On the codimension growth of simple color Lie superalgebras
aa r X i v : . [ m a t h . R A ] F e b Journal of Lie TheoryVolume (2012) ??–??c (cid:13) On the codimension growth of simple color Liesuperalgebras
Duˇsan Pagon, Duˇsan Repovˇs, and Mikhail Zaicev
Abstract.
We study polynomial identities of finite dimensional simple colorLie superalgebras over an algebraically closed field of characteristic zero gradedby the product of two cyclic groups of order 2 . We prove that the codimensionsof identities grow exponentially and the rate of exponent equals the dimensionof the algebra. A similar result is also obtained for graded identities and gradedcodimensions.
Mathematics Subject Classification 2000:
Primary 17B01, 17B75. Secondary17B20.
Key Words and Phrases: color Lie superalgebras, polynomial identities, codi-mensions, exponential growth.
1. Introduction
In this paper we begin to study numerical invariants of polynomial identitiesof finite dimensional simple color Lie superalgebras over an algebraically closedfield of characteristic zero. Identities play an important role in the study ofsimple algebras. It follows from the celebrated Amitsur-Levitzky Theorem (see,for example, [8, pp.16-18]) that two finite dimensional simple associative algebrasover an algebraically closed field are isomorphic if and only if they satisfy thesame polynomial identities. Similar results were later obtained for Lie algebras[13], Jordan algebras [3] and some other classes. Most recent results [16] wereproved for arbitrary finite dimensional simple algebras. In the associative case,finite dimensional graded simple algebras can also be uniquely defined by theirgraded identities [12].An alternative approach to the characterization of finite dimensional simplealgebras by their identities uses numerical invariants of identities of algebras.Given an algebra A , one can associate with it a sequence of integers { c n ( A ) } ,called codimensions of A (all definitions will be recalled in the next section). Ifdim A = d , then it is well-known that c n ( A ) ≤ d n +1 (see [10]). For associative Lieand Jordan algebras it is known that c n ( A ) grows asymptotically like t n , where t is an integer and 0 ≤ t ≤ d (see [5], [10], [15]). Moreover, t = d if and only if A is simple.In the present paper we study the asymptotics of codimensions of color Liesuperalgebras in the case when G = h a i × h b i ≃ Z ⊕ Z is the product of two Pagon, Repovˇs, and Zaicev cyclic groups of order two and a skew-symmetric bicharacter β : G × G → F ∗ isgiven by β ( a, a ) = β ( b, b ) = 1 , β ( a, b ) = −
1. For any finite dimensional simple Liealgebra B , the corresponding color Lie superalgebra L = F [ G ] ⊗ B is simple (see[2]). The main result of the paper asserts that the limit lim n →∞ n p c n ( L ) existsand equals dim A (see Theorem 4.2). All necessary information about polynomialidentities, codimensions and color Lie superalgebras can be found in [1], [8].
2. Preliminaries
Let F be a field and G a finite abelian group. An algebra L over F is said to be G -graded if L = M g ∈ G L g where L g is a subspace of L and L g L h ⊆ L gh . An element x ∈ L is said to behomogeneous if x ∈ L g for some g ∈ G and then we say that the degree of x inthe grading is g , deg x = g . Any element x ∈ L can be uniquely decomposed intoa sum x = x g + · · · + x g k , where x g ∈ L g , . . . , x g k ∈ L g k and g , . . . , g k ∈ G arepairwise distinct. A subspace V ⊆ L is said to be homogeneous or graded subspaceif for any x = x g + · · · + x g k ∈ V we have x g , . . . , x g k ∈ V . A subalgebra (ideal) H ⊆ L is said to be a graded subalgebra (ideal) if it is graded as a subspace.A map β : L × L → F ∗ is said to be a skew-symetric bicharacter if β ( gh, k ) = β ( g, k ) β ( h, k ) , β ( g, hk ) = β ( g, h ) β ( g, k ) , β ( g, h ) β ( h, g ) = 1 . A graded G -graded algebra L = L g ∈ G L g is called a color Lie superalgebraor, more precisely, a ( G, β )-color Lie superalgebra if for any homogeneous x, y, z ∈ L one has xy = − β ( x, y ) yx, ( xy ) z = x ( yz ) − β ( x, y ) y ( xz ) . Here, for convenience, we write β ( x, y ) instead of β (deg x, deg y ). Traditionally,the product in color Lie superalgebras is written as a Lie bracket, xy = [ x, y ]. Itis not difficult to see that β ( e, g ) = β ( g, e ) = 1, where e is the unit of G and β ( g, g ) = ± g ∈ G and for any bicharacter β . In the case G = Z , β (1 ,
