Graded identities of block-triangular matrices

Abstract

Let F be an infinite field and UT(d(1),..., d(n)) be the algebra of upper block-triangular matrices over F. In this paper we describe a basis for the C-graded polynomial identities of UT(d(1),..., d(n)), with an elementary grading induced by an n-tuple of elements of a group G such that the neutral component corresponds to the diagonal of UT(d(1),..., d(n)). In particular, we prove that the monomial identities of such algebra follow from the ones of degree up to 2n - 1. Our results generalize, for infinite fields of arbitrary characteristic, previous results in the literature which were obtained for fields of characteristic zero and for particular G-gradings. In the characteristic zero case we also generalize results for the algebra UT(d(1),..., d(n)) circle times C with a tensor product grading, where C is a color commutative algebra generating the variety of all color commutative algebras. (C) 2016 Elsevier Inc. All rights reserved.