Graded modules over simple Lie algebras with a grading

Abstract

In the past two decades there has been a considerable interest in describing all possible gradings by abelian groups on simple Lie algebras. Over an algebraically closed field of characteristic zero, the answer is nearly complete in the finite-dimensional case: fine gradings have been classified up to equivalence for all types, and arbitrary $G$-gradings, for a fixed group $G$, have been classified up to isomorphism except for types $E_6$, $E_7$ and $E_8$. For a given $G$-grading on a simple finite-dimensional Lie algebra $L$, we will discuss some recent classification results for finite-dimensional graded $L$-modules.