1) = − β ( g, g ) = 1 for all g ∈ G then a ( G, β )-color Lie superalgebra is called a color Lie algebra.By definition, a color Lie superalgebra is simple if it has no non-trivialgraded ideals. We study identical relations of (
G, β )-color Lie algebras in the case G = h a i × h b i ≃ Z ⊕ Z and β ( a, a ) = β ( b, b ) = 1 , β ( a, b ) = −
1. Recently,for these G and β all finite dimensional simple color Lie algebras were classifiedunder a certain weak restriction [2]. One of the series of finite dimensional simplealgebras can be represented in the following way.Let L = F [ G ] ⊗ B be a tensor product of the group ring F [ G ] with thecanonical G -grading, and a finite dimensional simple Lie algebra B with thetrivial grading. Then L is a G -graded algebra if we set deg( g ⊗ x ) = g forall g ∈ G, x ∈ B . agon, Repovˇs, and Zaicev i, j, k, l ∈ { , } , we define the product[ a i b j ⊗ x, a k b l ⊗ y ] = ( − j + k a i + k b j + l ⊗ [ x, y ] (1)in L . Then under the multiplication (1), an algebra L becomes a ( G, β )-color Liealgebra. Moreover, L is a simple color Lie algebra. Remark 2.1.
The group algebra F [ G ] with the multiplication( a i b j ) ∗ ( a k b l ) = ( − j + k a i + k b j + l (2)is isomorphic to M ( F ), the two-by-two matrix algebra over F , if we identify e, a, b, ab with (cid:18) (cid:19) , (cid:18) − (cid:19) , (cid:18) (cid:19) , (cid:18) −
11 0 (cid:19) , respectively. (cid:3) We study non-graded identities of such an algebra L .Next we recall the main notions of the theory of polynomial identities codi-mension growth (see [8]). Let F { X } be an absolutely free algebra over F with thecountable set of free generators X = { x , x , . . . } . A non-associative polynomial f = f ( x , . . . , x n ) is said to be an identity of F -algebra A if f ( a , . . . , a n ) = 0 forany a , . . . , a n ∈ A . The set of all identities of A forms an ideal Id( A ) of F { X } stable under all endomorphisms of F { X } . Denote by P n = P n ( x , . . . , x n ) the sub-space of F { X } of all multilinear polynomials in x , . . . , x n . Then P n ∩ Id( A ) is asubspace of all multilinear identities of A on variables x , . . . , x n . A non-negativeinteger c n ( A ) = dim P n P n ∩ Id( A )is called n th codimension of A . It is well-known [10, Proposition 2] that c n ( A ) ≤ d d +1 (3)as soon as dim A = d < ∞ . In particular, the sequence n p c n ( A ) is restricted. Inthe 1980’s, Amitsur conjectured that the limit lim n →∞ n p c n ( A ) exists and is aninteger for any associative PI-algebra A . Amitsur’s conjecture was confirmed forassociative [6],[7], finite dimensional Lie [17] and simple special Jordan algebras[10]. For general non-associative algebras a series of counterexamples with afractional rate of exponent were constructed in [4], [18]. If the limit exists wecall it the PI-exponent of A ,PI-exp( A ) = lim n →∞ n p c n ( A ) . Pagon, Repovˇs, and Zaicev
3. Multialternating polynomials
Multialternating polynomials play an exceptional role in computing PI-exponentsof simple algebras. In the associative and the Lie case one may choose multial-ternating polynomials among central polynomials constructed by Formanek andRazmyslov. In the Jordan case the existence of central polynomials is an openproblem. Nevertheless, Razmyslov’s approach (see [14]) allows one to constructthe required multialternating polynomials. We shall follow the Jordan case [9],[10]. Recall that B is a finite dimensional simple Lie algebra over an algebraicallyclosed field of characteristic zero, G = h a i × h b i ≃ Z ⊕ Z and β : G × G → F ∗ the skew-symmetric bicharacter on G . The simple color Lie algebra L is equal to F [ G ] ⊗ B and the multiplication on L is defined by (1).As in the Lie case we define the linear transformation ad x : L → L as theright multiplication by x , ad x : y [ y, x ]. Consider the Killing form ρ on L : ρ ( x, y ) = tr(ad x · ad y ) . Lemma 3.1.
