj K 1

morphism of the algebra Jl onto the matrix algebra C [9] (so-

called adjoint representation of rv) with generators C . Theorem 1C

implies that the algebra VJ, = BCA is isomorphic to the abstract Liealgebra /£ , Hence every semisimple Lie algebra ^ is iso-

morphic to an algebra formed by bilinear combinations of creation

and annihilation operators?'

*) A general discussion of the mapping of any lie algebra into the set of second-order polynomials

of a , b. is given in [10].

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in, LINEAR REPRESENTATIONS OF AN ABSTRACT LIE ALGEBRA

In this section we construct a set of representations for a given

Lie algebra % using theorem 1.

Let B be an algebra spanned by the creation operators b ,. . .,b, ,

"jv an arbitrary Lie algebra, C some matrix representation of V

of dimension k with generators C , and V^ the representation-

generating algebra of % corresponding to C , By </r we denoteQ

the direct sum of the linear spaces V«r and B . As the elements

of iff are polynomials of first-and second-order creation and annihil-

ation operators, it is natural to define a commutator between the

elements of \A~ as a commutator between polynomials of creation

and annihilation operators. Then it turns out that the algebra t*r is

a semidirect product of B and

A = S C+ ^ , (4)

where S\ is a subalgebra and S is an ideal of «fr . The last state-

ment follows from the commutation relations [b. a., b. ] = 6 b. .

Using now (2) we conclude that the representation F f l ofc

V^ ; and hence of "H > m a y b e realized in the space G = X S L,

where X is the universal enveloping algebra of B and L is a space

where some representation 5L^ of \Jv is given. In order to con-

struct this representation F H of 'H- in matrix form, let us introduce

a basis in G with elements denoted by (n1t.. .,n ) i O w n e r e

(n , . . . , n ) = b . . . b- is a basis in the space of the universal en-L K JL J&

veloping algebra X and | i > labels the basis vectors in L , Then

from eq. (2) we haveF (n n )| i> = (n 1 , . . . , n )• <? I i> + [V^fn^ . . . ,nfc)] |i> , (5)

i i i

and the operators F are a representation of the generators e.

because of V = C b a . Therefore,e. pq p q

^e'.{nl V 1 = Cpq EbP

aq' ( V ' * ' ' "k)] ( 6 )

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Calculating [b a ,{n , . . . , n )] directly we obtainp a l K

[b a , ( . . .n . . . n )] = n (.. ,n , . . .n ) , (7)p q»v p q q p+1 q-1

Eqs. (5), (6) and (7J) define completely the generators of the represent-ation F

e.l

The representation FTT , which is obviously infinite dimensional,

contains no more than a countable set of irreducible representations of

*$•£, . Indeed, the basis vectors (n . . .n ) • | i^ of G form a count-

able set X and^therefore, every set K of non-intersecting subsets

of 'f , in particular the set of basis vectors of the irreducible re-

presentations, is countable vi

It is important to note that formulae (5), (6) and (7) still form a

representation of #v. for a much wider space G , which contains G

as a subspace. The linear space G is constructed from G by

analytical continuation in the indices labelling the basis vectors of G .

The exact definition is given by the following.

Theorem 2 . Let cj{, be an algebra with generators e. , e , . . . , ef t

let C be some k-dimensional matrix representation of Ji and gC another

representation of ^ acting in the space spanned by j | iM, Denote

by G the linear space spanned by the elements (n.,., ., n ) • \ i/ ,

where n. , „.. , n are arbitrary complex numbers. With every

generator e. of $t we associate a linear operator F defined on1 e.the basis of G by

F e (n r • -^k) * I i> = (nr.. .,nfe) • £^ \ i> + d PPq (n ,.. .,n ) • | i>1 (8)

*' This may be connected with the fact that representations obtained from the realization of the

generators in terms of creation and annihilation operators usually generate discrete classes of re-

presentations for U(p, q) (see, for instance, [6,11]).

**^As in the quantum theory the vectors (n , . . . ,n ) |o > = b > . . .,b 10 > are interpreted as1 ' X it X « '

states with a definite number of particles, the above construction corresponds to an analytic continuation

in the particle number. (The author is indebted to Prof. Doebner for this remark.)

