Fine gradings and gradings by root systems on simple Lie algebras

Abstract

Given a fine abelian group grading $\Gamma :\mathcal{L}={⨁}_{g\in G}{\mathcal{L}}_g$ on a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero, with $G$ being the universal grading group, it is shown that the induced grading by the free group $G/\mathrm{tor}(G)$ on $\mathcal{L}$ is a grading by a (not necessarily reduced) root system.

Some consequences for the classification of fine gradings on the exceptional simple Lie algebras are deduced.