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Let U T n (F) be the algebra of the n × n upper triangular matrices and denote U T n (F) (-) the Lie algebra on the vector space of U T n (F) with respect to the usual bracket (commutator), over an infinite field F. In this paper, we give a positive answer to the Specht property for the ideal of the Z n -graded identities of U T n (F) (-) with the canonical grading when the characteristic p of F is 0 or is larger than n - 1 . Namely we prove that every ideal of graded identities in the free graded Lie algebra that contains the graded identities of U T n (F) (-) , is finitely based. Moreover we show that if F is an infinite field of characteristic p = 2 then the Z 3 -graded identities of U T 3 (-) (F) do not satisfy the Specht property. More precisely, we construct explicitly an ideal of graded identities containing that of U T 3 (-) (F) , and which is not finitely generated as an ideal of graded identities. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic zero and let R be an algebra that admits a regular grading by an abelian group H. Moreover, we consider G a group and let A be an algebra with a grading by the group G × H, we define the R-hull of A as the G × H-graded algebra given by R (A) = ⊕ (g , h) ∈ G × H A (g , h) ⊗ R h . In this paper we provide a basis for the graded identities (resp. central polynomials) of the R-hull of A, assuming that a (suitable) basis for the graded identities (resp. central polynomials) of the G × H-graded algebra A is known. In particular, for any a, b ∈ ℕ , we find a basis for the graded identities and the graded central polynomials for the algebra Ma,b(E), graded by the group G × ℤ 2 . Here E is the Grassmann algebra of an infinite dimensional F-vector space, equipped with its natural ℤ 2 -grading and the matrix algebra Ma+b(F) is equipped with an elementary grading by the group G × ℤ 2 , so that its neutral component coincides with the subspace of the diagonal matrices. We describe the isomorphism classes of gradings on Ma,b(E) that arise in this way and count the isomorphism classes of such gradings. Moreover, we give an alternative proof of the fact that the tensor product Ma,b(E) ⊗ Mr,s(E) is PI-equivalent to Mar+bs,as+br(E). Finally, when the grading group is ℤ 3 × ℤ 2 (resp. ℤ × ℤ 2 ), we present a complete description of a basis for the graded central polynomials for the algebra M2,1(E) (resp. Ma,b(E) in the case b = 1). [ABSTRACT FROM AUTHOR]

The graded exponent is an important invariant of group graded PI-algebras. In this paper we study a specific elementary grading on the algebra of upper triangular matrices UTn, compute its codimensions, and use this grading to find the asymptotic behaviour of the codimensions of any elementary grading on UTn, for any group. Moreover, we extend this to the Lie case, and obtain, for any elementary grading on the Lie algebra UTn(-), an upper bound and a lower bound for the asymptotic behaviour of its codimensions. Also, we obtain the graded exponent of any grading on UTn(-) and for any grading on the Jordan algebra UJn. It turns out that the graded exponent for UTn, considered as an associative, Jordan or Lie algebra, for any grading, coincides with the exponent of the ordinary case. In the associative case, the asymptotic behaviour of the codimensions of any grading on UTn coincides with the asymptotic behaviour of the ordinary codimensions. But this is not the case for the graded asymptotics of the codimensions of the Lie algebra UTn(-). [ABSTRACT FROM AUTHOR]

