On identities of infinite dimensional Lie superalgebras
aa r X i v : . [ m a t h . R A ] F e b ON IDENTITIES OF INFINITE DIMENSIONAL LIESUPERALGEBRAS
DUˇSAN REPOVˇS AND MIKHAIL ZAICEV
Abstract.
We study codimension growth of infinite dimensional Lie superal-gebras over an algebraically closed field of characteristic zero. We prove thatif a Lie superalgebra L is a Grassmann envelope of a finite dimensional simpleLie algebra then the PI-exponent of L exists and it is a positive integer. Introduction
We shall consider algebras over a field F of characteristic zero. One of theapproaches in the investigations of associative and non-associative algebras is tostudy numerical invariants associated with their identical relations. Given an alge-bra A , we can associate the sequence of its codimensions { c n ( A ) } n ∈ N (all notionsand definitions will be given in the next section).This sequence gives some information not only about identities of A but alsoabout structure of A . For example, A is nilpotent if and only if c n ( A ) = 0 for alllarge enough n . If A is an associative non-nilpotent F -algebra then A is commuta-tive if and only if c n ( A ) = 1 for all n ≥ A with a non-trivial polynomial identity the sequence c n ( A ) is exponentially bounded by the celebrated Regev’s Theorem [20] while c n ( A ) = n ! if A does not satisfy any non-trivial polynomial identity. In the non-associative case the sequence of codimensions may have even faster growth. Forexample, if A is an absolutely free algebra then c n ( A ) = a n n !where a n = 12 (cid:18) n − n − (cid:19) is the Catalan number, i.e. the number of all possible arrangements of brackets inthe word of length n .For Lie algebra L the sequence { c n ( L ) } n ∈ N is not exponentially bounded in gen-eral even if L satisfies non-trivial Lie identities (see for example [18]). Nevertheless,a class of Lie algebras with exponentially bounded codimensions is sufficiently wide.It includes in particular, all finite dimensional algebras [1, 11], Kac-Moody alge-bras [23, 24], infinite dimensional simple Lie algebras of Cartan type [15], Virasoroalgebra, and many others. Mathematics Subject Classification.
Primary 17C05, 16P90; Secondary 16R10.
Key words and phrases.
Polynomial identity, Lie algebra, codimensions, exponential growth.The first author was supported by the Slovenian Research Agency grants P1-0292-0101 andJ1-4144-0101. The second author was partially supported by RFBR grant No 13-01-00234a. Wethank the referee for several comments and suggestions.
In the case when { c n ( A ) } n ∈ N is exponentially bounded, the upper and the lowerlimits of the sequence { n p c n ( A ) } n ∈ N exist and a natural question arises: does theordinary limit lim n →∞ n p c n ( A )exist? In case of existence we call this limit exp ( A ) or PI-exponent of A .Amitsur conjectured in the 1980’s that for any associative P.I. algebra such alimit exists and it is a non-negative integer. This conjecture was confirmed first forverbally prime P.I. algebras in [4, 21], and later in the general case in [8, 9]. ForLie algebras a series of positive results was obtained for finite dimensional algebras[6, 7, 25], for algebras with nilpotent commutator subalgebras [17], for affine Kac-Moody algebras [23, 24], and some other classes (see [16]). For Lie superalgebrasthere exist only partial results [26, 27, 30, 31].On the other hand it was shown in [28] that there exists a Lie algebra L with3 . < lim inf n →∞ n p c n ( L ) ≤ lim sup n →∞ n p c n ( L ) < . . This algebra L is soluble and almost nilpotent, i.e. it contains a nilpotent idealof finite codimension. In the general non-associative case there exists, for any realnumber α >
1, an algebra A α such thatlim n →∞ n p c n ( A α ) = α. (see [5]). Note also that by a recent result [12] there exist finite dimensional Liesuperalgebras with a fractional limit n p c n ( L ).In the present paper we shall study Grassmann envelopes of finite dimensionalsimple Lie algebras. Our main result is the following theorem: Theorem 1.
Let L ⊕ L be a finite dimensional simple Lie algebra over an al-gebraically closed field F of characteristic zero with some Z -grading. Let also e L = L ⊗ G ⊕ L ⊗ G be the Grassmann envelope of L . Then the limit exp ( e L ) = lim n →∞ n q c n ( e L ) exists and is a positive integer. Moreover, exp ( e L ) = dim L . Another result of our paper concerns graded identities. Since any Lie superalge-bra L is Z -graded one can consider Z -graded identities of L and the correspondinggraded codimensions c grn ( L ). We shall prove that graded codimensions have similarproperties. Theorem 2.
