Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 23 (2013), No. 3, 669--689Copyright 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 [Abstract-pdf] 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 [ Fulltext-pdf  (364  KB)] for subscribers only.