intermediate growth in lie algebras and their enveloping algebras · 2017-02-03 · journal of...

Ž .JOURNAL OF ALGEBRA 179, 459]482 1996ARTICLE NO. 0020

Intermediate Growth in Lie Algebras and TheirEnveloping Algebras

V. M. Petrogradsky*

( )Department of Mechanics and Mathematics Algebra ,Branch of Moscow State Uni ersity in Ulyano¨sk,

432700 Le¨ Tolstoy 42, Ulyano¨sk, Russia

Communicated by Susan Montgomery

Received November 15, 1993

In this paper a series of dimensions is suggested which includes as first termsdimension of a vector space, Gelfand]Kirillov dimension, and superdimension. Interms of these dimensions we describe the change of a growth in transition from aLie algebra to its universal enveloping algebra. Also, we find the growth of freepolynilpotent finitely generated Lie algebras; as an application we specify thosealgebras with rational Hilbert]Poincare series. As a corollary we find an asymp-´totic growth of lower central series ranks for free polynilpotent finitely generatedgroups. Q 1996 Academic Press, Inc.

INTRODUCTION

Ž .Let A be a Lie associative algebra over a field K, generated by a finiteŽ . Ž .set X all algebras in this paper are finitely generated . Denote by A X, n

subspace, spanned by all monomials in X of length not exceeding nŽ .including identity if A is associative . Denote

g n s g X , n s dim A X , n ,Ž . Ž . Ž .A A K

l n s g n y g n y 1 ,Ž . Ž . Ž .A A A

where dim stands for the dimension of a vector space over K. Growth ofKŽ .a function g n is an important characteristic of A.A

q q � 4On functions f : N ª R , where R s a g R N a ) 0 , we consider theŽ . Ž .following partial order: f n $ g n iff there exist N ) 0, C ) 0, such that]

Ž . Ž . Ž . Ž . Ž . Ž .f n F g Cn , n G N. An equivalence f n ; g n means that f n % g n ,]Ž . Ž .f n $ g n . It is easy to verify that for another generating set X 9 we have]

Ž .* Supported in part by the International Science Foundation under Grant M 22000 1994 .

459

0021-8693r96 $12.00Copyright Q 1996 by Academic Press, Inc.

All rights of reproduction in any form reserved.

V. M. PETROGRADSKY460

Ž . Ž .an equivalent function g X, n ; g X 9, n of a growth rather than thisA Aequivalence.

Ž .If L is a Lie algebra with dim L s k then g n is a polynomialK UŽL.Ž . k k k 9function and g n ; n . Remark that for k / k9 one has n ¤ n . InUŽL.

Ž . kthe general case an arbitrary function f n is compared with functions nw xby computing upper and lower Gelfand]Kirillov dimensions 1 :

ln f n ln f nŽ . Ž .GK dim f n s lim ; GK dim f n s lim .Ž . Ž .

ln n ln nnª` nª`

It is easy to see that exponential growth is the highest possible growth forLie and associative algebras. Indeed, if L is a finitely generated free Lie

Ž . Ž .Ž . Ž . Ž w x.algebra then one has g n ; g U L n ; exp n see, e.g., 2 . If LieL D 'Ž . Ž . w xalgebra L is infinite dimensional then g n %exp n 3 , in particularUŽL. ]Ž . Ž .GK dim U L s `. For Witt algebra L s W one has exactly g n ;1 UŽL.

'Ž .exp n . Such a growth, lying between polynomial and exponential, iscalled intermediate. For study of such a growth, superdimension and lower

w xsuperdimension have been suggested 4 :

ln ln f n ln ln f nŽ . Ž .S dim f n s lim ; S dim f n s lim .Ž . Ž .

ln n ln nnª` nª`

Ž . Ž . Ž .If S dim f n s S dim f n s a , 0 - a - 1, then this means that f nŽ a .behaves like a function exp n . Other properties of these two dimensions

w x w xcan be found in monograph 5 and review 6 .Ž .Growth less then exponential in the sense $ is called subexponential.]

w xIn 3, 7 it is proved that subexponential growth of a Lie algebra impliessubexponential growth of its universal enveloping algebra. By use of thisresult it is easy to prove that any finitely generated solvable Lie algebra

w xhas subexponential growth 8 . Lie algebras of subexponential growth areof some interest since their universal enveloping algebras satisfy the Ore

w xcondition; that is, they have division rings of fractions 8 . There arefurther attempts to specify more presisely the change of a subexponentialgrowth in transition from a Lie algebra to its universal enveloping algebraw x Ž .7, 9 see comments below .

w xTo study this problem we recall a series of dimensions suggested in 10 .Denote by iteration

lnŽ1.n s ln n; lnŽqq1.n s ln lnŽq. n , q s 1, 2, . . . .Ž .

LIE ALGEBRAS AND ENVELOPING ALGEBRAS 461

qŽ .Consider a series of functions F n , q s 1, 2, 3, . . . , of a naturala

argument with parameter a g Rq:

F1 n s a ,Ž .a

D2 n s na ,Ž .a

F3 n s exp narŽaq1. ,Ž . Ž .a

nqF n s exp ; q s 4, 5, . . . .Ž .a 1raž /Žqy3.ln nŽ .

qŽ .Functions F n , q G 4, are defined and they are positive, beginning witha

some number.Ž .Suppose that f n is any positive valued function of a natural argument.

Ž . Ž .By an upper dimension of leël q, q s 1, 2, 3, . . . or q-dimension , wedenote

Dimq f n s inf a g RqN f n $ F q n .� 4Ž . Ž . Ž .a]

In the same way we define a lower dimension of leël q, q s 1, 2, 3, . . . :

q q qDim f n s sup a g R N f n % F n .� 4Ž . Ž . Ž .a]

Ž . Ž .For convenience we shall also use another partial ordering: f n < g nŽ . Ž .iff there exists N such that f n F g n , n G N. Then for any function

Ž .f n and arbitrary integer q these dimensions can also be computed asŽ .follows see Lemma 2 :

Dimq f n s inf a g RqN f n < F q n ;� 4Ž . Ž . Ž .a

q q qDim f n s sup a g R N f n 4 F n .� 4Ž . Ž . Ž .a

Ž .Suppose that A is a finitely generated algebra with g n as above. ByAŽ .q-dimension lower q-dimension , q s 1, 2, 3, . . . , of A we denote

q qq qDim A s Dim g n , Dim A s Dim g n .Ž . Ž .A A

By the remark above these dimensions we do not depend on generatingset X.

q Ž .It is easy to verify that Dim f n s a , a / 0, a / `, implies thats Ž . s Ž . qDim f n s `, s - q; Dim f n s 0, s ) q. So, if 0 - Dim A F

DimqA - ` then we say that A belongs to leël q.Remark that 1-dimension coincides with ordinary dimension of a vector

space over the field K : Dom1 A s Dim1 A s dim A. Dimensions of levelK2 are exactly upper and lower Gelfand]Kirillov dimensions. Dimensions of


w x Žlevel 3 correspond to superdimensions of 4 up to normalization see.Corollary 2 of Lemma 2 in the present paper . Dimensions of levels q s 4,

5, . . . correspond to growths which are subexponential but are greater thanŽ b .any function exp n , b - 1. Such growths have not been studied previ-

ously. Let us call them logarithmical growths.The first main result of this paper is as follows:

THEOREM 1. Let L be a finitely generated Lie algebra with Dimq L sa ) 0, and q be an arbitrary integer. Also, for q G 3 suppose that Dimq L s a

2 Ž .and for q s 2 suppose that Dim l n s a y 1, a G 1. ThenL

qq1 qq1Dim U L s Dim U L s a .Ž . Ž .