Let L = L ⊕ L be a finite dimensional simple Lie algebra over analgebraically closed field F of characteristic zero with some Z -grading. Let also e L = L ⊗ G ⊕ L ⊗ G be a Grassmann envelope of L . Then the limit exp gr ( e L ) = lim n →∞ n q c grn ( e L ) exists and is a non-negative integer. Moreover, exp gr ( e L ) = dim L . N IDENTITIES OF INFINITE DIMENSIONAL LIE SUPERALGEBRAS 3
In other words, both PI-exponent exp ( e L ) and graded PI-exponent exp gr ( e L ) exist,they are integers and they coincide. Note that for an arbitrary Z -graded algebrathe growth of ordinary codimensions and graded codimensions may differ. Forexample, if A = M k ( F ) ⊗ F Z with the canonical Z -grading induced from groupalgebra F Z , where M k ( F ) is full k × k matrix algebra, then exp ( A ) = k while exp gr ( A ) = 2 k (see [10] for details). In the Lie case one can take L = L ⊕ L tobe a two-dimensional metabelian algebra with L = < e >, L = < f > and withonly one non-trivial product [ e, f ] = f . Then c n ( L ) = n − n ≥ exp ( L ) = 1. On the other hand exp gr ( L ) = 2.2. The main constructions and definitions
Let A be an arbitrary non-associative algebra over a field F and let F { X } bean absolutely free F -algebra with a countable generating set X . A polynomial f = f ( x , . . . , x n ) is said to be an identity of A if f ( a , . . . , a n ) = 0 for any a , . . . , a n ∈ A . The set of all identities of L forms a T-ideal Id ( A ) in F { X } , that is an idealwhich is stable under all endomorphisms of F { X } . Denote by P n = P n ( x , . . . , x n )the subspace of all multilinear polynomials on x , . . . , x n in F { X } . Then P n ∩ Id ( A ) is a subspace of all multilinear identities of A of degree n . In the casewhen char F = 0, the T-ideal Id ( A ) is completely determined by the subspaces { P n ∩ Id ( A ) } , n = 1 , , . . . .For estimating how many identities an algebra A can have one can define theso-called n -th codimension of the identities of A or, for shortness, codimension of A : c n ( A ) = dim P n P n ∩ Id ( A ) , n = 1 , , . . . . As it was mentioned above, the class of associative and non-associative algebraswith exponentially bounded sequence { c n ( A ) } is sufficiently wide. In the case when c n ( A ) < a n for some real a , one can define the lower and the upper PI-exponentsof A as follows: exp ( A ) = lim inf n →∞ n p c n ( A ) , exp ( A ) = lim sup n →∞ n p c n ( A )and the ordinary PI-exponent(1) exp ( A ) = lim n →∞ n p c n ( A ) , provided that exp ( A ) = exp ( A ).For Z -graded algebras one can also consider graded identities. Let X and Y betwo infinite sets of variables and let F { X ∪ Y } be an absolutely free algebra gener-ated by X ∪ Y . If we suppose that all elements of X are even and all elements of Y are odd, i.e. deg( x ) = 0 , deg( y ) = 1 for any x ∈ X, y ∈ Y then F { X ∪ Y } can benaturally endowed by a Z -grading. A polynomial f = f ( x , . . . , x m , y , . . . , y n ) ∈ F { X ∪ Y } is said to be a graded identity of a superalgebra A = A ⊕ A if f ( a , . . . , a m , b , . . . , b n ) = 0 for all a , . . . , a m ∈ A , b , . . . , b n ∈ A . Fix 0 ≤ k ≤ n and denote by P k,n − k the subspace of F { X ∪ Y } spanned by all multilinear poly-nomials in x , . . . , x k ∈ X , y , . . . , y n − k ∈ Y . Then P k,n − k ∩ Id ( A ) is the set of allmultilinear polynomial identities of the superalgebra A = A ⊕ A in k even and n − k odd variables. D. REPOVˇS, AND M. ZAICEV