Let $$sl_2(K)$$ be the Lie algebra of the $$2\times 2$$ traceless matrices over an infinite field $$K$$ of characteristic different from 2, denote by $$R_m= R_m(sl_2(K))$$ the relatively free (also called universal) algebra of rank $$m$$ in the variety of Lie algebras generated by $$sl_2(K).$$ In this paper we compute the Gelfand-Kirillov dimension of the Lie algebra $$R_m(sl_2(K)).$$ It turns out that whenever $$m\ge 2$$ one has $$\mathrm{GK}\dim R_m = 3(m-1).$$ In order to compute it we use the explicit form of the Hilbert series of $$R_m$$ described by Drensky. This result is new for $$m>2$$ ; the case $$m=2$$ was dealt with by Bahturin in 1979. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic 0, and let UT( K) be the algebra of n × n upper triangular matrices over K. We denote by P the vector space of all multilinear polynomials of degree n in x, ..., x in the free associative algebra K( X). Then P is a left S-module where the symmetric group S acts on P by permuting the variables. The S-modules P and KS are canonically isomorphic, a fact that lets us employ the representation theory in the study of algebras with polynomial identities. Denote by A the alternative subgroup of S. One may study KA and its isomorphic copy in P with the corresponding action of A. Henke and Regev studied A-identities. They described the A-codimensions of the Grassmann algebra and conjectured a finite generating set of the A-identities for E. In an earlier paper we answered in the affirmative their conjecture. Another problem posed by Henke and Regev concerned the minimal degree of an A-identity satisfied by the full matrix algebra M( K). They asked whether this minimal degree equals 2 n+2. In this paper we show that this is not the case as long as n ≥ 6. Our main result consists in computing a lower bound for the minimal degree d( n) of an A-identity satisfied by the algebra UT( K). It turns out that, given any positive integer k, there exists n such that d( n) > 2 n + k for all n > n. Moreover, we compute d( n) for n ≤ 4 and for n = 6. It turns out that d(6) = 15 > 2 × 6 + 2. We have reasons to believe that for every n, d( n) = [(5 n + 1)/2] holds, where as usual [ x] stands for the integer part of the real number x, that is the largest integer ≤ x. [ABSTRACT FROM AUTHOR]

In this paper we describe the central polynomials for the infinite-dimensional unitary Grassmann algebra G over an infinite field F of characteristic ≠ 2. We exhibit a set of polynomials that generates the vector space C( G) of the central polynomials of G as a T-space. Using a deep result of Shchigolev we prove that if charF = p > 2 then the T-space C( G) is not finitely generated. Moreover, over such a field F, C( G) is a limit T-space, that is, C( G) is not a finitely generated T-space but every larger T-space W ≩ C( G) is. We obtain similar results for the infinite-dimensional nonunitary Grassmann algebra H as well. [ABSTRACT FROM AUTHOR]

LetM2(K) be the matrix algebra of order two over an infinite fieldK of characteristicp≠2. IfK is algebraically closed then, up to isomorphism, there are two involutions of first kind onM2(K), namely the transpose and the symplectic. IfK is not algebraically closed, studying *-identities it is still sufficient to consider only these two involutions. We describe bases of the polynomial identities with involution in each of these cases. [ABSTRACT FROM AUTHOR]

In this paper we study 2-graded polynomial identities. We describe bases of these identities satisfied by the matrix algebra of order twoM2(K), by the algebraM1,1(G), and by the algebraG ⊗KG. HereK is an arbitrary infinite field of characteristic not 2,G stands for the Grassmann (or exterior) algebra of an infinite dimensional vector space overK, andM1,1(G) is the algebra of all 2×2 matrices overG whose entries on the main diagonal are even elements ofG, and those on the second diagonal are odd elements ofG. The gradings on these three algebras are supposed to be the standard ones. It turns out that the graded identities of these three algebras are closely related, and furthermoreM1,1(G) andG ⊗G satisfy the same 2-graded identities provided that charK=0. When charK=p>2, then the algebraG ⊗G satisfies some additional 2-graded identities that are not identities forM1,1(G). The methods used in the proofs are based on appropriate constructions for the corresponding relatively free algebras, on combinatorial properties of permutations, and on a version of Specht’s commutator reduction. We hope that this paper is a step towards the description of the ordinary identities satisfied by the algebrasG ⊗G andM1,1(G) over an infinite field of positive characteristic. Note that in characteristic 0 such a description was given in [12] and in [10]. [ABSTRACT FROM AUTHOR]

In this note we exhibit bases of the polynomial identities satisfied by the Grassmann algebras over a field of positive characteristic. This allows us to answer the following question of Kemer: Does the infinite dimensional Grassmann algebra with 1, over an infinite field K of characteristic 3, satisfy all identities of the algebra M 2( K) of all 2×2 matrices over K? We give a negative answer to this question. Further, we show that certain finite dimensional Grassmann algebras do give a positive answer to Kemer's question. All this allows us to obtain some information about the identities satisfied by the algebra M 2( K) over an infinite field K of positive odd characteristic, and to conjecture bases of theidentities of M 2( K). [ABSTRACT FROM AUTHOR]