Fine gradings and gradings by root systems on simple Lie algebras

TL;DR

This paper explores how fine abelian group gradings on simple Lie algebras induce root system gradings, providing insights into their classification, especially for exceptional Lie algebras.

Contribution

It establishes that the grading by the free group derived from a fine abelian grading is a root system grading, advancing classification methods.

Findings

Induced gradings by free groups are root system gradings.

Implications for classifying fine gradings on exceptional Lie algebras.

Connections between abelian group gradings and root systems.

Abstract

Given a fine abelian group grading on a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero, with universal grading group $G$, it is shown that the induced grading by the free group $G/\tor(G)$ 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 drawn.