Ž .In fact this result is about generalized partitions Section 1 . The proof ofthis theorem as well as that of a theorem on the growth of solvable Lie

Ž . w xalgebras see the corollary below was only outlined in 10 . Now we notonly give detailed proofs but we also prove a more general result on thegrowth of any free polynilpotent finitely generated Lie algebra.

If L is a Lie algebra then by iteration the lower central series is defined1 iq1 w i xas L s L, L s L, L , i s 1, 2 . . . . Now L is called nilpotent of class s

sq1 � 4 s � 4iff L s 0 , L / 0 . All Lie algebras nilpotent of class s form theŽ .variety NN . Recall that L is polynilpotent with tuple s , . . . , s , s iff theres q 2 1

exists a chain of ideals

0 s L ; L ; ??? ; L ; L s Lqq1 q 2 1

with L rL g NN . All polynilpotent Lie algebras with fixed tuple form ai iq1 s1

variety denoted by NN . . . NN NN . If MM is a variety of Lie algebras then bys s sq 2 1Ž . ŽF MM, k we denote its free algebra of rank k this is an algebra generated

by k elements x , . . . , x and such that for all H g MM and any y , . . . , y g1 k 1 kŽ . .H there exists a homomorphism f : F ª H with f x s y , i s 1, . . . , k .i i

In the case s s ??? s s s 1 one has an AAq-variety of solvable Lieq 1w xalgebras of length q. For the theory of varieties see monograph 2 .

Our second main result is

Ž .THEOREM 2. Let L s F NN . . . NN NN , k , q G 2, be a free polynilpotents s sq 2 1

Lie algebra of rank k, k G 2. Thenq qDim L s Dim L s s dim F NN , k .Ž .2 K s1

w xOf course, by application of the Witt formula 2 , one hass1 m

adim F NN , k s m k ,Ž . Ý ÝK s ž /1 ams1 aNm

Ž .where m n is the Mobius function.¨


COROLLARY. Let L be a free sol able Lie algebra L g AAq of rank k. Then

q qDim L s Dim L s k .

Hilbert]Poincare series of the free algebra of a finite rank of AA 2 is´w xrational 11 . In Section 3 we specify those polynilpotent varieties that have

such a property. Namely, only NN NN have rational Hilbert]Poincare series.´c dw xM. I. Kargapolov raised the problem 2.18 in 13 to describe the lower

central series ranks for a free polynilpotent finitely generated group. Exactw xrecursive formulae were given in 14 . But those formulae do not give any

idea about the character of the growth of these ranks. In this paper wesuggest another answer to this problem by describing an asymptoticbehavior of these ranks.

w xSimilar formulae for growth appeared also in 9 . Let us comment onthem. They were developed for solvable Lie algebras L g AAq and they aretrue for q s 2, 3. But in the case q ) 3 there are mistakes in thecomputation of the derivative of the function C. So, the formulae in thiscase are wrong. Remark also that those methods of analytic number theorylead to difficult integrals.

Ž .Let W be a Witt algebra and WW s var W be a variety defined by alln n n

identical relations of this algebra. Free finitely generated algebras of thisw xvariety were studied by A. A. Kirillov and his students 15]17 . They

obtained some asymptotic formulae for these algebras. In terms of dimen-sions of different levels these results imply:

w x Ž . 3THEOREM 15]17 . Let L s F WW , k , k G n q 1. Then Dim L sn

Dim3 L s n.

It is interesting that for WW and polynilpotent varieties all free algebrasn

of finite rank belong to the same level. So, it makes sense to define theŽ .le¨el Lev MM of a variety MM as the maximum of levels of its free algebras

w xof finite rank. Enumeration of levels in this paper differs from that in 10Ž w x .namely, finite-dimensional algebras in 10 belong to the 0-level . This is

Ždone because we hope the following conjecture is true it is true for.polynilpotent varieties by Theorem 2 .

Ž . Ž . Ž .Conjecture. Lev MM ? MM s Lev MM q Lev MM .1 2 1 2

No doubt the study of noninteger dimensions for a series of levels isinteresting too; we leave it for a later project.


1. SERIES OF DIMENSIONS

Ž . Ž . Ž . Ž . Ž . Ž .Let us denote f n $ g n iff f n $ g n , g n $ f n . It is easy tou] ]prove the following lemma:

LEMMA 1. Strict inequalities hold for q, s G 1, a , b g Rq;

Ž . qŽ . qŽ .1 F n $ F n if a - b ;a b

Ž . qŽ . sŽ .2 F n $ F n if q - s;a b

Ž . qŽ . Ž .3 F n $ exp n .a

q Ž . s Ž .By this lemma Dim f n s a , a / 0, a / `, implies that Dim f n s `,s Ž .s - q; Dim f n s 0, s ) q. There are functions that lie between levels;

Ž . ln n ŽŽ .2 .for example, f n s n s exp ln n lies between levels 2 and 3, i.e.,2 Ž . 3 Ž .Dim f n s `, Dim f n s 0. This paper leaves open a question whether

there exist algebras lying between levels.Ž . d Ž .Nevertheless we shall use the notation f n ; g n iff some q G 1,

a / 0, a / `, the following equalities hold:

q qq qDim f s Dim f s Dim g s Dim g s a .

Ž .LEMMA 2. For any function f n and q G 1 one has

Ž . q Ž . � q Ž . qŽ .41 Dim f n s inf a g R N f n < F n ;a

Ž . q Ž . � q Ž . qŽ .42 Dim f n s sup a g R N f n 4 F n .a

Proof. Suppose that C ) 0 is fixed; then for any « ) 0 let us prove that

F q n < F q Cn < F q n , q G 1. 1Ž . Ž . Ž . Ž .ay« a aq«

For example, if q s 2,

F2 Cn s C ana < naq« s F2 n .Ž . Ž .a aq«

If q s 3, then for any « ) 0 we have

F3 Cn s exp C9narŽaq1. < exp nŽaq« .rŽaq«q1. s F3 n .Ž . Ž . Ž . Ž .a aq«

The case q ) 3 and lower bounds are treated in the same way. NowŽ .that assertion of the lemma follows from 1 . Indeed, if, for example,

q Ž . Ž . q Ž .Dim f n s a then ;« ) 0, 'N, C ) 0 such that f n F F Cn ,aq«

Ž . q Ž . q Ž .n G N. By 1 there exists N such that F Cn F F n , n G N .1 aq« aq2 « 1Ž . q Ž . � 4Thus f n F F n , n G max N, N ; therefore two infimums coincide.aq2 « 1

q Ž . q Ž .COROLLARY 1. Suppose that 0 - Dim f n s Dim f n s a - `,Ž . q Ž . Ž .q G 2. Then 'M: f n s F n , n G M, where lim « n s 0.aq« Žn. nª`


qŽ .Proof. Observe that F n is an increasing function of a , while n isa

Ž . Ž .fixed, and f n has subexponential growth. Hence « n exist. By thelemma, ;« ) 0, 'N such that

F q n F f n F F q n , n G N.Ž . Ž . Ž .ay« aq«

Ž .Therefore « n F « , n G N, thus completing the proof.

