
Journal of Lie Theory 23 (2013), No. 3, 669689 Copyright Heldermann Verlag 2013 The Structure of H(co)module Lie algebras Alexey S. Gordienko Dept. of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL, Canada A1C 5S7 gordienko.a.s@gmail.com [Abstractpdf] Let $L$ be a finite dimensional Lie algebra over a field of characteristic $0$. Then by the original Levi theorem, $L = B \oplus R$ where $R$ is the solvable radical and $B$ is some maximal semisimple subalgebra. We prove that if $L$ is an $H$(co)module algebra for a finite dimensional (co)semisimple Hopf algebra $H$, then $R$ is $H$(co)invariant and $B$ can be chosen to be $H$(co)invariant too. Moreover, the nilpotent radical $N$ of $L$ is $H$(co)invariant and there exists an $H$sub(co)module $S\subseteq R$ such that $R=S\oplus N$ and $[B,S]=0$. In addition, the $H$(co)invariant analog of the Weyl theorem is proved. In fact, under certain conditions, these results hold for an $H$comodule Lie algebra $L$, even if $H$ is infinite dimensional. In particular, if $L$ is a Lie algebra graded by an arbitrary group $G$, then $B$ can be chosen to be graded, and if $L$ is a Lie algebra with a rational action of a reductive affine algebraic group $G$ by automorphisms, then $B$ can be chosen to be $G$invariant. Also we prove that every finite dimensional semisimple $H$(co)module Lie algebra over a field of characteristic $0$ is a direct sum of its minimal $H$(co)invariant ideals. Keywords: Lie algebra, stability, Levi decomposition, radical, grading, Hopf algebra, Hopf algebra action, $H$module algebra, $H$comodule algebra. MSC: 17B05; 17B40, 17B55, 17B70, 16T05, 14L17 [ Fulltextpdf (364 KB)] for subscribers only. 