The Killing form is a symmetric non-degenerate bilinear form on L . Proof . Linearity and symmetry of ρ are obvious. Fix any basis C = { c , . . . , c d } of B where d = dim c and consider the basis¯ C = { e ⊗ c i , a ⊗ c i , b ⊗ c i , ab ⊗ c i | ≤ i ≤ d } of L . Let M be the matrix of ρ in this basis. Consider two basis elements x = g ⊗ c i , y = h ⊗ c j ∈ ¯ C with g, h ∈ G . If g = h then gh = e in G and ad x · ad y maps the homogeneous component L t to L ght = L t . Hence tr(ad x · ad y ) = 0.Conversely, if g = h then any homogeneous subspace L t is invariant under thead x · ad y -action. Moreover, if we order ¯ C in the following way¯ C = { e ⊗ c , . . . , e ⊗ c d , a ⊗ c , . . . , a ⊗ c d , b ⊗ c , . . . , b ⊗ c d , ab ⊗ c , . . . , ab ⊗ c d then M is a block-diagonal matrix with four blocks M , . . . , M on the maindiagonal and all M , . . . , M are matrices of the Killing form of B . Since theKilling form on B is non-degenerate, the matrix M and ρ are also non-degenerateand we have completed the proof of the lemma. (cid:3) Now we fix our simple color Lie algebra L , dim L = q = 4 dim B andconstruct multialternating polynomials which are not identities of L . In the restof this section we shall assume that F is algebraically closed.We shall use the following agreement. Given a set of indeterminates Y = { y , . . . , y n } , we denote by Alt Y the alternation on Y . That is, if f = f ( x , . . . , x m , y , . . . , y n ) is a polynomial multilinear on y , . . . , y n then Alt Y ( f ) = X σ ∈ S n (sgn σ ) f ( x , . . . , x m , y σ (1) , . . . , y σ ( n ) )where S n is the symmetric group and sgn σ is the sign of the permutation σ ∈ S n . agon, Repovˇs, and Zaicev Lemma 3.2.
Let B be a finite dimensional simple Lie algebra, dim L = d .Then there exists a left-normed monomial f = [ x , . . . , x t , y , x , . . . , x t , y , . . . , x d , . . . , x dt d , y d , x d +11 ] (4) with t , . . . , t d > such that Alt Y ( f ) is not an identity of B .Proof . By [14, Theorem 12.1] there exists a central polynomial for the pair( B, Ad B ) that is an associative polynomial w = w ( x , . . . , x d , . . . , x k , . . . , x kd )such that w is alternating on each set { x i , . . . , x id } and w (ad ¯ x , . . . , ad ¯ x kd ) = λE is a scalar limear map on B for any evaluation x ij ¯ x ij ∈ B . Moreover, λ = 0 assoon as ¯ x i , . . . , ¯ x id are linearly independent for any fixed 1 ≤ i ≤ d .Hence [ x , w ] is not an identity of B . Here we write [ x , w ] instead of w ( x ) = w (ad ¯ x , . . . , ad ¯ x kd )( x ). By interrupting the alternation on all sets except x k , . . . , x kd and renaming x k = y , . . . , x kd = y d we obtain a multilinear polynomialskew-symmetric on y , . . . , y d which is not an identity of B . By rewriting thispolynomial as a linear combination of left-normed monomials we can get at leastone monomial of the type (4) such that Alt Y ( g ) is not an identity of B but perhapsdoes not satisfy the condition t , . . . , t d > t , . . . , t d > t i = 0. For brevitywe assume t = 1 , t = 0. We again use the central polynomial. Replace g = [ x , y , y , . . . ] with g ′ = [ x , y , w ′ , y , . . . ]where w ′ is the central polynomial written in new variables e x ij and we apply w ′ ( e x ij )to [ x , y ]. Since w ′ is a central polynomial, one of the left-normed monomials f ′ of g ′ is also of the form (4) with the same t , t , . . . , t d but with t > Alt Y ( g ′ )is not an identity of B . By applying this procedure at most d times we obtain arequired polynomial (4). The existence of the last factor x d +11 is obvious. (cid:3) Using Lemma 3.2 we construct the first alternating polynomial for L = F [ G ] ⊗ B . Lemma 3.3.