Ž .COROLLARY 2. Suppose that f n is an increasing function.

Ž .1 Dimensions of leël 2 coincide with the Gelfand]Kirillo¨ dimen-sions

22Dim f n s GKdim f n , Dim f n s GKdim f n .Ž . Ž . Ž . Ž .

Ž .2 Dimensions of leël 3 coincide with superdimensions up to normal-ization:

Sdim f n Sdim f nŽ . Ž .33Dim f n s ; Dim f n s .Ž . Ž .1 y Sdim f n 1 y Sdim f nŽ . Ž .

Ž .3 Dimensions of leëls q s 4, 5, . . . can be computed as

lnŽqq1. nqDim f n s lim ,Ž .

ln n y ln ln f nŽ .nª`

lnŽqq1. nqDim f n s lim .Ž .ln n y ln ln f nŽ .nª`

Also, it is easy to verify that

LEMMA 3. The following equi alences hold:d

b bŽ . Ž . Ž .1 exp ln n n ; exp n , 0 - b - 1.ds q q qŽ . Ž . Ž . Ž .2 F n ? F Cn ; F n for all s - q, a , b g R , C ) 0.b a a

dq C qŽ . Ž Ž .. Ž .3 F n ; F n proïded that q G 3, C ) 0.a a

Ž . q Ž .LEMMA 4. Let g n be an increasing function with Dim g n s a ) 0,Ž . Ž . Ž .q G 2. Then for a function l n s g n y g n y 1 the following estimates

hold:Ž . 2 Ž .1 If q s 2 and a ) 1 then a y 1 F Dim l n F a .Ž . q Ž .2 If q ) 2 then Dim l n s a .

q Ž .Proof. An upper estimate Dim l n F a is evident in both cases.q Ž .Consider q s 2 and to obtain a contradiction suppose that Dim l n s


Ž .b - a y 1. Then for any d g b , a y 1 owing to Lemma 2 there exists NŽ . dsuch that l n F n , n G N. Therefore

dq1n n n q 1Ž .dg n s C q l i F C q i F C q ,Ž . Ž .Ý Ý

d q 1isN isN

Dim2g n F d q 1 - a ,Ž .

and we have a contradiction.q Ž .In the case q G 3 by way of contradiction suppose that Dim l n s b -

Ž .a . Then by taking any d g b , a and using Lemma 2 we come to acontradiction

nq qg n F C q F i F C q nF n ,Ž . Ž . Ž .Ý d d

isN

Dimqg n F d - a .Ž .

Remarks. It is evident that for a finitely generated algebra A either2 2 Ž .Dim A G 1 or Dim A s 0 i.e., A is finite dimensional ; this explains the

q Ž .condition a ) 1 in assertion 1. If we also add Dim g n s a to theq Ž .hypothesis of the lemma then we can say nothing about Dim l n . Indeed,

Ž .we may set l m s 1 for some subsequence m , i s 1, 2, . . . , withouti iŽ .seriously damaging the growth of g n . Nevertheless we can assert some-

thing.

Ž . q Ž .LEMMA 5. Let g n be an increasing function with Dim g n sq Ž . Ž . Ž . Ž .Dim g n s a ) 0, q G 2. Also let l n s g n y g n y 1 . Then

Ž .u w x;u ) 1 ;a 9 - a 'N ;N , N : N - N - N , N G N 'm g N , N1 2 1 2 2 1 1 2satisfying

Ž . Ž . a 9y11 if q s 2 and a ) a 9 ) 1 then l m G m ;Ž . Ž . q Ž .2 if q ) 2 then l m G F m .a 9

Ž . q Ž . Ž .Proof. By Corollary 1, g n s F n , n G N, where « n ª 0.aqcŽn. nª`

Suppose that we are given u ) 1. Now we fix C ) 0 and take N such that< Ž . Ž . < Ž .« N y « N - C ln u for all N , N ) N. Define function « x , x g1 2 1 2w . Ž . Ž .N, q` , by segments connecting « n y 1 and « n . Then

l n s g n y g n y 1 s g 9 j s F q j 9, j g n y 1, n .Ž . Ž . Ž . Ž . Ž . Ž .Ž .aq« Ž j .

2Ž .


Consider q s 2. Then

x aq« Ž x . 9 s x aq« Ž x .y1 a q « x q x ln x« 9 x . 3Ž . Ž . Ž . Ž .Ž .

Ž .uNow let N , N be taken such that N - N - N , N G N and1 2 1 2 2 1Ž . w xsuppose that x ln x« 9 x F yar4 for all x g N , N . Then1 2

a dxN N2 2« N y « N s y « 9 x dx GŽ . Ž . Ž .H H1 2 4 x ln xN N1 1

a ln N a2s ln G ln u ,ž /4 ln N 41

a contradiction provided that C s ar8 is fixed. Hence there exists x gw x Ž . w xN , N satisfying x ln x« 9 x ) yar4. If x g m y 1, m then we can1 2

Ž . Ž .also assume that for j g m y 1, m we have j ln j« 9 j ) yar3.< Ž . <We may also assume that N was taken such that ;M ) N « M -

� Ž . 4 Ž . Ž .min ar3, a y a 9 r2 . Now m is as required by 2 and 3 .

The case q s 3

a q « xŽ .3 bF x s exp x , b s b x s ,Ž . Ž . Ž .aq« Ž x . a q « x q 1Ž .

exp x b 9 s exp x b x by1 b q x ln xb9 x .Ž . Ž . Ž .Ž .

The above arguments prove that for some m the term in brackets isŽ .greater than some constant, and the assertion follows by 2 .

The case q ) 3

Xx xqF x 9 s exp ? ;Ž .Ž .aq« Ž x . b bž / ž /Ž s. Ž s.ln x ln xŽ . Ž .

1b s , s s q y 3;

a q « xŽ .Xx 1 b

Ž sq1.s ? 1 y y b9 x x ln xŽ .b b Ž s. Ž sy1.ž /ž /Ž s. Ž s. ln x ln x ??? ln xln x ln xŽ . Ž .1 1

2 Ž sq1.4 ? q b « 9x ln x .b ž /Ž s. 2ln xŽ .


Ž sq1.Ž . w xSuppose that « 9x ln x F yC, x g N , N , C ) 0; then1 2

dxN N2 2« N y « N s y « 9 x dx G CŽ . Ž . Ž .H H1 2 Ž sq1.x ln xN N1 1

dx ln NN 22G C s C ln ,H ž /x ln x ln NN 11

which is also a contradiction and we finish the proof as above.