One of the equivalent definitions of graded codimensions of A is c grn ( A ) = n X k =0 (cid:18) nk (cid:19) c k,n − k ( A ) , where c k,n − k ( A ) = dim P k,n − k P k,n − k ∩ Id ( A ) . Starting from a Z -graded algebra of some class (Lie, Jordan alternative, etc.)one can construct a Z -graded algebra of different class using the notion of theGrassmann envelope. Grasmann envelopes play an exceptional role in PI-theory.For example, any variety of associative algebras is generated by the Grassmannenvelope of some finite dimensional associative superalgebra [14]. In Lie case any so-called special variety is generated by the Grassmann envelope of a finitely generatedLie superalgera [22].We recall this construction for Lie and super Lie cases. Let G be the Grassmannalgebra generated by 1 and the infinite set { e , e , . . . } satisfying the followingrelations: e i e j = − e j e i , i, j = 1 , , . . . . It is known that G has a natural Z -grading G = G ⊕ G where G = Span < e i · · · e i n | n = 2 k, k = 0 , , . . . >,G = Span < e i · · · e i n | n = 2 k + 1 , k = 0 , , . . . > . Given a Lie algebra L with Z -grading L = L ⊕ L , its Grassmann envelope G ( L ) = L ⊗ G ⊕ L ⊗ G ⊂ L ⊗ G is a Lie superalgebra. Vice versa, if L = L ⊕ L is a Lie superalgebra then G ( L )is an ordinary Lie algebra with a Z -grading.3. Cocharacters of Grassmann envelopes
The main tool in studying codimensions asymptotics is representation theory ofsymmetric groups. We refer the reader to [13] for details. Symmetric group S n actsnaturally on multilinear polynomials in F { X } as(2) σf ( x , . . . , x n ) = f ( x σ (1) , . . . , x σ ( n ) ) . Hence P n is an F S n -module and P n ∩ Id ( L ) and also P n ( L ) = P n P n ∩ Id ( L )are F S n -modules. S n -character χ ( P n ( L )) is called n -th cocharacter of L and weshall write χ n ( L ) = χ ( P n ( L )) . Recall that any irreducible
F S n -module corresponds to a partition λ of n , λ ⊢ n , λ = ( λ , . . . , λ k ), where λ ≥ . . . ≥ λ k are positive integers and λ + · · · + λ k = n . By the Maschke Theorem, any finite dimensional F S n -module M decomposesinto the direct sum of irreducible components and hence its character χ ( M ) has adecomposition χ ( M ) = X λ ⊢ n m λ χ λ N IDENTITIES OF INFINITE DIMENSIONAL LIE SUPERALGEBRAS 5 where m λ are non-negative integers. In particular, for the algebra L we have(3) χ ( L ) = X λ ⊢ n m λ χ λ . Integers m λ in (3) are called multiplicities of χ λ in χ n ( L ) and d λ = deg χ λ = χ λ (1) are the dimensions of corresponding irreducible representations. Therefore(4) c n ( L ) = dim P n ( L ) = X λ ⊢ n m λ d λ . For any partition λ = ( λ , . . . , λ k ) ⊢ n one can construct Young diagram D λ containing λ boxes in the first row, λ boxes in the second row and so on: D λ = · · · · · ·· · · ...Given integers k, l, d ≥
0, we define the partition h ( k, l, d ) = ( l + d, . . . , l + d | {z } k , l, . . . , l | {z } d )of n = kl + d ( k + l ). The Young diagram associated with h ( k, l, d ) is hook shaped,and we define H ( k, l ), an infinite hook, as the union of all D λ with λ = h ( k, l, d ), d = 1 , , · · · . For shortness we will say that a partition λ ⊢ n lies in the hook H ( k, l ), λ ∈ H ( k, l ), if D λ ⊂ H ( k, l ). In other words, λ ∈ H ( k, l ) if λ = ( λ , · · · , λ t ) and λ k +1 ≤ l . According to this definition we will say that the cocharacter of L lies inthe hook H ( k, l ) if m λ = 0 in (3) as soon as λ H ( k, l ).A particular case of H ( k, l ) is an infinite strip H ( k, λ ∈ H ( k, λ k +1 = 0.The following fact is well-known and we state it without proof. Lemma 1.
