Group gradings on finite dimensional Lie algebras

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 $G$.