LEMMA 6. Suppose that x, y are integers. Then

x y ; x G 1, y G 0;x q y y 1Ž .1 F x½ž /y y ; x G 1, y G 2.

x q y y 1Ž . Ž .2 is an increasing function of each ¨ariable while othery

¨ariables are fixed.

x q y y 1 yŽ . Ž . ŽŽ Ž .. .Proof. 1 s x q y y 1 ??? x ry!F x becausey

Ž Ž .. w xx q t y 1 rt F x, t g 1, y . This proves the first inequality.The case x s 1 is trivial. If x G 2 then

y q x y 1 ??? y q 2Ž . Ž .Ž .x q y y 1 s y q 1 .Ž .ž /y x y 1 ??? 2Ž .2 Ž .It remains to remark that for y G 2 one has y q 1 F y and y q t rt F y,

t G 2.Ž .2 This follows from two decompositions of the binomial coefficient

above.

2m y 1Ž . Ž .LEMMA 7. 4 exp m .m

Proof. By applying the Stirling formula

1 1 12m y 1 2m 2 ms f 2 4 exp m .Ž .ž / ž /m m '2 2 p m

2. GROWTH IN ENVELOPING ALGEBRAS

Suppose that an algebra A is generated by a finite set X. By theX-length of an element x g A we denote

l x s min n N x g A X , n .� 4Ž . Ž .A

� 4Let L be a Lie algebra with an ordered basis as follows: X s u , . . . , u1 g Ž1.L� 4 Ž . Ž .and u , . . . , u forms a basis of L X, n modulo L X, n y 1 .g Žny1.q1 g Žn.L L


By the Poincare]Birkhoff]Witt theorem the standard monomials´u ??? u , i F i F ??? F ii i 1 2 r1 r

Ž . Ž w x.form a basis of U L . Also see 3, 7 ,

l u ??? u s l u q ??? ql u .Ž . Ž . Ž .UŽL. i i L i L i1 r 1 r

Ž .Thus, l n coincides with the number of solutions of the DiophantineUŽL.equation

1 ? x q x q ??? qx q 2 ? x q ??? qxŽ . Ž .11 12 1l Ž1. 21 2 l Ž2.L L

q ??? qm ? x q ??? qx q ??? qn ? x q ??? qx sn ,Ž . Ž .m1 m l Žm. n1 nl Žn.L L

� 4x g N j 0 .i j

Hence

l n s h y , . . . , y ,Ž . Ž .ÝUŽL. n 1 ny q ??? qmy q ??? qny sn1 m n

g n s h y , . . . , y ,Ž . Ž .ÝUŽL. n 1 ny q ??? qmy q ??? qny Fn1 m n

4Ž .n y q l m y 1Ž .m Lh y , . . . , y s .Ž . Łn 1 n yž /mms1

In general, if we have a sequence b g N, n s 1, 2, . . . , then we cannobtain another sequence a g N, n s 0, 1, 2, . . . :n

y q b y 1 y q b y 11 1 n na s ??? .Ýn y yž / ž /1 ny q ??? qmy q ??? qny sn1 m n

These sequences are connected as coefficients of the formal power seriesw x18

` `1ns a t .Ł Ý nbn n1 y tns1 Ž . ns0

Ž .In the case b s 1 one has a s r n number of partitions for n. Then nŽ . Ž . Ž . n Ž .number of summands for g n in 4 is equal to r n s Ý r m . By˜UŽL. ms1

w xthe well known asymptotic formula 18

1 2n d 3'r n f exp p ; exp n s F n ,Ž . Ž .Ž .( 1ž /' 34n 3 5Ž .d 3'r n - r n - nr n « r n ; exp n s F n ,Ž . Ž . Ž . Ž . Ž .˜ ˜ Ž . 1


Ž . Ž . Ž . Ž .where by f n f g n we shall denote lim f n rg n s 1. In the gen-nª`

eral case one has so-called generalized partitions; they are studied inay1 Ž arŽaq1.. Žanalytic number theory. If b , n then a , exp n . For then n

w x . w xexact meanings of , in both cases see 18 . Moreover, in 18 anasymptotic formula for a is given, but strong conditions on b aren nimposed. In Proposition 1 we give an elementary proof for this transition

Žwith weak assumptions for b as a result we also get a weak equivalencen.for a . Our approach also enables us to study faster growth and obtainn

information on generalized partitions that has not been known previously.ay1 w xThe case b , n has also been treated in 9 with analytic numbern

theory methods, but computations for faster growth in that paper areŽ .wrong see comments above .

w x Ž .Now we set y s nrm and consider in 4 terms of typem

w xnrm q l m y 1Ž .LH m s h 0, . . . , 0, y , 0, . . . , 0 s .Ž . Ž .n n m ž /w xnrm

By Lemma 6 we have the estimates

w xnrm¡ l m s f m , m s 1, . . . , n;Ž . Ž .Ž .L~H m F 6Ž . Ž .n Ž .l m¢ l Žm.L Lw x w xnrm F n s g m , m s 1, . . . , nr2 .Ž .

Now let us study the polynomial growth of a Lie algebra. Suppose thatŽ . w ay1 xl m s m , a ) 1. ThenL

ln mnrmay1f m F F m s m s exp a y 1 n ;Ž . Ž . Ž . Ž .ž /m

g m F G m s nm ay1 s exp ln n ? may1 .Ž . Ž . Ž .

Ž .By computing the derivative we conclude that F m decreases for m ) e;Ž .obviously G m increases for all m.

Ž . 2 � 4PROPOSITION 1. 1 Let Dim L s a G 1. Then max 1, a y 1 F3 Ž .Dim U L F a .

Ž . 2 Ž .2 If in addition we suppose that Dim l n s a y 1, thenL3 Ž .Dim U L s a .

2 Ž .Proof. Let us prove the upper estimate of claim 1. We have Dim l nLŽ . a 9F a . By Lemma 2, for any a 9 ) a there exists N such that l m F m ,L

3 Ž . 3 Ž .m G N. If we prove that Dim U L F a 9 then the estimate Dim U L F awill be proved since a 9 ) a was taken arbitrarily. For convenience weshall write a instead of a 9.


w 1rŽaq1.xSet w s n ; then by both assertions of Lemma 6

h y , . . . , y F H 1 ??? H N H N q 1 ??? H w y 1Ž . Ž . Ž . Ž . Ž .n 1 n n n n n

y yw nay1 ay1? w ??? n .Ž . Ž .