Let L be a finite dimensional algebra, dim L = d < ∞ . Then χ n ( L ) lies in the hook H ( d, for all n ≥ . (cid:3) Another important numerical invariant of the identities of L is the colength l n ( L ). By definition(5) l n ( L ) = X λ ⊢ n m λ where m λ are taken from (3). It easily follows from (4) and (5) that(6) max { d λ | m λ = 0 } ≤ c n ( L ) ≤ l n ( L ) · max { d λ | m λ = 0 } . For studying graded identities of L = L ⊕ L we need to act separately on evenand odd variables. More precisely, the space P k,n − k = P k,n − k ( x , . . . , x k , y , . . . , y n − k )is an S k × S n − k -module where symmetric groups S k , S n − k act on x , . . . , x k and y , . . . , y n − k ), respectively. Any irreducible S k × S n − k -module is a tensor productof S k -module and an S n − k -module and corresponds to the pair λ, µ of partitions, λ ⊢ k, µ ⊢ n − k . As before, the subspace P n − k ∩ Id ( L ) is an S k × S n − k -stablesubspace and one can consider the quotient P k,n − k ( L ) = P k,n − k P k,n − k ∩ Id ( L ) D. REPOVˇS, AND M. ZAICEV as an S k × S n − k -module. Its S k × S n − k -character χ k,n − k ( L ) = χ ( P k,n − k ( L )) isdecomposed into irreducible components.(7) χ k,n − k ( L ) = X λ ⊢ kµ ⊢ n − k m λ,µ χ λ,µ and we define the ( k, n − k )-colength of L as l k,n − k ( L ) = X λ ⊢ kµ ⊢ n − k m λ,µ with m λ,µ taken from (7).First, we prove some relations between graded and non-graded numerical in-variants. We begin by recalling the correspondence between multilinear homoge-neous polynomials in a free Z -graded Lie algebra and in a free Lie superalgebra.Let f = f ( x , . . . , x k , y , . . . , y m ) be a non-associative polynomial multilinear on x , . . . , x k , y , . . . , y m , where x , . . . , x k are supposed to be even and y , . . . , y m odd indeterminates. Then f is a linear combination of monomials from P k,m . Let M = M ( x , . . . , x k , y , . . . , y m ) be such a monomial. We fix positions of y , . . . , y m in M and write M for shortness in the following form M = X y σ (1) X · · · X m − y σ ( m ) X m where X , . . . , X m are some words (possibly empty) consisting of left and rightbrackets and indeterminates x , . . . , x k . Now we define a monomial f M on evenindeterminates x , . . . , x k and odd indeterminates y , . . . , y m from free Lie superal-gebra as f M = sgn( σ ) X y σ (1) X · · · X m − y σ ( m ) X m . Extending this map e by linearity we obtain a linear isomorphism P k,m → P k,m of two subspaces of a Z -graded free Lie algebra and a free Lie superalgebra, re-spectively. Although the monomials in P k,m are not linearly independent, it easilyfollows from Jacobi and super-Jacobi identities that the map e is well-defined. Simi-larly, we can define the inverse map from a free Lie superalgebra to a free Z -gradedLie algebra.Following the same argument as in the associative case (see [10, Lemma 3.4.7])we obtain for any Z -graded Lie algebra L and its Grassmann envelope G ( L ) = G ⊗ L ⊕ G ⊗ L the following result. Lemma 2.
Let f ∈ P k,m be a multilinear polynomial in the free Lie algebra. Then • f is a graded identity of L if and only if e f is a graded identity of G ( L ) ; and • ee f = f . (cid:3) The next lemma is an obvious generalization of Lemma 1.
Lemma 3.
Let L = L ⊕ L be a finite dimensional Lie algebra, dim L = k, dim L = l , and let χ q,n − q ( L ) = X λ ⊢ qµ ⊢ n − q m λ,µ χ λ,µ be its ( q, n − q ) -graded cocharacter. If m λ,µ = 0 then λ ∈ H ( k, and µ ∈ H ( l, . (cid:3) Using this remark we restrict the shape of the graded cocharacter of the Grass-mann envelope G ( L ). N IDENTITIES OF INFINITE DIMENSIONAL LIE SUPERALGEBRAS 7
Lemma 4.
Let L = L ⊕ L be a finite dimensional Lie algebra, dim L = k, dim L = l , and let e L be its Grassmann envelope. If (8) χ q,n − q ( e L ) = X λ ⊢ qµ ⊢ n − q m λ,µ χ λ,µ and m λ,µ = 0 in (8) then λ ∈ H ( k, and µ ∈ H (0 , l ) .Proof . Suppose m λ,µ = 0 in (8) for some λ ⊢ q, µ ⊢ n − q . Then there exists amultilinear polynomial g = g ( x , . . . , x q , y , . . . , y n − q ) such that f = e T λ e T µ g ( x , . . . , y n − q )is not a graded identity of e L , where e T λ ∈ F S q , e T µ ∈ F S n − q are essential idem-potents generating minimal left ideals in F S q , F S n − q , respectively. Inclusion λ ∈ H ( k,
0) immediately follows by Lemma 3 since L and G ( L ) have the same cochar-acters on even indeterminates. Since e T λ and e T µ commute, applying Lemma 4.8.6from [10] we get e f = ae T λ g, where a ∈ I µ ′ . Here µ ′ is the conjugated to µ partition of n − q and I µ ′ is theminimal two-sided ideal of F S n − q generated e T µ ′ . That is, I µ ′ has the character r · χ µ ′ , where r = d µ ′ = deg µ ′ .By Lemma 2, e f is not a graded identity of G ( e L ). Since ee h = h for any h ∈ P q,n − q ,we see that e f is not a graded identity of L and µ ′ ∈ H ( l,
0) by Lemma 3. In otherwords, the number of rows of Young diagram D µ ′ does not exceed l . This numberequals the number of columns of D µ hence µ ∈ H (0 , l ) and we are done. (cid:3) Using the previous lemma we restrict the shape of non-graded cocharacter of G ( L ). Lemma 5.
