Plamen Koshlukov

On the identities of the Grassmann algebras in characteristicp>0

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 fieldK of characteristic 3, satisfy all identities of the algebraM2(K) of all 2×2 matrices overK? We give a negative answer to this question. Further, we show that certain finite dimensional Grassmann algebras do give a positive answer to Kemers question. All this allows us to obtain some information about the identities satisfied by the algebraM2(K) over an infinite fieldK of positive odd characteristic, and to conjecture bases of theidentities ofM2(K).

Graded identities forT-prime algebras over fields of positive characteristic

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].

Graded polynomial identities for the Lie algebra sl(2)(K)

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 ...

GRADED IDENTITIES AND PI EQUIVALENCE OF ALGEBRAS IN POSITIVE CHARACTERISTIC

ABSTRACT 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 2a (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.

GRADED IDENTITIES FOR THE ALGEBRA OF n×n UPPER TRIANGULAR MATRICES OVER AN INFINITE FIELD

We consider the algebra Un(K) of n×n upper triangular matrices over an infinite field K equipped with its usual ℤn-grading. We describe a basis of the ideal of the graded polynomial identities for this algebra.

Finitely based ideals of weak polynomial identities

Let K be a field, char K≠2, and let Vk be a k-dimensional vector space over K equipped with a nondegenerate symmetric bilinear form. Denote Ck the Clifford algebra of Vk . We study the polynomial identities for the pair (Ck ,Vk ). A basis of the identities for this pair is found. It is proved that they are consequences of the single identity [x 2 ,y] = 0 when k = ∝ It is shown that when k < ∝ the identities for (Ck ,Vk ) follow from[x 2,y]=0 and Wk+1 = 0 where Wk+1 is an analog of the standard polynomial Stk+1 Denote M 2(K) the matrix algebra of order two over K, and let sl 2(K) be the Lie algebra of all traceless 2 × 2 matrices over K. As an application, new proof of the fact that the identity [x 2,y] = 0 is a basis of the weak Lie identities for the pair (M 2(K),sl 2(K) is given.

Identities with involution for the matrix algebra of order two in characteristicp

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.

A-IDENTITIES FOR THE GRASSMANN ALGEBRA : THE CONJECTURE OF HENKE AND REGEV

Let K be an algebraically closed field of characteristic 0, and let E be the infinite dimensional Grassmann (or exterior) algebra over K. Denote by P n the vector space of the multilinear polynomials of degree n in x 1 , ..., x n in the free associative algebra K(X). The symmetric group S n acts on the left-hand side on P n , thus turning it into an S n -module. This fact, although simple, plays an important role in the theory of PI algebras since one may study the identities satisfied by a given algebra by applying methods from the representation theory of the symmetric group. The S n -modules P n and KS n are canonically isomorphic. Letting An be the alternating group in S n , one may study KA n and its isomorphic copy in P n with the corresponding action of An. Henke and Regev described the A n -codimensions of the Grassmann algebra E, and conjectured a finite generating set of the An-identities for E. Here we answer their conjecture in the affirmative.

Ideals of identities of representations of nilpotent lie algebras

Let L be a Lie algebra, nilpotent of class 2, over an infinite field K, and suppose that the centre C of L is one dimensional; such Lie algebras are called Heisenberg algebras. Let ρ:L→hom KV be a finite dimensional representation of the Heisenberg algebra L such that ρ(C) contains non-singular linear transformations of V, and denote l(ρ) the ideal of identities for the representation ρ. We prove that the ideals of identities of representations containing I(ρ) and generated by multilinear polynomials satisfy the ACC. Let sl 2(L) be the Lie algebra of the traceless 2×2 matrices over K, and suppose the characteristic of K equals 2. As a corollary we obtain that the ideals of identities of representations of Lie algebras containing that of the regular representation of sl 2(K) and generated by multilinear polynomials, are finitely based. In addition we show that one cannot simply dispense with the condition of multilinearity. Namely, we show that the ACC is violated for the ideals of representations of Lie algebras (over an infinite field of characteristic 2) that contain the identities of the regular representation of sl 2(K).

Graded Central Polynomials for T-Prime Algebras

Let K be a field, char K = 0, and let E = E 0⊕ E 1 be the Grassmann algebra of infinite dimension over K, equipped with its natural ℤ2-grading. If G is a finite abelian group and R = ⨁ g∈G R (g) is a G-graded K-algebra, then the algebra R⊗ E can be G × ℤ2-graded by setting (R⊗ E)(g, i) = R (g) ⊗ E i . In this article we describe the graded central polynomials for the T-prime algebras M n (E)≅ M n (K)⊗ E. As a corollary we obtain the graded central polynomials for the algebras M a, b (E)⊗ E. As an application, we determine the ℤ2-graded identities and central polynomials for E⊗ E.