Graded modules over classical simple Lie algebras with a grading

TL;DR

This paper classifies simple modules over classical simple Lie algebras with a given abelian group grading, providing invariants and criteria for when modules admit such gradings.

Contribution

It offers a comprehensive classification of G-graded simple modules over classical Lie algebras, including explicit invariants and grading criteria.

Findings

Classification of simple G-graded modules for classical Lie algebras

Explicit invariants for module classification

Criteria for modules to admit G-gradings

Abstract

Given a grading by an abelian group G on a semisimple Lie algebra L over an algebraically closed field of characteristic 0, we classify up to isomorphism the simple objects in the category of finite-dimensional G-graded L-modules. The invariants appearing in this classification are computed in the case when L is simple classical (except for type D4, where a partial result is given). In particular, we obtain criteria to determine when a finite-dimensional simple L-module admits a G-grading making it a graded L-module.