Triangular matrices: group gradings and their graded polynomial identities

Abstract

This dissertation explores the algebra of upper triangular matrices and, more generally, the structure of block triangular matrix algebras, with a particular focus on gradings and graded polynomial identities. The work is situated within the broader framework of theory of Polynomial Identities. The primary objective is to classify group gradings on the algebra of upper triangular matrices and to describe the graded polynomial identities that these algebras satisfy. Building upon this classification, the study is extended to algebras of upper block-triangular matrices. In this broader context, we investigate the structure of a basis for the G-graded polynomial identities of the algebra $UT(d_1, \\ldots, d_n)$, induced by a specific family of gradings.