Graded monomial identities and almost non-degenerate gradings on matrices

Abstract

Let F be a field of characteristic zero, G be a group and Mn(F) be the algebra of matrices of size n with entries from F with a G-grading. Bahturin and Drensky proved that if the G grading on Mn(F) is elementary and the neutral component of Mn(F) is commutative, then the graded identities of Mn(F) follow from three basic types of identities and monomial identities of length ≥2 bounded by a function f(n) of n. In this paper we prove the best upper bound is f(n)=n. More generally, we prove that all the graded monomial identities of an elementary G-grading on Mn(F) follow from those of degree at most n. We also study gradings which satisfy no graded multilinear monomial identities but the trivial ones, which we call almost non-degenerate gradings. The description of non-degenerate elementary gradings on matrix algebras is reduced to the description of non-degenerate elementary gradings on matrix algebras that have commutative neutral component. We provide necessary conditions so that the grading on Mn(F) is almost non-degenerate and we apply the results on monomial identities to describe all almost non-degenerate Z-gradings on Mn(F) for n≤5.