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Let K be an infinite field of characteristic different from two and let U 1 be the Lie algebra of the derivations of the algebra of Laurent polynomials K [ t , t − 1 ]. The algebra U 1 admits a natural Z -grading. We provide a basis for the graded identities of U 1 and prove that they do not admit any finite basis. Moreover, we provide a basis for the identities of certain graded Lie algebras with a grading such that every homogeneous component has dimension ≤1, if a basis of the multilinear graded identities is known. As a consequence of this latter result we are able to provide a basis of the graded identities of the Lie algebra W 1 of the derivations of the polynomial ring K [ t ]. The Z -graded identities for W 1 , in characteristic 0, were described in [8]. As a consequence of our results, we give an alternative proof of the main result, Theorem 1, in [8] , and generalize it to positive characteristic. We also describe a basis of the graded identities for the special linear Lie algebra s l q (K) with the Pauli gradings where q is a prime number. [ABSTRACT FROM AUTHOR]

Kac–Moody algebras, g (A) , are Lie algebras defined by generators and relations given by generalized Cartan matrices A. In this paper, we study the graded identities for Kac–Moody algebras when the matrix A is diagonal. More precisely, we provide a basis for the graded identities of g (A) equipped with its natural grading, the grading of Cartan type. These results are obtained over an arbitrary infinite field. We also compute the graded codimensions for these algebras and provide a basis for the vector space of the multihomogeneous polynomials of any given multidegree in the relatively free algebra. As the base field is infinite we have a vector space basis of the relatively free algebra. As a consequence of our results, we give an alternative proof of Theorem 17 in [15] , and generalize it to characteristic two. Finally, we also describe a basis of the graded identities for the Heisenberg algebra with its natural grading, over any field. [ABSTRACT FROM AUTHOR]

The Virasoro algebra, defined by the basis elements { L n , c ˆ } n ∈ Z with commutation relations [ L m , L n ] = (m − n) L m + n + δ m + n , 0 ⋅ (C m c ˆ) and [ L m , c ˆ ] = 0 , is an infinite-dimensional Lie algebra with many applications in various areas of Mathematics and Theoretical Physics. Here the symbol δ i , j denotes the Kronecker delta and C m = (m (m 2 − 1)) / 12. This algebra admits a natural Z -grading. Over an infinite field of characteristic different from 2 and 3, we describe the graded identities of the Virasoro algebra for this grading. It turns out that all these Z -graded identities are consequences of a collection of polynomials of degree 2, 3 and 4 and that they do not admit a finite basis. [ABSTRACT FROM AUTHOR]

Let A be an algebra over a field of characteristic 0 and assume A is graded by a finite group G . We study combinatorial and asymptotic properties of the G -graded polynomial identities of A provided A is of polynomial growth of the sequence of its graded codimensions. Roughly speaking this means that the ideal of graded identities is “very large”. We relate the polynomial growth of the codimensions to the module structure of the multilinear elements in the relatively free G -graded algebra in the variety generated by A . We describe the irreducible modules that can appear in the decomposition, we show that their multiplicities are eventually constant depending on the shape obtained by the corresponding multipartition after removing its first row. We relate, moreover, the polynomial growth to the colengths. Finally we describe in detail the algebras whose graded codimensions are of linear growth. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic 0 and let W 1 be the Lie algebra of the derivations of the polynomial ring K [ t ] . The algebra W 1 admits a natural Z -grading. We describe the graded identities of W 1 for this grading. It turns out that all these Z -graded identities are consequences of a collection of polynomials of degree 1, 2 and 3 and that they do not admit a finite basis. Recall that the “ordinary” (non-graded) identities of W 1 coincide with the identities of the Lie algebra of the vector fields on the line and it is a long-standing open problem to find a basis for these identities. We hope that our paper might be a step to solving this problem. [ABSTRACT FROM AUTHOR]

Abstract: The Jordan algebra of the symmetric matrices of order two over a field K has two natural gradings by , the cyclic group of order 2. We describe the graded polynomial identities for these two gradings when the base field is infinite and of characteristic different from 2. We exhibit bases for these identities in each of the two cases. In one of the cases we perform a series of computations in order to reduce the problem to dealing with associators while in the other case one employs methods and results from Invariant theory. Moreover we extend the latter grading to a -grading on , the Jordan algebra of a symmetric bilinear form in a vector space of dimension n (). We call this grading the scalar one since its even part consists only of the scalars. As a by-product we obtain finite bases of the -graded identities for . In fact the last result describes the weak Jordan polynomial identities for the pair . [ABSTRACT FROM AUTHOR]

Abstract: Let be an arbitrary abelian group and let and be two finite dimensional -graded simple algebras over an algebraically closed field such that the orders of all finite subgroups of are invertible in . We prove that and are isomorphic if and only if they satisfy the same -graded identities. We also describe all isomorphism classes of finite dimensional -graded simple algebras. [Copyright &y& Elsevier]