There exists a multilinear polynomial f = f ( x , . . . , x q , y , . . . , y k ) which is not not vanishing on L and is alternating on x , . . . , x q .Proof . Let f be the monomial obtained in Lemma 3.2. Then there existsan evaluation ϕ : X → B , ϕ ( x ij ) = ¯ x ij , ϕ ( y i ) = ¯ y i , such that ϕ ( h ) = 0 where h = Alt Y ( f ). Given 1 ≤ i, j ≤
2, we consider the evaluation ϕ ij : X → L of thefollowing type: ϕ ij ( y k ) = E ij ⊗ ¯ y k , ϕ ij ( x kt k ) = E i ⊗ ¯ x kt k , ϕ ij ( x k +11 ) = E j ⊗ ¯ x k +11 , ≤ k ≤ d, Pagon, Repovˇs, and Zaicev and ϕ ij ( x rs ) = E ⊗ ¯ x rs for all remaining x rs where E ij ’s are matrix units of F [ G ] ≃ M ( F ) (see Remark2.1). Then ϕ ij ( h ) = E ⊗ ϕ ( h ) = 0in L . Now we write h on four disjoint sets of indeterminates, h = h ( X , Y ) , . . . , h = h ( X , Y ) . Since B is simple, the polynomial H = [ h , z , . . . , z r , h , z , . . . , z r , . . . , h ]is not an identity of L for some r , . . . , r ≥
0. Moreover, ϕ ( Alt ( H )) = 4 d ! · [ ϕ ( h ) , ¯ z , . . . , ¯ z r , ϕ ( h ) , . . . , ϕ ( h )] (5)where ϕ | X ,Y = ϕ , . . . , ϕ | X ,Y = ϕ , ϕ ( z γδ ) = ¯ z γδ and the right hand side of(5) is non-zero for some ¯ z γδ ∈ L . Here Alt on the left hand side of (5) means thealternation on Y ∪ . . . ∪ Y . Since | Y ∪ . . . ∪ Y | = 4 d = dim L = q , we havecompleted the proof of the lemma. (cid:3) For extending the number of alternating sets of variables we shall use thefollowing technical lemma.
Lemma 3.4.
Let f = f ( x , . . . , x m , y , . . . , y k ) be a multilinear polynomialalternating on x , . . . , x m . Then, for v, z ∈ X , the polynomial g = m X i =1 f ( x , . . . , x i − , [ x i , v, z ] , x i +1 , . . . , x m , y , . . . , y k ) is also alternating on x , . . . , x m .Proof . Clearly it is enough to check that g is alternating on x r , x s , 1 ≤ r
Let Y = Y ∪ Y ∪ · · · ∪ Y r ⊆ X be a disjoint union with r ≥ and Y eventually empty. Let f = f ( x , . . . , x q , Y ) be a multilinear polynomialalternating on each Y i , ≤ i ≤ r , and on x , . . . , x q . Then, for any k ≥ andfor any v , z , . . . , v k , z k ∈ X , there exists a multilinear polynomial g = g ( x , . . . , x q , v , z , . . . , v k , z k , Y ) such that, for any evaluation ϕ : X → L , ϕ ( x i ) = ¯ x i , ≤ i ≤ q , ϕ ( v j ) = ¯ v j , ϕ ( z j ) = ¯ z j , ≤ j ≤ k , ϕ ( y ) = ¯ y , for y ∈ Y , we have ϕ ( g ) = g (¯ x , . . . , ¯ x q , ¯ v , ¯ z , . . . , ¯ v k , ¯ z k , ¯ Y )= tr ( ad v · ad z ) · · · tr ( ad v k · ad z k ) f (¯ x , . . . , ¯ x q , ¯ Y ) . Moreover g is alternating on each set Y i , ≤ i ≤ r , and on x , . . . , x q .Proof . The proof is by induction of k . Suppose first that k = 1 and define g = g ( x , . . . , x q , v, z, Y ) = q X i =1 f ( x , . . . , [ x i , v, z ] , . . . , x q , Y ) . Then g is alternating on each set Y i , ≤ i ≤ r and, by Lemma 3.4, is alsoalternating on x , . . . , x q . Consider an evaluation ϕ : X → L such that ϕ ( x i ) = ¯ x i ,1 ≤ i ≤ q , ϕ ( v ) = ¯ v , ϕ ( z ) = ¯ z , ϕ ( y ) = ¯ y , for y ∈ Y . Suppose first that theelements ¯ x , . . . , ¯ x q are linearly dependent over F . Then, since f and g arealternating on x , . . . , x q , it follows that ϕ ( f ) = ϕ ( g ) = 0 and we are done.Therefore we may assume that ¯ x , . . . , ¯ x q are linearly independent over F and so, since dim L = q , they form a basis of L . Hence for all i = 1 , . . . , q , wewrite [¯ x i ¯ v, ¯ z ] = α ii ¯ x i + X j = i α ij ¯ x j , for some scalars α ij ∈ F . Since f is alternating on x , . . . , x q , f (¯ x , . . . , [¯ x i , ¯ v, ¯ z ] , . . . , ¯ x q , ¯ Y ) = α ii f (¯ x , . . . , ¯ x i , . . . , ¯ x q , ¯ Y ) . Therefore g (¯ x , . . . , ¯ x q , ¯ v, ¯ z, ¯ Y ) = ( α + · · · + α qq ) f (¯ x , . . . , ¯ x q , ¯ Y ) , and, since α + · · · + α qq = tr(ad v · ad z ), the lemma is thus proved in case k = 1.Now let k > g = g ( x , . . . , x q , v , z , . . . , v k − , z k − , Y ) be amultilinear polynomial satisfying the conclusion of the lemma. Then we write g = Pagon, Repovˇs, and Zaicev g ( x , . . . , x q , Y ′ ), where Y ′ = Y ′ ∪ Y ∪· · ·∪ Y r and Y ′ = Y ∪{ v , z , . . . , v k − , z k − } .If we now apply to g the same arguments as in the case k = 1, we obtain apolynomial satisfying the conclusion of the lemma. (cid:3) Now we are ready to construct the required multialternating polynomialfor our simple color Lie algebra L , dim L = q . Recall that F is an algebraicallyclosed field of characteristic zero. Proposition 3.6.
