Gradings on the algebra of triangular matrices as a Lie algebra: revisited

TL;DR

This paper revisits the classification of group gradings on the Lie algebra of upper triangular matrices, offering streamlined proofs, a complete classification, and detailed analysis in characteristic 2.

Contribution

It provides a simplified, comprehensive classification of gradings on triangular matrix Lie algebras, including the case of characteristic 2, with improved methods and practical isomorphism considerations.

Findings

Complete classification of gradings on triangular matrix Lie algebras.

Streamlined proofs simplifying previous results.

Detailed analysis of characteristic 2 case.

Abstract

We investigate the group gradings on the algebra of upper triangular matrices over an arbitrary field, viewed as a Lie algebra. These results were obtained a few years early by the same authors. We provide streamlined proofs, and present a complete classification of isomorphism classes of the gradings. We also provide a classification of the practical isomorphism classes of the gradings, which is a better alternative way to consider these gradings up to being essentially the same object. Finally, we investigate in details the case where the characteristic of the base field is $2$, a topic that was neglected in previous works.