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Let D be a Noetherian infinite integral domain, denote by M 2 (D) and by s l 2 (D) the 2 × 2 matrix algebra and the Lie algebra of the traceless matrices in M 2 (D) , respectively. In this paper we study the weak polynomial identities for the natural grading by the cyclic group Z 2 of order 2 on M 2 (D) and on s l 2 (D). We describe a finite basis of the graded polynomial identities for the pair (M 2 (D) , s l 2 (D)). Moreover we prove that the ideal of the graded identities for this pair satisfies the Specht property, that is every ideal of graded identities of pairs (associative algebra, Lie algebra), satisfying the graded identities for (M 2 (D) , s l 2 (D)) , is finitely generated. The polynomial identities for M 2 (D) are known if D is any field of characteristic different from 2. The identities for the Lie algebra s l 2 (D) are known when D is an infinite field. The identities for the pair we consider were first described by Razmyslov when D is a field of characteristic 0, and afterwards by the second author when D is an infinite field. The graded identities for the pair (M 2 (D) , g l 2 (D)) were also described, by Krasilnikov and the second author. In order to obtain these results we use certain graded analogues of the generic matrices, and also techniques developed by G. Higman concerning partially well ordered sets. [ABSTRACT FROM AUTHOR]

Let A = B + C be an associative algebra graded by a group G , which is a sum of two homogeneous subalgebras B and C. We prove that if B is an ideal of A , and both B and C satisfy graded polynomials identities, then the same happens for the algebra A. We also introduce the notion of graded semi-identity for the algebra A graded by a finite group and we give sufficient conditions on such semi-identities in order to obtain the existence of graded identities on A. We also provide an example where both subalgebras B and C satisfy graded identities while A = B + C does not. Thus the theorem proved by Kȩpczyk in 2016 does not transfer to the case of group graded associative algebras. A variation of our example shows that a similar statement holds in the case of group graded Lie algebras. We note that there is no known analogue of Kȩpczyk's theorem for Lie algebras. [ABSTRACT FROM AUTHOR]

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]

Let K be any field of characteristic two and let $U_1$ and $W_1$ be the Lie algebras of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$ and of the polynomial ring K [ t ], respectively. The algebras $U_1$ and $W_1$ are equipped with natural $\mathbb{Z}$ -gradings. In this paper, we provide bases for the graded identities of $U_1$ and $W_1$ , and we prove that they do not admit any finite basis. [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]

In this paper we study matrix algebras with a degenerate trace in the framework of the theory of polynomial identities. The first part is devoted to the study of the algebra D n of n × n diagonal matrices. We prove that, in case of a degenerate trace, all its trace identities follow by the commutativity law and by pure trace identities. Moreover we relate the trace identities of D n + 1 endowed with a degenerate trace, to those of D n with the corresponding trace. This allows us to determine the generators of the trace T-ideal of D 3. In the second part we study commutative subalgebras of M k (F) , denoted by C k of the type F + J that can be endowed with the so-called strange traces: tr (a + j) = α a + β j , for any a + j ∈ C k , α , β ∈ F. Here J is the radical of C k. In case β = 0 such a trace is degenerate, and we study the trace identities satisfied by the algebra C k , for every k ≥ 2. Moreover we prove that these algebras generate the so-called minimal varieties of polynomial growth. In the last part of the paper, devoted to the study of varieties of polynomial growth, we completely classify the subvarieties of the varieties of algebras of almost polynomial growth introduced in ([7]). [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic zero. We study the asymptotic behavior of the codimensions for polynomial identities of representations of Lie algebras, also called weak identities. These identities are related to pairs of the form (A,L)$(A,L)$ where A is an associative enveloping algebra for the Lie algebra L. We obtain a characterization of ideals of weak identities with polynomial growth of the codimensions in terms of their cocharacter sequence. Recall that such a characterization was obtained by Kemer in [12] for associative algebras and by Benediktovich and Zalesskii in [2] for Lie algebras. We prove that the pairs (UT2,UT2(−))$\Big (UT_2,UT_2^{(-)}\Big)$, (E,E(−))$\big (E,E^{(-)}\big)$ and (M2,sl2)$\big (M_2,sl_2\big)$ generate varieties of pairs of almost polynomial growth. Here E denotes the infinite dimensional Grassmann algebra with 1. Also UT2$UT_2$ is the associative subalgebra of M2 (the 2 × 2 matrices over the field K) consisting of upper triangular matrices and sl2$sl_2$ is the Lie subalgebra of M2(−)$M_2^{(-)}$ of the traceless matrices. [ABSTRACT FROM AUTHOR]

