# Group gradings on upper block triangular matrices

## TL;DR

This paper extends the classification of group gradings on upper block triangular matrix algebras to non-abelian groups and more general fields, broadening the understanding of algebraic structures under group actions.

## Contribution

It generalizes previous results by Valenti and Zaicev to include non-commutative groups and fields that are not necessarily algebraically closed.

## Key findings

- Classification holds for non-abelian groups
- Results apply to fields with characteristic zero or large enough
- Extends isomorphism to broader algebraic settings

## Abstract

It was proved by Valenti and Zaicev, in 2011, that, if $G$ is an abelian group and $K$ is an algebraically closed field of characteristic zero, then any $G$-grading on the algebra of upper block triangular matrices over $K$ is isomorphic to a tensor product $M_n(K)\otimes UT(n_1,n_2,\ldots,n_d)$, where $UT(n_1,n_2,\ldots,n_d)$ is endowed with an elementary grading and $M_n(K)$ is provided with a division grading. In this paper, we prove the validity of the same result for a non necessarily commutative group and over an adequate field (characteristic either zero or large enough), not necessarily algebraically closed.