Let L = L ⊕ L be a finite dimensional Lie algebra, dim L = k, dim L = l , and let χ ( e L ) = X λ ⊢ n m λ χ λ be the n -th (non-graded) cocharacter of e L = G ( L ) . Then m λ = 0 only if λ ∈ H ( k, l ) .Proof . Suppose f ∈ P n is not an identity of e L . Since f is multilinear we mayassume that f ( x , . . . , x q , y , . . . , y n − q ) ∈ P q,n − q is not an identity of e L for some0 ≤ q ≤ n . Moreover, we can consider only the case when a graded polynomial f generates in P q,n − q an irreducible S q × S n − q -submodule M with the character( χ λ , χ µ ), λ ⊢ q, µ ⊢ n − q .Now we lift the S q × S n − q -action up to an S n -action and consider a decompositionof F S n M into irreducible components: χ ( F S n M ) = X ν ⊢ n m ν χ ν . Since λ lies in the hook H ( k, k by Lemma 4and µ lies in H (0 , l ), the vertical strip of width l , it follows from the Littlewood-Richardson rule for induced representations ([13, 2.8.13], see also [10, Theorem2.3.9]) that m ν = 0 as soon as ν H ( k, l ) and we have completed the proof. (cid:3) D. REPOVˇS, AND M. ZAICEV
Lemma 6.
Let G ( L ) = e L = e L ⊕ e L be the Grassmann envelope of a finite di-mensional Lie algebra L = L ⊕ L with dim L = k, dim L = l . Then its colengthsequence { l n ( e L ) } is polynomially bounded.Proof. We use the notation { z , z , . . . } for non-graded indeterminates here since { x , x , . . . } were even variables in the previous statements.Let(9) χ ( e L ) = X λ ⊢ n m λ χ λ be the n -th cocharacter of e L . By Lemma 5 we have λ ∈ H ( k, l ) as soon as m λ = 0in (9). Fix λ ⊢ n with m λ = m = 0 and consider the F S n -submodule(10) W ⊕ · · · ⊕ W m ⊆ P n ( e L )with χ ( W i ) = χ λ , for all i = 1 , . . . , m .We shall prove that(11) m ≤ ( k + l )2 kl n k + l in (10). Denote by λ ′ , . . . , λ ′ l the heights of the first l columns of the Young diagram D λ . Clearly, it suffices to prove the inequality (11) only for λ with λ k > l and λ ′ l > k .Otherwise λ ∈ H ( k ′ , l ′ ) with k ′ ≤ k, l ′ ≤ l and k ′ + l ′ < k + l .Denote µ = λ ′ − k, . . . , µ l = λ ′ l − k. Then λ + · · · + λ k + µ + · · · + µ l = n .It is well-known (see, for example, [29]) that one can choose multilinear f ∈ W , . . . , f m ∈ W m such that F S n f = W , . . . , F S n f m = W m and each f i , i =1 , . . . , m , is symmetric on k sets of indeterminates of orders λ , . . . , λ k and is alter-nating on l sets of orders µ , . . . , µ l .According to this decomposition into symmetric and alternating sets we rename z , . . . , z n as follows(12) { z , . . . , z n } = { z , . . . , z λ , . . . , z k , . . . , z kλ k , ¯ z , . . . , ¯ z µ , . . . , ¯ z l , . . . , ¯ z lµ l } , where each f i is symmetric on any set { z j , . . . , z jλ j } , j = 1 , . . . , k , and is alternatingon any set { ¯ z s , . . . , ¯ z sµ s } , s = 1 , . . . , l .We shall find δ , . . . , δ m ∈ F such that f = δ f + · · · + δ m f m is an identity of e L if (11) does not hold. Note that for any δ , . . . , δ m ∈ F a poly-nomial f is also symmetric on each subset { z i , . . . , z iλ i } , ≤ i ≤ k , and alternatingon each subset { ¯ z s , . . . , ¯ z sµ s } , s = 1 , . . . , l .Let E = { e , . . . , e k + l } be a homogeneous basis of L with E = { e , . . . , e k } ⊂ L , E = { e k +1 , . . . , e k + l } ⊂ L . Then f is an identity of e L if and only if ϕ ( f ) = 0 forany evaluation ϕ : Z → e L such that ϕ ( z i ) = g i ⊗ a i , ≤ i ≤ n, where a i is a basiselement from E and g i ∈ G has the same parity as a i and g · · · g n = 0 in G .Note also that ϕ ( f ) = 0 implies ϕ ′ ( f ) = 0 for any evaluation ϕ ′ such that ϕ ′ ( z i ) = g ′ i ⊗ a i , ≤ i ≤ n provided that g · · · g n = 0.Using these two remarks we shall find an upper bound for the number of evalu-ations for asking the question whether f is an identity of e L or not. N IDENTITIES OF INFINITE DIMENSIONAL LIE SUPERALGEBRAS 9