Let F be an infinite field of characteristic different from 2, and let E be the Grassmann algebra of an infinite dimensional F -vector space L. In this paper we study the Z -graded polynomial identities of E with respect to certain Z -grading such that the vector space L is homogeneous in the grading. More precisely, we construct three types of Z -gradings on E , denoted by E ∞ , E k ⁎ and E k , and we give the explicit form of the corresponding Z -graded polynomial identities. We show that the homogeneous superalgebras E ∞ , E k ⁎ and E k studied in [12] can be obtained from E ∞ , E k ⁎ and E k as quotient gradings. Moreover we exhibit several other types of homogeneous Z -gradings on E , and describe their graded identities. [ABSTRACT FROM AUTHOR]

Let G be an arbitrary group and let K be a field of characteristic different from 2. We classify the G -gradings on the Jordan algebra UJ n of upper triangular matrices of order n over K . It turns out that there are, up to a graded isomorphism, two families of gradings: the elementary gradings (analogous to the ones in the associative case), and the so called mirror type (MT) gradings. Moreover we prove that the G -gradings on UJ n are uniquely determined, up to a graded isomorphism, by the graded identities they satisfy. [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]

Upper triangular, and more generally, block-triangular matrices, are rather important in Linear Algebra, and also in Ring theory, namely in the theory of PI algebras (algebras that satisfy polynomial identities). The group gradings on such algebras have been extensively studied during the last decades. In this paper we prove that for any group grading on a block-triangular matrix algebra, over an arbitrary field, the Jacobson radical is a graded (homogeneous) ideal. As noted by F. Yasumura [Arch. Math. (Basel) 110 (2018), pp. 327–332] this yields the classification of the group gradings on these algebras and confirms a conjecture made by A. Valenti and M. Zaicev [Arch. Math. (Basel) 89 (2007), pp. 33–40]. [ABSTRACT FROM AUTHOR]

The algebras U T n of the n × n upper triangular matrices over a field K are of significant importance in the theory of algebras with polynomial identities. Group gradings on algebras appear in various areas and provide an indispensable tool in the study of the algebraic and combinatorial properties of the algebras in question. We classify the group gradings on the Lie algebra U T n ( − ) . It was proved by Valenti and Zaicev in 2007 that every group grading on the associative algebra U T n is isomorphic to an elementary grading. The elementary gradings on U T n are also well understood, see [6] . It follows from our results that there are nonelementary gradings on U T n ( − ) . Thus the gradings on the Lie algebra U T n ( − ) are much more intricate than those in the associative case. [ABSTRACT FROM AUTHOR]

The algebras U T n ( K ) of the upper triangular matrices over a field K are of significant importance in the theory of algebras with polynomial identities. Group gradings on algebras appear in various areas and provide an indispensable tool in the study of the algebraic and combinatorial properties of the algebras in question. In this paper we consider the Lie algebra U T n ( K ) ( − ) of all upper triangular matrices of order n . We study the group gradings on this algebra. It turns out that the gradings on the Lie algebra U T n ( K ) are much more intricate than those in the associative case. In this paper we describe the elementary gradings on the Lie algebra U T n ( K ) ( − ) . Finally we study the canonical grading on U T n ( K ) ( − ) by the cyclic group Z n of order n . We produce a (finite) basis of the graded polynomial identities satisfied by this grading. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic zero and let E be the Grassmann algebra of an infinite dimensional F -vector space L. In this paper we study the superalgebra structures (that is the Z 2 -gradings) that the algebra E admits. By using the duality between superalgebras and automorphisms of order at most 2 we prove that in many cases the Z 2 -graded polynomial identities for such structures coincide with the Z 2 -graded polynomial identities of the "typical" cases E ∞ , E k ⁎ and E k where the vector space L is homogeneous. Recall that these cases were completely described by Di Vincenzo and da Silva in [13]. Moreover we exhibit a wide range of non-homogeneous Z 2 -gradings on E that are Z 2 -isomorphic to E ∞ , E k ⁎ and E k. In particular we construct a Z 2 -grading on E with only one homogeneous generator in L which is Z 2 -isomorphic to the natural Z 2 -grading on E , here denoted by E c a n. [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]

