On the module structure of a group action on a Lie algebra

Abstract

AbstractLet G be a finite group, K a field, and V a finite-dimensional K G-module. Write L(V) for the free Lie algebra on V; similarly, let M ( V) be the free metabelian Lie algebra. The action of G extends naturally to these algebras, so they become KG-modules, which are direct sums of finite-dimensional submodules. This paper explores whether indecomposable direct summands of such a KG-module (for some specific choices of G, K and V) must fall into finitely many isomorphism classes. Of course this is not a question unless there exist infinitely many isomorphism classes of indecomposable KG-modules (that is, K has positive characteristic p and the Sylow p-subgroups of G are non-cyclic) and dim V > 1.The first two results show that the answer is positive for M(V) when K is finite and dim V = 2, but negative when G is the Klein four-group, the characteristic of K is 2, and V is the unique 3-dimnsional submodule of the regular module D. In the third result, G is again the Klein four-group, K is any field of charateristic 2 with more than 2 elements, V is any faithfull module of dimension 2, and B is the unique 3-dimensional quotient of D; the answer is positive for L(V) if and only if it is positive for each of L(B), L(D), and L(V ⊗ V).