Group gradings on finite dimensional Lie algebras

TL;DR

This paper investigates how noncommutative group gradings influence the structure of finite dimensional Lie algebras over algebraically closed fields of characteristic zero, revealing that such gradings impose specific decompositions and support conditions.

Contribution

It demonstrates that non-abelian group gradings lead to a graded solvable radical and a homogeneous Levi decomposition with commutative supports.

Findings

The solvable radical is G-graded.

Existence of a homogeneous Levi subalgebra with graded simple summands.

Supports of simple components are commutative subsets of G.

Abstract

We study gradings by noncommutative groups on finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It is shown that if $L$ is gradeg by a non-abelian finite group $G$ then the solvable radical $R$ of $L$ is $G$-graded and there exists a Levi subalgebra $B=H_1\oplus\cdots\oplus H_m$ homogeneous in $G$-grading with graded simple summands $H_1, \ldots, H_m$. All supports $Supp~H_i, i=1\ldots, m$, are commutative subsets of $G$.