On Z2-graded identities of the generalized Grassmann envelope of the upper triangular matrices UTk,l(F)

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 as the superalgebra . Note that when E is the canonical grading of E then is the Grassmann envelope of A. In this work we describe the generators of the T2-ideal, , of the Z2-graded polynomial identities of the superalgebras , 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 . 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.