Graded identities forT-prime algebras over fields of positive characteristic

Abstract

In this paper we study 2-graded polynomial identities. We describe bases of these identities satisfied by the matrix algebra of order twoM ₂(K), by the algebraM _1,1(G), and by the algebraG ⊗ _K G. 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, andM _1,1(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 furthermoreM _1,1(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 forM _1,1(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 andM _1,1(G) over an infinite field of positive characteristic. Note that in characteristic 0 such a description was given in [12] and in [10].