For any k ≥ there exists a multilinear polynomial g k = g k ( x (1)1 , . . . , x (1) q , . . . , x (2 k +1)1 , . . . , x (2 k +1) q , y , . . . , y N ) satisfying the following conditions:1) g k is alternating on each set { x ( i )1 , . . . , x ( i ) q } , ≤ i ≤ k + 1 ;2) g k is not the identity of L ;3) the integer N does not depend on k .Proof . Let f = f ( x , . . . , x q , y , . . . , y m ) be the multilinear polynomial fromLemma 3.3. Hence f is alternating on x , . . . , x q and does not vanish on L .Suppose first that k = 1 and write Y = { y , . . . , y m } . By Lemma 3.5 thereexists a multilinear polynomial g = g ( x , . . . , x q , v (1)1 , z (1)1 , . . . , v (1) q , z (1) q , Y )such that g (¯ x , . . . , ¯ x q , ¯ v (1)1 , ¯ z (1)1 , . . . , ¯ v (1) q , ¯ z (1) q , ¯ Y )= tr(ad ¯ v (1)1 · ad ¯ z (1)1 ) · · · tr(ad ¯ v (1) q · ad ¯ z (1) q ) f (¯ x , . . . , ¯ x q , ¯ Y ) . Now, for any σ, τ ∈ S q , define the polynomial g σ,τ = g σ,τ ( x , . . . , x q , v (1)1 , z (1)1 , . . . , v (1) q , z (1) q , Y )= g ( x , . . . , x q , v (1) σ (1) , z (1) τ (1) , . . . , v (1) σ ( q ) , z (1) τ ( q ) , Y ) . Then set g ( x , . . . , x q , v (1)1 , z (1)1 , . . . , v (1) q , z (1) q , Y ) = 1 q ! X σ,τ ∈ S q (sgn σ )(sgn τ ) g σ,τ . The polynomial g is alternating on each of the sets { x , . . . x q } , { v (1)1 , . . . , v (1) q } and { z (1)1 , . . . , z (1) q } . Next we show that for any evaluation ϕ , ϕ ( g ) = det ¯ ρ · ϕ ( f ) , agon, Repovˇs, and Zaicev ρ = ρ (¯ v (1)1 , ¯ z (1)1 ) · · · ρ (¯ v (1)1 , ¯ z (1) q )... ... ρ (¯ v (1) q , ¯ z (1)1 ) · · · ρ (¯ v (1) q , ¯ z (1) q ) . By Lemma 3.5, ϕ ( g ) = γϕ ( f )for any evaluation ϕ : X → L , where γ = 1 q ! X σ,τ ∈ S q (sgn σ )(sgn τ ) ρ (¯ v (1) σ (1) , ¯ z (1) τ (1) ) · · · ρ (¯ v (1) σ ( q ) , ¯ z (1) τ ( q ) ) . We fix σ ∈ S q and we compute the sum γ σ = X τ ∈ S q (sgn τ ) ρ (¯ v (1) σ (1) , ¯ z (1) τ (1) ) · · · ρ (¯ v (1) σ ( q ) , ¯ z (1) τ ( q ) ) . Write simply ¯ v (1) σ ( i ) = a i , ¯ z (1) i = b i , i = 1 , . . . , q . Then γ σ = X τ ∈ S q (sgn τ ) ρ ( a , b τ (1) ) · · · ρ ( a q , b τ ( q ) ) = det ρ ( a , b ) · · · ρ ( a , b q )... ... ρ ( a q , b ) · · · ρ ( a q , b q ) = (sgn σ )det ρ ( a σ − (1) , b ) · · · ρ ( a σ − (1) , b q )... ... ρ ( a σ − ( q ) , b ) · · · ρ ( a σ − ( q ) , b q ) = (sgn σ )det ¯ ρ . Hence γ = 1 q ! X σ ∈ S q (sgn σ ) γ σ = det ¯ ρ and ϕ ( g ) = det ¯ ρ · ϕ ( f ). Thus, since ρ is a non degenerate form, g does notvanish in L . This completes the proof in case k = 1.If k >
1, by the inductive hypothesis there exists a multilinear polynomial g k − ( x , . . . , x q , v (1)1 , z (1)1 , . . . , v (1) q , z (1) q , . . . , v ( k − , z ( k − , . . . , v ( k − q , z ( k − q , Y )satisfying the conclusion of the theorem. We now write g k − = g k − ( x , . . . , . . . , x q , Y ′ ) , where Y ′ = Y ∪ { v (1)1 , z (1)1 , . . . , v (1) q , z (1) q , . . . , v ( k − , z ( k − , . . . , v ( k − q , z ( k − q } andwe apply Lemma 3.5 and the previous arguments to g k − . In this way we canconstruct the polynomial g k and, for any evaluation ϕ , we have ϕ ( g k ) = det ¯ ρ k · ϕ ( g k − ) = det ¯ ρ · · · det ¯ ρ k · ϕ ( f ) , where ¯ ρ s = ρ (¯ v ( s )1 , ¯ z ( s )1 ) · · · ρ (¯ v ( s )1 , ¯ z ( s ) q )... ... ρ (¯ v ( s ) q , ¯ z ( s )1 ) · · · ρ (¯ v ( s ) q , ¯ z ( s ) q ) , for all 1 ≤ s ≤ k . This completes the proof of the proposition. (cid:3) Pagon, Repovˇs, and Zaicev
