Graded monomial identities and almost non-degenerate gradings on matrices

Author(s)

Centrone, Lucio ; Diniz, Diogo ; Mello, Thiago Castilho de

• Document type: Journal article
• Source: Linear Algebra and its Applications; v. 653, p. 20-pg., 2022-08-24.

Abstract

Let F be a field of characteristic zero, G be a group and M-n(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 M-n(F) is elementary and the neutral component of M-n(F) is commutative, then the graded identities of M-n(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 M-n(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 M-n(F) is almost non-degenerate and we apply the results on monomial identities to describe all almost non-degenerate Z-gradings on M-n(F) for n <= 5. (C) 2022 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 18/02108-7 - Identities in (non) associative algebras and related themes.
• Grantee: Lucio Centrone
• Support Opportunities: Regular Research Grants
• FAPESP's process: 18/15627-2 - Gradings, automorphisms and identities in algebras
• Grantee: Thiago Castilho de Mello
• Support Opportunities: Regular Research Grants
• FAPESP's process: 18/23690-6 - Structures, representations, and applications of algebraic systems
• Grantee: Ivan Chestakov
• Support Opportunities: Research Projects - Thematic Grants