Tensor products of matrix algebras over the Grassmann algebra (Q1814038)

Let \(E\) be the exterior (or Grassmann) algebra of an infinite-dimensional vector space over a field \(F\) of characteristic \(0\) and let \(E=E_ 0\oplus E_ 1\) be the natural \(\mathbb{Z}_ 2\)-grading of \(E\). Let \[ M_{k,\ell}(E)= \begin{pmatrix} M_{k\times k}(E_ 0) & M_{k\times\ell}(E_ 1)\\ M_{\ell\times k}(E_ 1) & M_{\ell\times\ell}(E_ 0)\\ \end{pmatrix}, \] where \(M_{p\times q}(R)\) is the vector space of \(p\times q\) matrices with entries from \(R\). \textit{A. R. Kemer} [Izv. Akad. Nauk SSSR, Ser. Mat. 48, 1042-1059 (1984; Zbl 0586.16010)] has established that the \(T\)-ideals of the polynomial identities of the algebras \(M_ k(F)\), \(M_ k(E)\), \(M_{k,\ell}(E)\) are the only non-trivial \(T\)-prime ideals of the free algebra. Kemer has also given a rule for expressing the \(T\)-ideal of the identities for the tensor products \(M_{k,\ell}(E)\otimes M_{p,q}(E)\), \(M_{k,\ell}(E)\otimes E\) and \(E\otimes E\). The main purpose of the paper under review is to give direct and detailed combinatorial proofs of the Kemer result on the tensor products. As a consequence of the developed method the author identifies the polynomial identities of some other subalgebras of \(M_ k(E)\) with those of some already known algebras.