Consider first one symmetric subset Z = { z , . . . , z λ } . If ϕ ( z i ) = g ⊗ e, ϕ ( z j ) = h ⊗ e, for some i = j with e ∈ E , then ϕ ( f ) = 0 , as follows from the symmetryon Z . Hence we need to check only evaluations with at most r ≤ l odd values ϕ ( z i ) = g ⊗ e t , . . . , ϕ ( z i r ) = g r ⊗ e t r , where e t , . . . , e t r ∈ E are distinct. Since Z is the symmetric set of variables, the result of evaluation ϕ does not depend (upto the sign) on the choice of i , . . . , i r . Hence we have (cid:0) lr (cid:1) possibilities.Given 0 ≤ r ≤ l , we estimate the number of evaluations of remaining λ − r variables in the even component of e L . First, let r = 0 and ϕ ( z i ) = g i ⊗ a i , a i ∈ E , ≤ i ≤ λ . If e appears in the row ( a , . . . , a λ ) exactly α times, e appears α times and so on, then the result of such substitution depends only on α , . . . , α k since f is symmetric on Z . Hence we have no more than ( λ + 1) k variants since0 ≤ α , . . . , α k ≤ λ . In particular, we need at most ( n + 1) k evaluations if r = 0.Let now r = 1. We can replace by odd element an arbitrary variable from Z and get (up to the sign) the same value ϕ ( f ) since f is symmetric on Z . Supposesay, that ϕ ( z λ ) = h ⊗ e, e ∈ E , and ϕ ( z ) = g ⊗ a , . . . , ϕ ( z λ − ) = g λ − ⊗ a λ − , where all a j are even. If α , . . . , α k are the same integers as in the case r = 0 thenthe result of the substitution also depends only on α , . . . , α k . Hence for r = 1 wehave at most (cid:18) l (cid:19) λ k ≤ (cid:18) l (cid:19) ( n + 1) k variants for ϕ since 0 ≤ α , . . . , α k ≤ λ − ≤ r ≤ l we have at most (cid:18) lr (cid:19) ( λ + 1 − r ) k ≤ (cid:18) lr (cid:19) ( n + 1) k variants. Therefore for evaluating all variables from Z it suffices l X r =0 (cid:18) lr (cid:19) ( n + 1) k = 2 l ( n + 1) k substitutions and for all symmetric variables we need at most(2 l ( n + 1) k ) k substitutions.Now consider the alternating set Z ′ = { ¯ z , . . . , ¯ z µ } . If ϕ (¯ z i ) = g ⊗ e, ϕ (¯ z j ) = h ⊗ e, for some i = j with the same e ∈ E , then ϕ ( f ) = 0, hence we can chooseonly 0 ≤ r ≤ k distinct basis elements b , . . . , b r ∈ E for values of ¯ z i , . . . , ¯ z i r ofthe type g i ⊗ b i . Up to the sign, the result of the substitution does not depend on i , . . . , i r and we have only (cid:0) kr (cid:1) options.Suppose now that all ϕ (¯ z i ) , ≤ i ≤ r , are fixed even values. Let ϕ (¯ z r +1 ) = g ⊗ b , . . . , ϕ (¯ z µ ) = g µ − r ⊗ b µ − r , b . . . , b µ − r ∈ E . Then (up to the sign) the result of ϕ depends only on the number of entries of e k +1 , . . . , e k + l into the row ( b , . . . , b µ − r ). Hence we have at most ( µ − r + 1) l variants for substitution of odd variables. As in the symmetric case we have thefollowing upper bound k X r =0 (cid:18) kr (cid:19) ( n + 1) l = 2 k ( n + 1) l for one subset and (2 k ( n + 1) l ) l for all skew variables. We have proved that one can find T ≤ kl ( n + 1) l + k evaluations ϕ , . . . , ϕ T such that the relations(13) ϕ ( f ) = · · · = ϕ T ( f ) = 0imply ϕ ( f ) = 0 for any evaluation ϕ , that is f is an identity of e L . Recall that f = δ f + · · · + δ m f m . Therefore for any evaluation ϕ the equality ϕ ( f ) = 0 canbe viewed as a system of k + l homogeneous linear equations in the algebra e L onunknown coefficients δ , . . . , δ m . If (11) does not hold then the system (13) has anon-trivial solution ¯ δ , . . . , ¯ δ m and f = ¯ δ f + · · · + ¯ δ m f m is an identity of e L , acontradiction.We have proved the inequality (11). From this inequality it follows that allmultiplicities in (9) are bounded by ( k + l )2 kl n k + l . Finally note that the numberof partitions λ ∈ H ( k, l ) is bounded by n k + l . Hence l n ( e L ) < ( k + l )2 kl n k + l + kl and we have thus completed the proof. (cid:3) As a corollary of previous results we obtain the following:
Proposition 1.