Ž . Ž . Ž .Remark that H 1 ??? H N s p n is a polynomial of a fixed degree. Byn nŽ . Ž .applying 6 and the fact that G m is increasing

wy1dw a rŽaq1. a Žaq1.H m F G w F exp ln n ? n ; exp n .Ž . Ž . Ž . Ž .Ž .Ł n

msNq1

If we take the logarithm of the formula

y y yw m nay1 ay1 ay1w ??? m ??? nŽ . Ž . Ž .

then we get a linear function of variables y , . . . , y . Now recall that thesew nvariables belong to the simplex

wy q ??? qmy q ??? qny F n; y G 0, . . . , y G 0.w m n w n

Ž .Hence, the maximum of 7 is achieved at one of its vertexes, and thus it isŽ . Ž . Ž . Ž .among F w , . . . , F n . Since F m decreases 7 is bounded by the

number

a y 1d darŽaq1. a rŽaq1.F w ; exp ln n ? n ; exp n .Ž . Ž .ž /a q 1

Ž . Ž . Ž .The number of terms for g n in 4 is equal to r n . If we take into˜UŽL.Ž .account 5 and the fact that a ) 1 then by Lemma 3 we get the desired

estimate

dw a rŽaq1.g n F p n G w F w r n ; exp n .Ž . Ž . Ž . Ž . Ž . Ž .Ž . ˜UŽL.

3 Ž .Now let us prove the lower estimate of claim 1. Estimate Dim U L G 1Ž . Ž . 2 Ž .follows from g Gr n and 5 . By Lemma 4 one has Dim l n )UŽL. L

a y 1. Then for any a 9 - a there exists a sequence t , i s 1, 2, . . . , suchiŽ . w a 9y1 x 3 Ž .that l t G t . It is enough to prove that Dim U L G a 9 y 1. ForL i i

w axconvenience we omit the prime. Now we fix t s t and set n s n s t qi i1. By applying Lemmas 6 and 7 we have

w ay1 x2 t y 1 day1 Žay1.r aw xg n G H t G G exp t ; exp n .Ž . Ž . Ž .Ž .UŽL. n ay1ž /w xt


Ž .Thus, sequence t generates sequence n , such that 8 holds, thus provingi ithe estimate.

3 Ž .In claim 2 it only remains to prove that Dim U L G a . If a s 1 thenŽ . Ž . Ž .by g n G r n and 5 the result follows. Now let a ) 1; then for anyUŽL.

Ž . w a 9y1 x1 - a 9 - a there exists N such that l m G m , m G N. Now weL3 Ž .want to prove that Dim U L G a 9. Let us omit the prime and set

h s h 0, . . . , 0, y , . . . , y , 0, . . . , 0 ,Ž .n N t

w ay1 x w 1rŽaq1. xy s m , m s N , . . . , t ; t s n .m

Ž . tBut this term enters 4 provided that Ý my F n. Indeedms N m

aq1t t t q 1Ž .ay1 aw xm m F m F - n.Ý Ý

a q 1msN ms1

Then

t ay1w x2 m y 1ay1 ay1w x w xg n G h G G exp N q ??? q tŽ . Ž .ŁUŽL. ay1ž /w xmmsN

tay1 a a rŽaq1.G exp m y t s exp t C t s exp n C nŽ . Ž .Ž . Ž .Ýž /

msN

;d exp narŽaq1. ,Ž .

Ž . Ž .where C m , C n are some functions approaching nonzero constants.This completes the proof.

COROLLARY. Suppose that the conditions of the proposition are imposed.Then in each claim we can pro¨e stronger lower estimates.

Ž . 3 Ž .1 a y 1 F Dim l n .UŽL.

Ž . 3 Ž .2 Dim l n s a .UŽL.

Ž .Proof. Let us prove claim 1. Recall that 8 is equal to the number ofmonomials of degree less than n. Letter x , by construction, does not1occur in these monomials. In order to obtain monomials of degree exactlyn we multiply them on the left by an appropriate power of x because1l x s 1. It is evident that this set of monomials remains linearlyUŽL.independent. Claim 2 is treated in the same way.

Analogous corollaries can also be added to Propositions 2 and 3 below.

Ž . 3 4 Ž .PROPOSITION 2. 1 Suppose that Dim Lsa)0. Then Dim U L sa .


Ž . 3 4 Ž .2 If in addition Dim L s a then Dim U L s a .

Ž .Proof. Denote b s ar a q 1 - 1 and d s 1rb s 1 q 1ra .Let us prove an upper estimate. Without loss of generality we consider

Ž . Ž b . Ž . Ž . Ž .that l m F exp m , m G N. For f m , g m from 6 we haveL

nrmb y1rŽaq1.f m F F m s exp m s exp nm ,Ž . Ž . Ž . Ž .Ž .g m F G m s nexpŽm b . s exp ln n exp m b .Ž . Ž . Ž .Ž .

w x Ž . Ž .For all m g 1, n function F m decreases while G m increases.Ž .dLet us consider the point w s ln n y 2d ln ln n , integral parts of

numbers we no longer use for simplicity. As in Proposition 1 we have theestimate

wh y , . . . , y F p n G w F w ,Ž . Ž . Ž . Ž .n 1 n

Ž .where p n is some polynomial. Other factors are evaluated as follows:

n ndF w s exp ; exp .Ž . 1ra 1raž / ž /ln n y 2d ln ln n ln nŽ . Ž .

dln n ? n ln n y 2d ln ln n nŽ .wG w s exp F exp .Ž . 2 d 1raž /ž /ln n ln nŽ . Ž .

Ž . Ž Ž ..Since the number of summands in 4 is small it is equal to r n these˜4 Ž .estimates prove an upper bound Dim U L F a .

3 Ž .By Lemma 4 one has Dim l n s a . As above without loss ofUŽL.generality it is enough to prove that the existence of a sequence m , i s 1,i

Ž . Ž b . 4 Ž .2, . . . , satisfying l m G exp m q 1 implies Dim U L G a . Now weL i iŽ b .fix m s m and set n s exp m r2 , y s nrm. For convenience we omiti m

Ž .brackets denoting integral parts of numbers. If we denote n s l m y 1Lthen by the Stirling formula

nqymn q y n q yŽ .l m q y y 1Ž . m mL mH m s fŽ .n n y (myž / n y 2pn ym m m

n ym1 q y rn 1 q nry Q mŽ . Ž . Ž .m m nG ) ,2pn y 2pn y' 'm m

9Ž .

nrml m y 1Ž .L

Q m s .Ž .n ž /n


Ž b .For the denominator n y - exp n n; therefore by Lemma 3 we aremŽ . Ž 2 .d 2interested only in Q m . Now we recall that m s ln n , thus n G n ,n

ln n nddn rŽ2 ln n.Q m G n s exp n ; exp .Ž .n 1q1ra 1raž /ž /2 ln n ln nŽ . Ž .

In order to prove claim 2 let us reverse our construction above. Fixu ) 1, a 9 - a and omit the prime for simplicity. Suppose that we are

Ž 2 .d w u xgiven n and set m s ln n . By Lemma 5 there exist h g m, m ,b uŽ . Ž .l h G exp h provided that n is taken big enough. Then h s m ,L

2 duŽ . Ž . Ž .1 - u - u for simplicity we omit the bar , h s ln n . We have by 9

n n 2u y 1Ž .u2Q h G exp ln n yln n G exp .Ž . Ž . .ž /n du uraž /2 ž /2 ln nŽ .ln nŽ .