Abstract: In this paper we study the graded identities satisfied by the superalgebras over the Grassmann algebra and by their tensor products. These algebras play a crucial role in the theory developed by A. Kemer that led to the solution of the long standing Specht problem. It is well known that over a field of characteristic 0, the algebras and satisfy the same ordinary polynomial identities. By means of describing the corresponding graded identities we prove that the T-ideal of the former algebra is contained in the T-ideal of the latter. Furthermore the inclusion is proper at least in case . Finally we deal with the graded identities satisfied by algebras of type and relate these graded identities to the ones of tensor powers of the Grassmann algebra. Our proofs are combinatorial and rely on the relationship between graded and ordinary identities as well as on appropriate models for the corresponding relatively free graded algebras. [Copyright &y& Elsevier]

Abstract: Let K be a field, . We study the polynomial identities satisfied by -graded tensor products of T-prime algebras. Regev and Seeman proved that in a series of cases such tensor products are PI equivalent to T-prime algebras; they conjectured that this is always the case. We deal here with the remaining cases and thus confirm Regev and Seeman''s conjecture. For some “small” algebras we can remove the restriction on the characteristic of the base field, and we show that the behaviour of the corresponding graded tensor products is quite similar to that for the usual (ungraded) tensor products. Finally we consider β-graded tensor products (also called commutation factors) and their identities. We show that Regev''s theorem holds for β-graded tensor products whenever the gradings are by finite abelian groups. Furthermore we study the PI equivalence of β-graded tensor products of T-prime algebras. [Copyright &y& Elsevier]

The Lie algebra sl2(K) over a field K has a natural grading by ℤ2, the cyclic group of order 2. We describe the graded polynomial identities for this grading when the base field is infinite and of characteristic different from 2. We exhibit a basis of these identities that consists of one polynomial. In order to obtain this basis we employ methods and results from Invariant theory. As a by-product we obtain finite bases of the graded identities for sl2(K) graded by other groups such as ℤ2 × ℤ2, and by the integers ℤ. [ABSTRACT FROM AUTHOR]

Abstract: The verbally prime algebras are well understood in characteristic 0 while over a field of positive characteristic little is known about them. In previous papers we discussed some sharp differences between these two cases for the characteristic, and we showed that the so-called Tensor Product Theorem is in part no longer valid in the second case. In this paper we study the Gelfand–Kirillov dimension of the relatively free algebras of verbally prime and related algebras. We compute the GK dimensions of several algebras and thus obtain a new proof of the fact that the algebras and are not PI equivalent in characteristic . Furthermore we show that the following algebras are not PI equivalent in positive characteristic: and ; and when , , and ; and finally, and . Here E stands for the infinite-dimensional Grassmann algebra with 1, and is the subalgebra of of the block matrices with blocks and on the main diagonal with entries from , and off-diagonal entries from ; is the natural grading on E. [Copyright &y& Elsevier]

Let K be an infinite field and let UTn(K) denote the algebra of n×n upper triangular matrices over K. We describe all elementary gradings on this algebra. Further we describe the generators of the ideals of graded polynomial identities of UTn(K) and we produce linear bases of the corresponding relatively free graded algebras. We prove that one can distinguish the elementary gradings by their graded identities. We describe bases of the graded polynomial identities in several “typical” cases. Although in these cases we consider elementary gradings by cyclic groups, the same methods serve for elementary gradings by any finite group. [Copyright &y& Elsevier]

We consider the algebra U[sub n](K) of n×n upper triangular matrices over an infinite field K equipped with its usual ℤ[sub n]-grading. We describe a basis of the ideal of the graded polynomial identities for this algebra. [ABSTRACT FROM AUTHOR]

The algebras M a , b ( E ) ? E and M a + b ( E ) are PI equivalent over a field of characteristic 0 where E is the infinite-dimensional Grassmann algebra. This result is a part of the well-known tensor product theorem. It was first proved by Kemer in 1984–1987 (see Kemer 1991); other proofs of it were given by Regev (1990), and in several particular cases, by Di Vincenzo (1992), and by the authors (2004). Using graded polynomial identities, we obtain a new elementary proof of this fact and show that it fails for the T-ideals of the algebras M 1, 1 ( E ) ? E and M 2 ( E ) when the base field is infinite and of characteristic p > 2. The algebra M a , a ( E ) ? E satisfies certain graded identities that are not satisfied by M 2 a ( E ). In another paper we proved that the algebras M 1, 1 ( E ) and E ? E are not PI equivalent in positive characteristic, while they do satisfy the same multilinear identities. [ABSTRACT FROM AUTHOR]

In this paper we study tensor products of T-prime T-ideals over infinite fields. The behaviour of these tensor products over a field of characteristic 0 was described by Kemer. First we show, using methods due to Regev, that such a description holds if one restricts oneself to multilinear polynomials only. Second, applying graded polynomial identities, we prove that the tensor product theorem fails for the T-ideals of the algebras M1,1(E) and E⊗E where E is the infinite-dimensional Grassmann algebra; M1,1(E) consists of the 2×2 matrices over E having even (i.e., central) elements of E, and the other diagonal consisting of odd (anticommuting) elements of E. Note that these proofs do not depend on the structure theory of T-ideals but are “elementary” ones. All this comes to show once more that the structure theory of T-ideals is essentially about the multilinear polynomial identities. [Copyright &y& Elsevier]