Gradings on the Lie algebra $D_4$ revisited

TL;DR

This paper classifies all group gradings on the simple Lie algebra of type D4 over algebraically closed fields, including fine gradings and G-gradings, and studies graded modules for each G-grading.

Contribution

It provides a comprehensive classification of group gradings on D4 Lie algebras and analyzes associated graded modules, extending previous work on Lie algebra gradings.

Findings

Classification of fine gradings up to equivalence

Classification of G-gradings up to isomorphism

Analysis of graded modules for each G-grading

Abstract

We classify group gradings on the simple Lie algebra $L$ of type $D_4$ over an algebraically closed field of characteristic different from 2: fine gradings up to equivalence and $G$-gradings, with a fixed group $G$, up to isomorphism. For each $G$-grading on $L$, we also study graded $L$-modules (assuming characteristic 0).