Now by u ª 1, a 9 ª a this estimate along with Lemma 2 yields4 Ž .Dim U L G a .

Ž . q qq1 Ž .PROPOSITION 3. 1 Let Dim L s a ) 0, q G 4. Then Dim U L sa .

Ž . q qq1 Ž .2 If in addition we suppose that Dim L s a then Dim U L s a .

Proof. We denote s s q y 3, b s 1ra . Let us prove an upper estimateŽ . q Ž .for g n . By Lemma 4, Dim l n s a . Without loss of generality weUŽL. L

consider that

ml m F exp , m G N.Ž .L bž /Ž s.ln mŽ .

Ž .As above we use the estimates 6 , where

nf m F F m s exp ;Ž . Ž . bž /Ž s.ln mŽ .

mg m F G m s exp ln n exp .Ž . Ž . bž /Ž s.ž /ln mŽ .

Ž . Ž .It is evident that F m is decreasing and G m is increasing. Fix w s ln n;then

nF w s exp ;Ž . bž /Ž sq1.ln nŽ .

Ž sq1. ybw Žln n. 'G w s exp ln n ln n ? n < exp n .Ž . Ž . Ž .


Ž .By analogy with Proposition 1 these estimates along with 5 yield thedesired bound

ndwg n F p n G w F w r n ; exp .Ž . Ž . Ž . Ž . Ž .˜UŽL. bž /Ž sq1.ln nŽ .

Now we prove the lower estimate. Without loss of generality there existsŽ . Ž Ž Ž s. .yb .a sequence m , i s 1, 2, . . . , satisfying l m G exp m ln m q 1.i L i i i

Ž . Ž .We fix some m s m and set n s exp mr2 . Recall the estimate 9 iniProposition 2:

nrml m y 1 rnŽ .Ž .Ž .L

g n G H m G .Ž . Ž .UŽL. n 2pl n n' Ž .L

The denominator is relatively small, as above. For the numerator we have

nrm 2l m y 1 n ln nŽ . Ž .L G exp y ln n2 bž / 2n Ž s.� 0� 0ln nŽ . ln ln nŽ .Ž .

1 1s exp n y

b2ž /ln nŽ s.� 0ln ln nŽ .Ž .nd

; exp .bž /Ž sq1.ln nŽ .

This bound yields the result.In order to prove claim 2 we fix u ) 1, a 9 - a , also omitting the dash

as above. For a given n we set m s ln n. By Lemma 5 without loss ofw u x Ž . Ž Ž Ž s. .y1ra .generality there exist h g m, m , l h G exp h ln h . Then h sL

u uŽ . Ž .m , 1 - u - u for convenience we omit the bar , h s ln n . Then oneŽ .has for 9

1 1 nQ h G exp n y G exp .Ž .n b uy1 1raž /u sq1ž /Ž s.� 0ln nŽ . C ln nŽ .ln ln nŽ .Ž .

qq1 Ž .By u ª 1, a 9 ª a this estimate yields Dim U L G a .

w xProof of Theorem 1. The case q s 1 is well known 1 . The cases q s 2,q s 3, and q G 4 were proved in Propositions 1, 2, and 3, respectively.


3. GROWTH OF FREE POLYNILPOTENT LIE ALGEBRASAND GROUPS

In this paper n-fold commutators are right-normed:

w x w xy , . . . , y , y s y , . . . , y , y ??? .1 ny1 n 1 ny1 n

Let A s ` A be a graded algebra. Then a Hilbert]Poincare[ ns0 nŽ . ` nseries is defined as H t s Ý dim A t . Suppose that MM is a multi-A ns0 K n

Ž .homogeneous variety of Lie algebras and L s F MM, k is freely generated� 4by X s x , . . . , x . Then we have a graded algebra1 k

`

² :w xL s L , L s x , . . . , x , x N x g X , dim L s l n .Ž .[ Kn n i i i i K n L1 ny1 n jns1

In particular, this is the case for polynilpotent varieties. In this theory it isinteresting when a Hilbert]Poincare series is the rational function. A´

2 w xHilbert]Poincare series is a rational for AA 11 :´

kt y 12H t s 1 q kt q , L s F AA , k . 10Ž . Ž . Ž .L k1 y tŽ .

Now we want to prove rationality for NN NN . First consider a special case:c d

Ž .LEMMA 8. Let L s F NN AA, k . Thenc

am r ac 1 m kt y 1

H t s kt q m 1 q ,Ž . Ý ÝL kž / m r až /m a 1 y tŽ .ms1 aNm

Ž .where m n is a Mobius function.¨� 4Proof. Let F be a free Lie algebra freely generated by X s x , . . . , x .1 k

2 w xCommutator subalgebra F is a free Lie algebra freely generated by 2 ,

w xY s y s x , . . . , x , x N i G i G ??? G i - i , n G 2 .� 4j i i i 1 2 ny1 n1 ny1 n

Ž . ` nBy generating a function of some set L we mean h t s Ý h t , whereL ns1 nh is the number of elements in L of length n with respect to X. Now setnŽ . Ž .h t s h t . There exists one-to-one correspondence between Y and theY

Ž .basis for the commutator ideal of a free metabelian algebra. Thus by 10Ž . Ž . Ž .kwe have h t s 1 q kt y 1 r 1 y t .

We proceed as in the proof of Witt’s formula by means of Shirshov’sw x Ž . Ž .basis 11 . Let C t a s 1, 2, . . . be the generating function for a basis ofa

the ath factor of a lower central series of the algebra L2. Then a


Žgenerating function of all associative Y-words of length m consideredŽ 2 ..lying in U L can be presented in two ways:

m m r ah t s a ? C t .Ž . Ž .Ž . Ý aaNm

m Ž Ž 1r m..m Ž 1r a.If we denote t s z then h z s Ý a ? C z . By applying theaN m aMobius inversion formula one has¨

m a1r m 1r am ? C z s m h zŽ . Ž .Ž .Ým ž /aaNm

1 m am r a« C t s m h t .Ž . Ž .Ž .Ým ž /m aaNm

Ž 2 .cq1 Ž . c Ž .Remark that L ( Fr F . Now H t s kt q Ý C t completes theL is1 iproof.

Now consider the general case.

Ž .PROPOSITION 4. Let L s F NN NN , k . Thenc d

Ž . Ž .1 H t is a rational function of typeL

P tŽ .H t s ,Ž .L NQ t 1 y tŽ . Ž .

Ž . Ž .where N s c dim F NN , k and all roots of polynomial Q t are roots ofK dunity distinct from 1. The fraction abo¨e is assumed irreducible.

Ž . Ž . Ž .2 g n and l n are polynomials of degrees N and N y 1, respec-L Lti ely.

Ž . 2 2 2 Ž . 2 Ž .3 Dim L s Dim L s N, Dim l n s Dim l n s N y 1.L L

Proof. Let F be a free algebra freely generated by x , . . . , x . An ideal1 kF dq1 is a free Lie algebra freely generated by some set. Let us find agenerating function for it.