. 7 -

where

p q q p+1 q-1

Then the operators F form a linear representation of the algebra

Hin G . j

To prove the theorem we note that from (9)

mn pq pmq _ ppnpn qm

From this, one easily obtains

which, together with the relation

F , .. = ttP• + |3F^ , (11)aa + &b a b

{where or, p a re arbi t rary complex numbers) proves the theorem.

For integer values of n., equations (8) and (9) coincide with

(5)-(7) and P q(n , . . . , n ) is the commutator [b a , (n , . . . , n )] .X It p CJ 1 ri

We shall now discuss some properties of the representation

(a) FTJ is infinite dimensional. The basis vectors of G con-

stitute a non-countable set and hence the space G resolves into a

continuous set of invariant subspaces (see also (6)). Therefore, the

representation F^ contains many more irreducible representationsrl

than F~ ; in fact it may also contain the representations of therlcontinuous classes.

(b) To show explicitly that G is reducible, let G C G be a

subspace spanhed by those basis vectors with the property that

n ^ . . ,+n, = N . From (9) it follows that GM is invariant under It.

Hence G reduces to an infinite direct sum of invariant subspaces

v G = © G ^

N N

- 8 -

where the summation is over the field of complex numbers.

The subspace GN is also an infinite-dimensional reducible

space. To show this, let GJ*J 1* • • •» k> e Q denote the subspace

generated by the operators P (p, q = 1, . . . , k) from the vector

(n 1 , . . . , n ) | i y V|i^ e L . Then two given subspaces Gw andI K IN

GN (n denotes the set (n1 > . . . , n )), for which at least one of the

numbers Q. = n. - n! (i = 1 , . . . , k) is not an integer, do not intersect.

n G

It is obvious that a continuous set of subspaces G exist which do

not intersect each other and which are invariant under d\ . There-

fore, the space G^ also resolves into a continuous direct sum of

subspaces GN " ^ * ' ' ' . From (9) it follows that the basis for

G ! * • • • * k is countable. Further reduction of GN into irreduc-

ible subspaces should be regarded separately for every specific Lie

algebra and is,in general,very involved.

(c) If all n. (i = l , . . . , n ) are positive integers, the space

G spans a finite-dimensional representation of the algebra

. This is a consequence of the relation (pp q)n (. . . n . . . ) = 0 if

n > nq

IV. SOME EXAMPLES

In order to demonstrate the method developed above we treat

the well-known [4] simple examples SU(2) and SU(1,1), We emphasise

that this section contains no new result. Let % be an algebra with

generators HQ , H and H , which satisfy the well-known com-

mutation relations

[H+,Hg] = -H+ , [H_,H3] = H_ and [H+ H j = 2§2H3 , (12)

- 9 -

where 5 = 1 and i for SU(2) and SU(1,1)> respectively. We choose

L = I 0> to be the trivial representation for 7i » i. e., cL. 10> = 0/and for C we use the lowest non-trivial representation;

* . . * ) . C H • « ( ! ! ) • < * - . ( ; ; ) . < » >

Denoting the basis vectors (m, n)- |0> by |m, n ) , from (8) and (9)

we obtain

F = - (m-n ) | m, n ) ,H3 2

F = ?n |.m + l , n - 1 ) , (14)+

and

where m and n are arbitrary complex numbers. In accordance

with the theorem stated above, this representation is finite dimensional

only for non-negative integral values of m and n . In order to

write the relations (14) in a more familiar form, let us first introduce

new variables

m + n , m - n ,, _.3 = — ^ — and m3 = - T - . (15)

Then, denoting | m , n ) = | j + m , j - m ) by | j , m > , we3

obtain from (14)

F H l j 'and

], mg

The representation (16) is reducible. Acting with the Casimir

operator C on an arbitrary vector I j ,m > , we obtain

C 2 | j . m >= - | [?2H+H_ + g2H_H++ 2Hg] | j ,m > = - j(j + 1)) j , m

Therefore the representation (16) is irreducible for fixed values of

j and m = m (mod 1). For integer values of j and m ,

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such that j > |m 1 , we obtain finite-dimensional representations. In

all other cases the representations are infinite dimensional.

Up to now all representations were realized in linear spaces.