4. PI-exponents of simple color Lie algebras
For computing PI-exponents of simple color Lie algebras we need to get a reason-able lower bound of codimension growth.
Proposition 4.1.
Let L be as in the previous section. Then for all n ≥ ,there exist constants C > and t such that Cn t q n ≤ c n ( L ) , (6) where q = dim L .Proof . The main tool for proving the inequality (6) is the representationtheory of symmetric groups. We refer reader to [11] for details of this theory.Recall that P m is a subspace of F { X } consisting of all multilinear poly-nomials in x , . . . , x m and Id( L ) is the ideal of all multilinear identities of L ofdegree m . One can define the S m -action on P m b setting σf ( x , . . . , x, m ) = f ( x σ (1 , . . . , x σ ( m ) . Then P m becomes an F [ S m ]-module and P m ( L ) = P m /P m ∩ Id( L ) is its submod-ule. By Mashke’s Theorem P m ( L ) is the direct sum of irreducible componentsand for proving the inequality (6) it is sufficient to find at least one irreduciblecomponent with the dimension greater or equal to Cn t q n . Slightly modifying thisapproarch we first consider P n + N ( L ) where n = (2 k + 1) q + N and k, N are as inProposition 3.6.Recall that there exists 1-1 correspondence between isomorphism classesof irreducible S n -representations and partitions of n (or Young diagrams with n boxes). A partition λ ⊢ n is an ordered set of integers λ = ( λ , . . . , λ t ) satisfying λ ≥ . . . ≥ λ t > λ + · · · + λ t = n . The corresponding Young diagram D λ is a tableau with n boxes. The first row of D λ contains λ boxes, the second rowcontains λ boxes, and so on. Young tableau T λ is the diagram D λ filled up byintegers 1 , . . . , n .Given a Young tableau T λ of shape λ ⊢ n , let R T λ and C T λ denote thesubgroups of S n stabilizing the rows and the columns of T λ , respectively. If weset ¯ R T λ = X σ ∈ R Tλ σ and ¯ C T λ = X τ ∈ C Tλ (sgn τ ) τ. then the element e T λ = ¯ R T λ ¯ C T λ is an essential idempotent of the group algebra F S n (i.e. e T λ = γe T λ for some 0 = γ ∈ F ) and F [ S n ] e T λ is an irreducible left F [ S n ]-module associated to λ .By Proposition 3.6, for any fixed k ≥ g k = g k ( x (1)1 , . . . , x (1) q , . . . , x (2 k +1)1 , . . . , x (2 k +1) q , y , . . . , y N )such that g k is alternating on each set of indeterminates { x ( i )1 , . . . , x ( i ) q } , 1 ≤ i ≤ k + 1, and g k is not a polynomial identity of L . Rename the variables and write g k = h ( x , . . . , x q (2 k +1) , Y ) , agon, Repovˇs, and Zaicev Y = { y , . . . , y N } .Since h Id( L ), there exists a partition λ = ( λ , . . . , λ m ) ⊢ n and aYoung tableau T λ such that F [ S n ] e T λ h Id( L ). Our next goal is to show that λ = ((2 k + 1) q ) is a rectangle of width 2 k + 1 and height q .If λ ≥ k + 2, then e T λ h is a polynomial symmetric in at least 2 k + 2variables among x , . . . , x n . But for any σ ∈ ¯ R T λ these variables in σ ¯ C T λ aredivided into 2 k +1 disjoint alternating subsets. It follows that σ ¯ C T λ h is alternatingand symmetric in at least two variables and so, e T λ h = 0 is the zero polynomial,a contradiction. Thus λ ≤ k + 1.Suppose now that m ≥ q + 1. Since the first column of T λ is of height atleast q + 1, the polynomial ¯ C T λ h is alternating in at least q + 1 variables among x , . . . , x n . Since dim L = q we get that for any σ , σ ¯ C T λ h ≡ L and so, also e T λ h = ¯ R T λ ¯ C T λ h ≡ L , a contradiction.We have proved that F [ S n ] e T λ h Id( L ), for some Young tableau T λ ofshape λ = ((2 k + 1) q ). From the Hook formula for dimensions of irreduciblerepresentations of S n (see [11]) and Stirling formula for factorials it follows thatdim F [ S n ] e T λ h ≥ q !