Ž .Let G s G X be the free groupoid of nonassociative monomials in the� 4set X s x , . . . , x and ) be the operation in it. For convenience let us1 k

Ž .identify elements of G with their canonical images in F X under anextention of the mapping x ¬ x , i s 1, . . . , k. For ¨ g G by deg ¨ wei idenote its degree with the respect to X.

Remark that SS s G l F dq1 is a two-sided ideal in G. Suppose that R isŽ .the Hall basis for F X such that x - x - ??? - x . Then u g R _ SS ,1 2 k

¨ g R l SS implies that u - ¨ by a degree argument. An element w g SS

is called SS-reducible iff w s u)¨ , u g SS , ¨ g SS ; otherwise it is SS-irre-


ducible. The set Y of all SS-irreducible elements in R l SS is the freedq1 w xgenerating system in F 2, 2.4.2 . Denote Z s R _ SS . If w g Y, w s

u)¨ , u, ¨ g R then by definition of the Hall basis, u - ¨ , and thereforeu g Z. There are two possibilities. Either ¨ g Z and our process stops or¨ f Z, in which case ¨ s ¨ )¨ , ¨ - ¨ , u G ¨ , thus ¨ g Z. In the last1 2 1 2 1 1case we repeat this process to ¨ . Finally, w g Y can be uniquely pre-2sented as

w xw s z , . . . , z , z , z g Z, n G 2;i i i i1 ny1 n j

w x w xz G z G ??? G z - z , z , z f Z, z , z g R .i i i i i i i i1 2 ny1 n ny1 n ny1 n

Conversely, one easily verifies that any such w belongs to R and isSS-irreducible, thus w g Y.

Ž . ` Ž . i Ž . d Ž . iSuppose that H t s Ý c i t ; then h t s Ý c i t , where k isF is1 k Z is1omitted for convenience. Consider a finite set

w x w xJ s j s a, b N a, b g Z, a, b g R , a, b f Z .� 4Ž .

Ž . �If ¨ g Z then by f ¨ we denote the cardinality of the set z g Z N4 w xdeg z s deg ¨ , z G ¨ . When z , z runs over J we geti iny 1 n

deg aqdeg b dt 1h t s .Ž . Ý ŁY Ž . Ž .f a c qqdeg a qsdeg aq1 1 y tŽ .1 y tŽ . Ž .js a ,b gJ

Obviously, maximal multiplicity of the root 1 in the denominator comesŽ .from terms with j s x , x , k s 2, . . . k. These terms yield1 k

d 12k y 1 t ;Ž . Ł Ž .c qqqs1 1 y tŽ .

Ž . d Ž . Žtherefore multiplicity of 1 in h t is equal to Ý c i s dim F NN ,Y is1 k K d.k . Also, other roots of the denominator are roots of unity.

Ž . Ž dq1.cq1As in the lemma above for L s F NN NN , k , one has L ( Fr F .c dAlso,

c 1 m am r aH t s h t q m h t .Ž . Ž . Ž .Ž .Ý ÝL Z Yž /m ams1 aNm

Ž .Now substitution of h t yields assertion 1. For assertion 2 we shouldYw xapply the well known fact on the growth of rational functions 6 . Now

assertion 3 is evident.

Ž .Proof of Theorem 2. Let L s F NN ??? NN NN , k , q G 2, be generateds s sq 2 1� 4 Ž .by X s x , . . . , x . Consider the auxiliary algebras A s F NN , k , c s s ,1 k c q


� 4 Ž .with free generators Z s z , . . . , z , and B s F NN ??? NN , k with free1 k s sqy 1 1� 4generators Y s y , . . . , y . Now we can construct a wreath product W s A1 k

w xwr B 2 . This algebra possesses the following properties: as a vectorNNc

space W ( A [ B, A is an ideal in W and A is a free algebra in NN withcthe following free generators:

w x¨ , . . . , ¨ , ¨ , z ¨ F ??? F ¨ F ¨ , n G 1, 1 F j F k ,i i i j i i in 2 1 n 2 1

where ¨ are basis elements for B ordered as in Section 1. The mappingi pŽ .C x s y q z , i s 1, . . . , k can be extended to monomorphism C: L ª Wi i i

w x2 .We prove our theorem by induction on q. The base of induction q s 2

is proved in Proposition 4. If q ) 2 then by the inductive hypothesis

qy1 qy1Dim B s Dim B s s dim F NN , k .Ž .2 K s1

By applying Theorem 1 we obtain

q qDim U B s Dim U B s s dim F NN , k . 11Ž . Ž . Ž .Ž .2 K s1

�Let us prove an upper bound. W is generated by X s y , . . . ,14y , z , . . . , z . Consider all products in X of length at most n. Either theyk 1 k

Žcontain some z or they do not. Elements of the second type belong B Y,j.n . As for the elements of the first type, by use of the Jacobi identity and

anticommutativity we present them as linear combinations of the elements

w x w xw , . . . , w , w , p F c, w s y , . . . , y , z ,p 2 1 s si s1 js s

p

0 F i F n y 1, p q i F n.Ýs sss1

² :Let B s u N i s 1, 2, . . . be a linearly ordered basis as in Section 1. ByKiw xrearrangement as in the Poincare]Birkhoff]Witt theorem 8 , w are in´ i

turn presented as linear combinations of

w xu , . . . , u , u , z , u F ??? F u , l u q ??? ql u F n y 1.Ž . Ž .i i i j i i B i B it 2 1 t 1 t 1

The dimension of a linear span of these elements is evaluated byŽ .g Y, n y 1 . ThusUŽB .

cg X , n F g X , n F kg Y , n q g Y , n .Ž . Ž . Ž . Ž .Ž .L W UŽB . B

Ž . Ž .Finally, Lemma 3 along with 11 yields the result since U B is at least oflevel 3.


q ˜ ŽNow let us prove the lower bound for Dim L. Set L s F NN NN ??? NN ,1 s sqy 1 1

˜.k ; then our L is naturally mapped onto L. Therefore it is sufficient toŽ .consider the case s s 1. Then A is abelian and it is the free U B -moduleq

w x w xunder an action b ? z s b, z , b g B, z g A 2 :

A s U B ? z [ ??? [ U B ? z . 12Ž . Ž . Ž .1 k

s q1 s q11 qy1Ž Ž . . Ž .Pick up 0 / ¨ g ??? L ??? . Then 0 / C ¨ g A. Suppose thatŽ .¨ g L X, N . Then we have

w xC x , . . . , x , a s y q z , . . . , y q z , C ¨Ž . Ž . Ž .Ž .i i i i i i1 s 1 1 s s

s y , . . . , y , C ¨ .Ž .i i1 s

Hence

C L X , n = U B Y , n y N ? C ¨ .Ž . Ž . Ž . Ž .Ž . Ž .