Let L = L ⊕ L be a finite dimensional Z -graded Lie algebrawith dim L = k, dim L = l and let e L = G ( L ) be its Grassmann envelope. Thenthere exist constants α, β ∈ R such that c n ( e L ) ≤ αn β ( k + l ) n . In particular, exp ( e L ) = lim sup n →∞ n q c n ( e L ) ≤ k + l. Proof . By [10, Lemma 6.2.5], there exist constants C and r such that X λ ∈ H ( k,l ) d λ ≤ Cn r ( k + l ) n for all n = 1 , , . . . . In particular,max { d λ | λ ⊢ n, λ ∈ H ( k, l ) } ≤ Cn r ( k + l ) n . Now Lemma 6 and the inequality (6) complete the proof. (cid:3) Existence of PI-exponents
Proposition 2.
Let L be a finite dimensional simple Lie algebra over an alge-braically closed field of characteristic zero with some Z -grading, L = L ⊕ L , dim L = k, dim L = l . Let also e L = G ( L ) be its Grassmann envelope. Then thereexist constants γ > , δ ∈ R such that c n ( e L ) ≥ γn δ ( k + l ) n . In particular, exp ( e L ) = lim inf n →∞ n q c n ( e L ) ≥ k + l. N IDENTITIES OF INFINITE DIMENSIONAL LIE SUPERALGEBRAS 11
Proof . Denote d = k + l = dim L . By [19, Theorem 12.1], for the adjoint represen-tation of L there exists a multilinear asssociative polynomial h = h ( u , . . . , u d , . . . ,u m , . . . , u md ) alternating on each subset of indeterminates { u i , . . . , u id } , 1 ≤ i ≤ m ,such that under any evaluation ϕ : u ij → ad b ij , b ij ∈ L , the value ϕ ( h ) is a scalarlinear transformation of L and ϕ ( h ) = 0 for some h . It follows that for any integer t ≥ f t = f t ( u , . . . , u d , . . . , u mt , . . . , u mtd , w )alternating on each set { u i , . . . , u id } , 1 ≤ i ≤ mt such that ϕ ( f t ) = 0 for someevaluation ϕ : { u , . . . , u mtd , w } → L ∪ L . Since f t is multilinear and alternatingon each set { u i , . . . , u id } and d = dim L + dim L it follows that for any t ≥ f t = f t ( x , . . . , x k , . . . , x mt , . . . , x mtk , y , . . . , y l , . . . , y mt , . . . , y mtl , w )which is not a graded identity of L and it is alternating on each subset { x i , . . . , x ik } and on each subset { y i , . . . , y il } , 1 ≤ i ≤ mt , where x ij ’s are even and y ij ’s are oddvariables. The latter indeterminate w can be taken of arbitrary parity, say, w = x is even.Consider an S p × S q -action on P p +1 ,q = P p +1 ,q ( x , x , . . . , x mtk , y , . . . , y mtl ) , where p = mtk, q = mtl and S p , S q act on { x ij } , { y ij } , respectively. It follows fromLemma 3 that the S p × S q -character of the submodule generated by f in P p +1 ,q liesin the pair of of strips H ( k, H ( l, χ ( F [ S p × F q ] f ) = X λ ⊢ pµ ⊢ q m λ,µ χ λ,µ with m λ,µ = 0, unless λ ∈ H ( k, , µ ∈ H ( l, λ is a partition of mtk with at most k rows. On the other hand, f depends on mt alternating subsets ofeven indeterminates of order k each. It is well-known that in this case m λ,µ = 0 if λ = ( λ , λ , . . . ) and λ ≥ mt + 1. It follows that only rectangular partition(14) λ = ( mt, . . . , mt | {z } k )can appear in F [ S p × F q ] f with non-zero multiplicity. Similarly,(15) µ = ( mt, . . . , mt | {z } l )if m λ,µ = 0. Hence we can assume that f has the form f = e T λ e T µ g ( x , . . . , y mtl , w )with λ and µ of the types (14), (15), respectively.By Lemma 2, the polynomial e f is not an identity of the Lie superalgebra e L = G ( L ) and by Lemma 4.8.6 from [10], the graded polynomial e f generates in P p +1 ,q ( e L )an irreducible S p × S q -submodule with the character ( χ λ , χ µ ′ ), where µ ′ = ( l, . . . , l | {z } mt )is conjugated to a µ partition of mtl . First we apply