In this paper we study the G -identities for the Lie algebra sl 2 ( C ) over the complex field C . If sl 2 ( C ) is acted on faithfully by a finite group G then G is isomorphic to one of the following groups: C n , D n , A 4 , S 4 , A 5 . Here C n is the cyclic group of order n , D n is the dihedral group (of order 2 n ), A n and S n stand for the alternating and for the symmetric group permuting n letters, respectively. In each one of the above cases we describe a basis of the G -identities for sl 2 ( C ) . In order to do that we use the explicit form of the graded identities for sl 2 ( C ) as well as properties of the group algebras of the corresponding groups. The same problem for the associative algebra M 2 ( C ) of the 2 × 2 matrices was settled by A. Berele in 2004. [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]

We discuss sortie new results concerning the representations of one of the simple Jordan pairs as well as some examples of special and of exceptional varieties of Jordan pairs. These results are the Jordan pair counterparts of the classical theorems for Jordan algebras. [ABSTRACT FROM AUTHOR]

Abstract: Verbally prime algebras are important in PI theory. They were described by Kemer over a field K of characteristic zero: 0 and (the trivial ones), . Here is the free associative algebra of infinite rank, with free generators denotes the infinite dimensional Grassmann algebra over and are the matrices over K and over E, respectively. The algebras are subalgebras of , see their definition below. The generic (also called relatively free) algebras of these algebras have been studied extensively. Procesi described the generic algebra of and lots of its properties. Models for the generic algebras of and are also known but their structure remains quite unclear. In this paper we study the generic algebra of in two generators, over a field of characteristic 0. In an earlier paper we proved that its centre is a direct sum of the field and a nilpotent ideal (of the generic algebra), and we gave a detailed description of this centre. Those results were obtained assuming the base field infinite and of characteristic different from 2. In this paper we study the polynomial identities satisfied by this generic algebra. We exhibit a basis of its polynomial identities. It turns out that this algebra is PI equivalent to a 5-dimensional algebra of certain upper triangular matrices. The identities of the latter algebra have been studied; these were described by Gordienko. As an application of our results we describe the subvarieties of the variety of unitary algebras generated by the generic algebra in two generators of . Also we describe the polynomial identities in two variables of the algebra . [Copyright &y& Elsevier]

Abstract: Verbally prime algebras are important in PI theory. They are well known over a field K of characteristic zero: 0 and (the trivial ones), , , . Here is the free associative algebra with free generators T, E is the infinite dimensional Grassmann algebra over K, and are the matrices over K and over E, respectively. Moreover are certain subalgebras of , defined below. The generic algebras of these algebras have been studied extensively. Procesi gave a very tight description of the generic algebra of . The situation is rather unclear for the remaining nontrivial verbally prime algebras. In this paper we study the centre of the generic algebra of in two generators. We prove that this centre is a direct sum of the field and a nilpotent ideal (of the generic algebra). We describe the centre of this algebra. As a corollary we obtain that this centre contains nonscalar elements thus we answer a question posed by Berele. [Copyright &y& Elsevier]

Let E be the Grassmann algebra of an infinite-dimensional vector space L over a field of characteristic zero. In this paper, we study the Z -gradings on E having the form E = E (r 1 , r 2 , r 3) (v 1 , v 2 , v 3) , in which each element of a basis of L has Z -degree r 1 , r 2 , or r 3 . We provide a criterion for the support of this structure to coincide with a subgroup of the group Z , and we describe the graded identities for the corresponding gradings. We strongly use Elementary Number Theory as a tool, providing an interesting connection between this classical part of Mathematics, and PI Theory. Our results are generalizations of the approach presented in Brandão A, Fidelis C, Guimarães A. Z -gradings of full support on the Grassmann algebra. J Algebra. 2022;601:332–353. DOI:10.1016/j.jalgebra.2022.03.014. See also in arXiv preprint, arXiv:2009.01870v1, 2020]. [ABSTRACT FROM AUTHOR]

Abstract: The associative algebras of the upper triangular matrices of order play an important role in PI theory. Recently it was suggested that the Jordan algebra obtained by has an extremal behaviour with respect to its codimension growth. In this paper we study the polynomial identities of . We describe a basis of the identities of when the field is infinite and of characteristic different from 2 and from 3. Moreover we give a description of all possible gradings on by the cyclic group of order 2, and in each of the three gradings we find bases of the corresponding graded identities. Note that in the graded case we need only an infinite field , . [Copyright &y& Elsevier]

