Graded simple Lie algebras and graded simple representations

TL;DR

This paper classifies $Q$-graded simple Lie algebras over algebraically closed fields of characteristic zero by reducing the problem to gradings on simple Lie algebras, and extends results to graded modules and semisimple algebras.

Contribution

It provides a full classification of finite-dimensional $Q$-graded simple Lie algebras and relates graded modules to gradings on simple modules, extending classical theorems.

Findings

Complete classification of finite-dimensional $Q$-graded simple Lie algebras.

Reduction of module classification to gradings on simple modules.

Graded analogue of the Weyl Theorem for semisimple algebras.

Abstract

For any finitely generated abelian group $Q$, we reduce the problem of classification of $Q$-graded simple Lie algebras over an algebraically closed field of "good" characteristic to the problem of classification of gradings on simple Lie algebras. In particular, we obtain the full classification of finite-dimensional $Q$-graded simple Lie algebras over any algebraically closed field of characteristic $0$ based on the recent classification of gradings on finite dimensional simple Lie algebras. We also reduce classification of simple graded modules over any $Q$-graded Lie algebra (not necessarily simple) to classification of gradings on simple modules. For finite-dimensional $Q$-graded semisimple algebras we obtain a graded analogue of the Weyl Theorem.