(2 πn ) q q n . It easily follows from the simplicity of tensor factor B in L = F [ G ] ⊗ B that c n ′ ( L ) ≥ c n ( L ) as soon as n ′ > n. (7)Hence c m ( L ) ≥ C ′ ( m − N ) q q m − N ≥ C ′′ m q q m for some constants C ′ , C ′′ for any m = q (2 k + 1) + N , k = 1 , , . . . . Finally,applying again the inequality (7) we get (6) for all n . (cid:3) Combining the inequality (3) and Proposition 4.1 we immediately obtainthe main result of the paper.
Theorem 4.2.
Let F be an algebraically closed field of characteristic zero andlet L = F [ G ] ⊗ B be a finite dimensional color Lie superalgebra over F where G = h a i × h b i ≃ Z ⊕ Z with the skew-symmetric bicharacter β defined by β ( a ) = β ( b ) = 1 , β ( a, b ) = − and B is a finite dimensional simple Lie algebrawith the trivial G -grading. Then the PI-exponent of L exists and exp ( L ) = dim L .
5. Graded identities of simple color Lie algebras
In conclusion we discuss codimensions behavior of algebras defined distinct bichar-acters and asymptotics of graded codimensions. We begin by an easy remark.
Remark 5.1. If L = F [ G ] ⊗ B and G ≃ Z ⊕ Z with the trivial bicharacter β , that is β ≡
1, then PI-exp( L ) = d = dim L , where d = dim B .2 Pagon, Repovˇs, and Zaicev
Proof . Since F [ G ] is a commutative ring in this case, L is an ordinary Liealgebra with the same identities as B . In particular, PI-exp( L ) = PI-exp( B ) =dim B (see [5] or [17]). (cid:3) Remark 5.1 shows that ordinary codimensions behavior strongly dependson bicharacter β . On the other hand one can consider graded identities of L since L is a G -graded algebra.Recall that if we define G -grading on an infinite generating set Y , i.e. split Y to a disjoint union Y = ∪ g ∈ G Y g , deg y g = g ∀ y g ∈ Y g , then F { Y } can beendowed by the induced grading if we set deg( y i · · · y i m ) = deg y i · · · deg y i m forany arrangement of brackets. The polynomial f ( y g , . . . , y g n n ) is called a gradedidentity of L if f ( u , . . . , u n ) = 0, as soon as deg u = deg y g = g , . . . , deg u n =deg y g n n = g n .Since F { Y } is a graded algebra, the subspace of multilinear polynomials P n should be replaced by a graded subspace M k + ··· + k = n P k ,k ,k ,k where P k ,k ,k ,k is a subspace of multilinear polynomials f on y g , . . . , y g k , . . . , y g k , . . . , y g k k . Graded codimensions are defined as c grn ( L ) = X k ≥ ,k ≥ ,k ≥ ,k ≥ k k k k n (cid:18) nk , k , k , k (cid:19) dim P k ,k ,k ,k P k ,k ,k ,k ∩ Id( L )(see [8] for details). For our class of algebras graded codimensions behavior doesnot depend on bicharacter β defining color on L = F [ G ] ⊗ B . In the proof of thenext result we shall use the following easy observation. Remark 5.2. If Lie ( X ) is a free Lie algebra on the countable set of generatorsand B is an arbibrary Lie algebras then P n P n ∩ Id( B ) = V n V n ∩ Id Lie ( B )where V n is a subspace of Lie ( X ) of all multilinear polynomials in variables x , . . . , x n and Id Lie ( B ) is the ideal of Lie identities of B in Lie ( X ). (cid:3) We need this remark since all previous results concerning codimensiongrowth of Lie algebras were proved for Lie codimensions.
