Graded identities of \(M_n(E)\) and their generalizations over infinite fields (Q2120931)

The paper under review considers the graded polynomial identities satisfied by algebras over infinite fields. Let \(G\) be a group and \(A\) a finite dimensional associative algebra graded by \(G\). Assume \(H\) is an additive abelian group and \(\beta\) a skew-symmetric bicharacter on \(H\). Let \(C\) be an associative algebra that is \(H\)-graded and \(\beta\)-commutative (that is homogeneous elements of \(C\) commute up to a scalar which is the value of \(\beta\)). The author describes the graded identities of \(A\otimes C\) in terms of those of \(A\). The description, as it is common in such situations, is up to graded monomial identities. Since the Grassmann algebra \(E\) of infinite dimension is \(\beta\)-commutative the author applies the methods so far developed in order to describe the graded identities of \(M_n(E)\). The grading on \(M_n(E)\) is assumed to be the induced from an elementary grading on the matrix algebra where the neutral component coincides with the main diagonal, and the grading on \(E\) is the natural one. The author proves that the degrees of the monomial identities one needs in this case are bounded by \(2d-1\). Moreover he obtains analogous results for the tensor product \(M_q(F)\otimes UT(d_1, \ldots, d_n)\). Here \(F\) is the base field, and \(UT(d_1, \ldots, d_n)\) is the corresponding block triangular matrix algebra. The algebra \(M_q(F)\) is equipped with the Pauli grading, and \(UT(d_1, \ldots, d_n)\) has an elementary grading such that its main diagonal coincides with the neutral component.