On [formula omitted]-graded identities of the generalized Grassmann envelope of the upper triangular matrices [formula omitted

Abstract

Let F be a field of characteristic zero and let E be the unitary Grassmann algebra generated by an infinite-dimensional F-vector space L. Denote by E=E(0)⊕E(1) an arbitrary Z2-grading on E such that the subspace L is homogeneous. Given a superalgebra A=A(0)⊕A(1), define its generalized Grassmann envelope A⊗̂E as the superalgebra A⊗̂E=(A(0)⊗E(0))⊕(A(1)⊗E(1)). Note that when E is the canonical grading of E then A⊗̂E is the Grassmann envelope of A. In this work we describe the generators of the T2-ideal, Idgr(UTk,l(F)⊗̂E), of the Z2-graded polynomial identities of the superalgebras UTk,l(F)⊗̂E, as well as linear bases of the corresponding relatively free graded algebras. Here, given k⩾1, l⩾0, UTk,l(F) is the algebra of (k+l)×(k+l) upper triangular matrices over F with the Z2-grading UTk+l(F)=(UTk(F)00UTl(F))⊕(0Mk×l(F)00). In order to prove our result we obtain a similar description corresponding to the T-ideals Id(UTn(E)) and Id(UTn(Gr)) of ordinary polynomial identities, where Gr is the Grassmann algebra generated by an r-dimensional vector space.