Theorem 5.3.
Let F be an algebraically closed field of characteristic zero andlet L = F [ G ] ⊗ B be a finite dimensional color Lie superalgebra over F where agon, Repovˇs, and Zaicev G = h a i × h b i ≃ Z ⊕ Z with any skey-symmetric bicharacter β and B isa finite dimensional simple Lie algebra with the trivial G -grading. Then gradedPI-exponent of L PI-exp gr ( L ) = lim n →∞ n p c grn ( L ) exists and is equal to | G | dim L = 4 dim L .Proof . First we prove thatdim P k ,k ,k ,k P k ,k ,k ,k ∩ Id( L ) = c n ( B ) . (8)Note that any multilinear Lie polynomial in x , . . . , x n can be written as alinear combination of left-normed monomials m σ = [ x , x σ (2) , . . . , x σ ( n ) ] , where σ is a permutation of 2 , . . . , n . If w , . . . , w n are homogeneous elementsof the color Lie superalgebra L = F [ G ] ⊗ B then any multilinear polynomialexpression on w , . . . , w n is also a linear combination of left-normed products[ w , w σ (2) , . . . , w σ ( n ) ]. Since we are interested in graded identities of L it is sufficientto consider only left-normed monomials and their linear combinations.Let ( g , . . . , g n ) be a n-tuple of elements of G such that( g , . . . , g n ) = ( e, . . . , e | {z } k , a, . . . , a | {z } k , b, . . . , b | {z } k , ab, . . . , ab | {z } k ) . Then m σ ( g ⊗ x , . . . , g n ⊗ x n ) = g · · · g n ⊗ λ σ m σ ( x , . . . , x n )in F [ G ] ⊗ F { X } where λ σ = ± σ for given g , . . . , g n .Given a multilinear polynomial f = f ( x , . . . , x n ) = X σ ∈ S n − α σ m σ of F { X } we denote by e f the element e f = e f ( x , . . . , x n ) = X σ ∈ S n − λ σ α σ m σ . Then for any w , . . . , w k , . . . , w , . . . , w k ∈ B we have f ( e ⊗ w , . . . , e ⊗ w k , . . . , ab ⊗ w , . . . , ab ⊗ w k )= a k b k ( ab ) k ⊗ e f ( w , . . . , w k , . . . , w , . . . , w k ) . In particular, f is a graded identity of L , f ∈ P k ,k ,k ,k ∩ Id( L ) if and onlyif f is an identity of the Lie algebra B . Now, if c n ( B ) = N and m σ , . . . , m σ N Pagon, Repovˇs, and Zaicev is a basis of V n in Lie ( X ) modulo Id Lie ( B ) then also m σ , . . . , m σ N is a basis of P k ,...,k modulo Id( L ) in F { X } and we have proved the relation (8). Hence c grn ( L ) = X k ≥ ,k ≥ ,k ≥ ,k ≥ k k k k n (cid:18) nk , k , k , k (cid:19) dim P k ,k ,k ,k P k ,k ,k ,k ∩ Id( L )= c n ( B ) X k ≥ ,k ≥ ,k ≥ ,k ≥ k k k k n (cid:18) nk , k , k , k (cid:19) = 4 n c n ( B )and we have completed the proof since lim n →∞ n p c n ( B ) = dim B by [5]. (cid:3) The first and the second author were supported by the Slovenian ResearchAgency grants P1-0292-0101-04 and J1-9643-0101. The third author was partiallysupported by RFBR grant No 09-01-00303 and SSC-1983.2008.1.
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