On ℤ2-graded identities of the super tensor product of UT 2(F) by the Grassmann algebra

Abstract

Let F be a field of characteristic zero and E be the unitary Grassmann algebra generated over an infinite-dimensional F-vector space L. Denote by \(\mathcal{E} = \mathcal{E}^{(0)} \oplus \mathcal{E}^{(1)}\) an arbitrary ℤ2-grading of E such that the subspace L is homogeneous. Given a superalgebra A = A(0) ⊕ A(1), define the superalgebra \(A\hat \otimes \mathcal{E}\) by \(A\hat \otimes \mathcal{E} = (A^{(0)} \otimes \mathcal{E}^{(0)} ) \oplus (A^{(1)} \otimes \mathcal{E}^{(1)} )\). Note that when E is the canonical grading of E then \(A\hat \otimes \mathcal{E}\) is the Grassmann envelope of A. In this work we find bases of ℤ2-graded identities and we describe the ℤ2-graded codimension and cocharacter sequences for the superalgebras \(UT_2 (F)\hat \otimes \mathcal{E}\), when the algebra UT2(F) of 2 ×2 upper triangular matrices over F is endowed with its canonical grading.