Ž .Suppose that f , . . . , f are standard basis monomials for U B . Then we1 tŽ . Ž .assert that f ? C ¨ , . . . , f ? C ¨ are linearly independent. Indeed, to1 t

t Ž .obtain a contradiction suppose that Ý s f ? C ¨ s 0, s g K. Then byis1 i i iŽ . Ž . Ž .12 we have C ¨ s b ? z q ??? qb ? z , b g U B , where some b is1 1 k k k jnonzero. By taking a projection on the jth component we haveŽ t . Ž .Ý s f b ? z s 0, a contradiction with the fact that U B is the integralis1 i i j jdomain. Hence

dg X , n G g Y , n y N ; g n .Ž . Ž . Ž .L UŽB . UŽB .

Ž .By 11 this inequality completes the proof.

COROLLARY 1. Let L be a free sol able Lie algebra L g AAq of rank k.Then

q qDim L s Dim L s k .

Ž .COROLLARY 2. A Hilbert]Poincare series for L s F NN ??? NN , k is´ s sq 1

rational iff q s 2.

Proof. Rationality for q s 2 is proved in Proposition 4. It is well knownw xthat a rational function has either polynomial or exponential growth 6 .

Ž .Thus by the theorem and Lemma 1 in case q G 3 function H t is notLrational.

Now suppose that K is a field of characteristic zero. Let G be anw xarbitrary group, K G be its group algebra, and D be an augmentation

w x n Žideal in K G . The graded algebra associated to the filtration D n s 0,


.1, . . . is defined as

` `n nq1A s A G s A s D rD .Ž . Ž .[ [n

ns0 ns0

In group theory an important characteristic of a group is the growth of theŽ . Ž n nq1. w xsequence a s a G s dim D rD 19 . This is one of many ways ton n K

define growth in group theory.Ž . �Now consider dimension subgroups D s D G s g g G N g y 1 gn n

n4D . They are characteristic subgroups and form central filtration. In astandard way Lie algebra is constructed:

` `

L s L G s L s D rD m K .Ž . Ž .[ [n n nq1 Zns1 ns1

Ž .Denote by b s dim D rD K s rank D rD rank of themn K n nq1 Z n nq1abelian group D rD .n nq1

Ž .Now suppose that G is a free polynilpotent group. Then L G is thefree polynilpotent Lie-algebra corresponding to the same tuple and with

w xthe same number of generators 12 . Also, it is well known that in our casedimension subgroups D coincide with terms of a lower central seriesnŽ . Ž .g G we hope that there is no confusion with other uses of g and then

n nq1 Ž Ž ..map gD ¬ 1 q g g D rD is extended to isomorphism U L G (nq1Ž . w x Ž . Ž .A G 20 recall that char K s 0 . Now remark that b s l n , a sn L n

Ž .l n .UŽL.

COROLLARY 3. Let G be a free polynilpotent group in ¨ariety NN ??? NN ,s sq 1Ž .q G 2, of rank k. Let g G , n s 1, 2, . . . , be terms of the lower centraln

series.

Ž . Ž . Ž .1 Suppose that b s rank g G rg G ; thenn n nq1

s dim F NN , k y 1, q s 2Ž .2 K s1q qDim b s Dim b sn n ½ s dim F NN , k , q G 3.Ž .2 K s1

Ž .2 Suppose that K is the field of zero characteristic, D is the fundamen-w x n nq1tal ideal in K G , and a s dim D rD , n s 0, 1, 2, . . . . Thenn K

qq1 qq1Dim a s Dim a s s dim F NN , k .Ž .n n 2 K s1

Proof. This follows from Theorems 1 and 2 by the arguments above.


ACKNOWLEDGMENTS

The author is grateful to Yu. A. Bachturin, V. A. Ufnarovsky, and M. V. Zaicev forconversations that stimulated his interest in growth theory.

REFERENCES

1. I. M. Gelfand and A. A. Kirillov, Sur les corps lies aux algebres enveloppantes des`Ž .algebres de Lie, Publ. Math. IHES 31 1966 , 509]523.`

2. Yu. A. Bahturin, ‘‘Identical Relations in Lie Algebras,’’ VNV Sci. Press, Utrecht, 1987.3. M. K. Smith, Universal enveloping algebras with subexponential but not polynomially

Ž .bounded growth, Proc. Amer. Math. Soc. 60 1976 , 22]24.4. W. Borho and H. Kraft, Uber die Gelfand]Kirillov-Dimension, Math. Ann. 220, No. 1

Ž .1976 , 1]24.5. G. R. Krause and T. H. Lenagan, ‘‘Growth of Algebras and Gelfand]Kirillov Dimension,’’

Pitman, London, 1985.6. V. A. Ufnarovskiy, ‘‘Combinatorial and Asymptotic Methods in Algebra,’’ VINITI Sovre-

mennie Problemi Matematiki, Fundam. Napravleniia, Vol. 57, Moscow, 1989.7. V. A. Ufnarovskiy, On growth of algebras, Vestnik Mosko¨. Uni . Ser. 1 Mat. Mekh. 4

Ž .1978 , 59]65.Ž .8. A. I. Lichtman, Growth in enveloping algebras, Israel J. Math., 47, No. 4 1984 , 297]304.

Ž .9. A. E. Berezniy, Discrete subexponential groups, Zap. Nauchn. Sem. LOMI 123 1983 ,155]166.

10. V. M. Petrogradsky, On some types of intermediate growth in Lie algebras, Uspekhi Mat.Ž .Nauk 5 1993 , 181]182.

11. Yu. A. Bahturin, A. A. Mikhalev, V. M. Petrogradsky, and M. V. Zaicev, Infinitedimensional Lie superalgebras, in ‘‘de Gruyter Expositions in Mathematics, Vol. 7,’’de Gruyter, Berlin, 1992.

12. A. L. Shmelkin, Free polynilpotent groups, Iz¨. Acad. Nauk. Armyan. SSR Ser. Mat. 28,Ž .No. 1 1964 , 91]122.

13. K. Tetrad, ‘‘Unsolved Problems in Group Theory,’’ Nauka, Novosibirsk, 1967.14. G. V. Egorychev, ‘‘Integral Presentation and Computation of Combinatorial Sums,’’

Nauka, Moscow, 1977.15. A. A. Kirillov and M. L. Kontsevich, Growth in Lie algebras generated by two general

Ž .vector fields on the line, Vestnik Mosko¨. Uni . Ser. 1 Mat. Mekh. 4 1983 , 15]20.16. A. A. Kirillov and A. I. Molev, On algebraic structure of Lie algebra of vector fields,

Preprint No. 168, Keldysh Inst. Prikl. Math., 1985.17. A. I. Molev, On growth of some algebras of vector fields, Uspekhi Mat. Nauk 41, No. 2

Ž .1986 , 205]206.18. G. E. Andrews, ‘‘The Theorie of Partitions,’’ Addison]Wesley, Reading, MA, 1976.19. R. I. Grigorchuk, On Hilbert]Poincare series of graded algebras associated with groups,´

Ž .Mat. Sb. 180, No. 2 1989 , 207]225.20. I. B. S. Passi, ‘‘Group Rings and Their Augmentation Ideals,’’ Lecture Notes in Mathe-

matics, Vol. 715, Springer-Verlag, BerlinrHeidelbergrNew York, 1979.

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