In order to discuss hermitian or integrable representations, which

are usually important for physical applications, it is necessary to

introduce an appropriately chosen scalar product. It is convenient

to define an orthonormalized basis ? . m = a j m( jm > and to determine

the normalization constants a j m from the hermiticity conditions for

the SU(2) or SU(1,1). algebras

VvFrom (17) we conclude that m is a real number. Imposing condition

(18) one easily obtains the foUowing results. (For a more detailed

version of the argument see the clear exposition in f4].),

(a) For SU(2), i. e. ? = 1 , eq. (18) can be satisfied only for finite

representations if the constants a,^ satisfy the equation

(19)aJ

/ j - m + H

where in (19) some phase convention has been used. Substituting (19)

into (16) we obtain the well-known formulae for hermitian represent-

ations of the algebra of the rotation group [12] ,

(j + m + l) ( j -m)5 2 Y j , (20)

and

where m = - I, ..,, + t and f = 1

-11-

(b) For SU(1,1) , i. e, 0 = i , there are three classes of solutions

for a. .

1st a. =1 and j = -— + ia , where a is an arbitrary complex1 Jim 2

number. This is the principal series [13] . Let us note that

m runs over all values m0 ± k , (k = 0,1, . . . ) where mn is

an arbitrary real number *}

2nd,a.jm— _ i i±±__j—- ^21)

The representation is hermitian for integer values j and m

such that

, m = j - k , k = 0 ,1 ,2 , . . . (22)

j = - |j 1 , m = -j + k . (23)

These are the two discrete classes 3j and "$> . The

explicit expressions for these representations are given by eqs. (20)

for £ = -1 . Thus, by using the method developed in the previous

section, we found all hermitian representations of SU(2) and SU(1,1).

V. CONCLUSION

We have found an explicit method to construct at least some

classes of representations of any arbitrary abstract Lie algebra.

Using this method, we have found all hermitian irreducible represent-

ations for the Lie algebras SU(2) and SU(1,1) . . However we have

no result as to whether, for an arbitrary Lie algebra f{, , the re-

presentation F ' contains all irreducible representations ofH

This shows that representations of SU<1,1) algebra are labelled by two numben j and m

-12-

ACKNOWLEDGMENTS

It is a pleasure to thank Professor H. D. Doebner for many

stimulating conversations concerning this work and for critical dis-

cussions of a preliminary version of this manuscript.

The author is indebted to Professors Abdus Salam and

P. Budini and the IAEA for the hospitality kindly extended to him

at the International Centre for Theoretical Physics, Trieste.

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REFERENCES

[I] B. KURSUNOGLU "Symmetry and Strong Interaction-

Symmetry Principles at High Energy", Proceedings of the

Second Coral Gables Conference, San Francisco and London,

1965, p. 160.

[2] Y. DOTHA.N, M. GELL-MANN and Y. NE'EMAN, Phys.

Letters 1J7, 148 (1965).

[3] I. T. TODOROV, "Non-Compact Groups and Dynamical

Symmetries" (in Russian), Proceedings of the International

Spring School for Theoretical Physics of the Joint Institute for

Nuclear Research, Yalta, 1966, p. 23 7.

[4] A.O. BARUT and C. FRONSDAL, Proc. Roy. Soc* 287A; 532

(1965).

[5] H.D. DOEBNER and O. MELSHEIMER, ICTP, Trieste,

preprint IC/67/28 (to be published in J. Math. Phys.).

[6] ABDUS SALAM and J. STRATHDEE, Phys. Rev. JU8, 1352

(1966).

[7] G. BISIACCHI and R.M. SANTILLI, "Oscillator-like Realization

of the U(l, 1) and (2 + 1)-Lorentz Algebras on Real and Complex

Minkowski Spaces", Institute of Physics, University of Torino

preprint.

[8] L. S. PONTRJAGIN, "Topological Groups", Princeton University

Press, Princeton, NJ, 1958.

[9j D.P. ZHELOBENKO, "Lectures on Theory of Lie Groups"

(in Russian), Dubna, 1965.

[10] H.D. DOEBNER and C.D. PALEV, "Embedding of Lie Algebras

in Second * Order Polynomials of Heisenberg Algebra Generators",

to appear as a preprint.

[II] I.T. TODOROV, ICTP, Trieste, preprint IC/66/71.

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[12] M.A. NAIMARK, "Linear Representations of the Lorentz

Group", Pergamon P r e s s , London, 1964.

[13] V. BARGMANN, Math. Rev. H), 583 and 584 (1949).

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