Abstract: In this paper we consider the algebra endowed with the involution induced by the transposition superinvolution of the superalgebra of -matrices over the field . We study the -polynomial identities for this algebra in the case of characteristic zero. We describe a finite set generating the ideal of its -identities. We also consider , the algebra of matrices over the Grassmann algebra . We prove that for a large class of involutions defined on it any -polynomial identity is indeed a polynomial identity. A similar result holds for the verbally prime algebra . [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: We describe the -graded central polynomials for the matrix algebra of order two, , and for the algebras and over an infinite field , . Here is the infinite-dimensional Grassmann algebra, and stands for the algebra of the matrices whose entries on the diagonal belong to , the centre of , and the off-diagonal entries lie in , the anticommutative part of . It turns out that in characteristic 0 the graded central polynomials for and are the same (it is well known that these two algebras satisfy the same polynomial identities when ). On the contrary, this is not the case in characteristic . We describe systems of generators for the -graded central polynomials for all these algebras. Finally we give a generating set of the central polynomials with involution for . We consider the transpose and the symplectic involutions. [Copyright &y& Elsevier]

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]

Abstract: In this paper we describe completely the involutions of the first kind of the algebra of upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for when n is even and a single class in the odd case. Furthermore we consider the algebra of the upper triangular matrices over an infinite field F of characteristic different from 2. For every involution ∗, we describe the ∗-polynomial identities for this algebra. We exhibit bases of the corresponding ideals of identities with involution, and compute the Hilbert (or Poincaré) series and the codimension sequences of the respective relatively free algebras. Then we consider the ∗-polynomial identities for the algebra over a field of characteristic zero. We describe a finite generating set of the ideal of ∗-identities for this algebra. These generators are quite a few, and their degrees are relatively large. It seems to us that the problem of describing the ∗-identities for the algebra of the upper triangular matrices may be much more complicated than in the case of ordinary polynomial identities. [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 exhibit minimal bases of the polynomial identities for the matrix algebra M2(K) of order two over an infinite field K of characteristic p≠2. We show that when p=3 the T-ideal of this algebra is generated by three independent identities, and when p>3 one needs only two identities: the standard identity of degree four and the Hall identity. Note that the same holds when the base field is of characteristic 0. Furthermore, using the exact form of the basis of the identities for M2(K) we give finite minimal set of generators of the T-space of the central polynomials for the algebra M2(K). The set of generators depends on the characteristic of the field as well. [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]

Let K be a finite field of characteristic p>2, and let M2(K) be the matrix algebra of order two over K. We describe up to a graded isomorphism the 2-gradings of M2(K). It turns out that there are only two nonisomorphic nontrivial such gradings. Furthermore, we exhibit finite bases of the graded polynomial identities for each one of these two gradings. One can distinguish these two gradings by means of the graded polynomial identities they satisfy. [Copyright &y& Elsevier]

In this paper we study algebras with trace and their trace polynomial identities over a field of characteristic 0. We consider two commutative matrix algebras: D 2 , the algebra of 2 × 2 diagonal matrices and C 2 , the algebra of 2 × 2 matrices generated by e 11 + e 22 and e 12. We describe all possible traces on these algebras and we study the corresponding trace codimensions. Moreover we characterize the varieties with trace of polynomial growth generated by a finite dimensional algebra. As a consequence, we see that the growth of a variety with trace is either polynomial or exponential. [ABSTRACT FROM AUTHOR]

Let F be an infinite field of characteristic different from 2, and let E be the Grassmann algebra of a countable of dimensional F -vector space L. In this paper, we study the graded central polynomials of gradings on E by the groups ℤ 2 and ℤ , where the basis of the vector space L is homogeneous. More specifically, we provide a basis for the T G -space of graded central polynomials for E , where the group G is ℤ 2 and ℤ. [ABSTRACT FROM AUTHOR]

Abstract Let K be a field of characteristic zero and let J be a Jordan algebra with a formal trace. We prove that the algebra J can be embedded into a Jordan algebra of a non-degenerate symmetric bilinear form over some associative and commutative K -algebra C if and only if J satisfies all trace identities of the Jordan algebra of a non-degenerate symmetric bilinear form over the field K. This is an extension of results of Procesi and Berele concerning the analogous problem for the associative matrix algebras and the matrix algebras with involution. As a consequence of these results we also prove that the ideal of all trace identities of the Jordan algebra of a non-degenerate symmetric bilinear form over K satisfies the Specht property. [ABSTRACT FROM AUTHOR]

Let be a field of characteristic 0 and let . The algebra admits a natural grading by the cyclic group of order 2. In this paper, we describe the -graded A-identities for . Recall that an A-identity for an algebra is a multilinear polynomial identity for that algebra which is a linear combination of the monomials where runs over all even permutations of that is , the th alternating group. We first introduce the notion of an A-identity in the case of graded polynomials, then we describe the graded A-identities for , and finally we compute the corresponding graded A-codimensions. [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]