\(\mathbb{Z}_{k+l}\times\mathbb{Z}_2\)-graded polynomial identities for \(M_{k,l}(E)\otimes E\). (Q2569615)

Summary: Let \(K\) be a field of characteristic zero, and \(E\) be the Grassmann algebra over an infinite-dimensional \(K\)-vector space. We endow \(M_{k,l}(E)\otimes E\) with a \(\mathbb{Z}_{k+l}\times\mathbb{Z}_2\)-grading, and determine a generating set for the ideal of its graded polynomial identities. As a consequence, we prove that \(M_{k,l}(E)\times E\) and \(M_{k+l}(E)\) are PI-equivalent with respect to this grading. In particular, this leads to their ordinary PI-equivalence, a classical result obtained by Kemer.