Littelwood-Richardson rule and induce this S p × S q -module up to S n -module. Then we induce the obtained S n -module up to S n +1 -module, where n = p + q = mt ( k + l ). It follows from the Littelwood-Richardson rule that the induced S n +1 -module can contain only simple submodule corresponding to partitions ν ⊢ n + 1 such that the Young diagram D ν contains a subdiagram D ν , where ν = h ( k, l, t ) = ( l + t , . . . , l + t | {z } k , l, . . . , l | {z } t )is a finite hook with t ≥ l − k, mt − kl . Since we are interested with asymptoticof codimensions, we may assume that mt − kl > l − k and then t = mt − kl . Inparticular, ν is a partition of n = ( k + l ) t + kl . Then n + 1 − n = ( k + l − kl + 1and by [10, Lemma 6.2.4] d ν ≤ d ν ≤ n c d ν where c = ( k + l − kl + 1 and d h ( k,l,t ) ≃ an b ( k + l ) n if n → ∞ for some constants a, b , by Lemma 6.2.5 from [10]. Here the relation f ( n ) ≃ g ( n )means that lim n →∞ f ( n ) g ( n ) = 1. Since c n +1 ( e L ) ≥ d ν we get the inequality(16) c n +1 ( e L ) ≥ α ( n + 1) β ( k + l ) n +1 for all n = m ( k + l ) t , t = 1 , , . . . for some constants α > β .Since Lie algebra L is simple, the Grassmann envelope e L is a centerless Liesuperalgebra. It is not difficult to see that c r +1 ( e L ) ≥ c r ( e L ) in this case for all r ≥
1. Hence by (16) we have c n + j ( e L ) ≥ α ( n + 1) β ( k + l ) n +1 for any 1 ≤ j ≤ m ( k + l ). Since n = m ( k + l ) t one can find constants γ > δ such that c r ( e L ) ≥ γr δ ( k + l ) r for all positive integer r and we have completed the proof. (cid:3) Theorem 1 now easily follows from Propositions 1 and 2.
Proof of Theorem 2 . First we obtain an upper bound for c grn ( e L ): c grn ( e L ) = n X q =0 (cid:18) nq (cid:19) c q,n − q ( e L ) , where(17) c q,n − q ( e L ) = X λ ⊢ qµ ⊢ n − q m λ,µ d λ,µ and d λ,µ = deg χ λ,µ = deg χ λ · deg χ µ = d λ d µ . Moreover, λ ∈ H ( k, µ ∈ H (0 , l )by Lemma 4. Applying Lemma 6.2.5 from [10], we obtain X λ ∈ H ( k, λ ⊢ q d λ ≤ Cn r k q , X µ ∈ H (0 ,l ) µ ⊢ n − q d µ ≤ Cn r l n − q for some constants C, r and hence(18) X λ ∈ H ( k, ,λ ⊢ qµ ∈ H (0 ,l ) ,µ ⊢ n − q d λ d µ ≤ C n r k q l n − q N IDENTITIES OF INFINITE DIMENSIONAL LIE SUPERALGEBRAS 13
On the other hand, graded colength l q,n − q ( e L ) = X λ ⊢ qµ ⊢ n − q m λ,µ is not greater than non-graded colength l n ( e L ). Since l n ( e L ) is polynomially boundedby Lemma 6, one can find a polynomial ϕ ( n ) such that(19) m λ,µ ≤ ϕ ( n )for any m λ,µ in (17). It now follows from (17), (18) and (19) that for ψ ( n ) = C n r ϕ ( n ) we have(20) c grn ( e L ) ≤ ψ ( n ) n X q =1 (cid:18) nq (cid:19) k q l n − q = ψ ( n )( k + l ) n and we have obtained an upper bound for c grn (( e L )).On the other hand, in [2, Lemma 3.1] it is proved that for any associative G -graded algebra A , where G is a finite group, an ordinary n -th codimension is lessthan or equal to the graded n -th codimension, for any n . Proof of this lemma doesnot use associativity. Hence(21) c grn ( e L ) ≥ c n ( e L ) . Theorem 2 now follows from (20), (21) and Proposition 2 and we have completedthe proof. (cid:3)
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Algebra Logic
Duˇsan Repovˇs, Faculty of Mathematics and Physics, and Faculty of Education, Uni-versity of Ljubljana, P. O. B. 2964, Ljubljana, 1001, Slovenia
E-mail address : [email protected] Mikhail Zaicev, Department of Algebra, Faculty of Mathematics and Mechanics,Moscow State University, Moscow,119992 Russia
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