J. Aust. Math. Soc.  82 (2007), 237-248
On the module structure of a group action on a Lie algebra

Athanassios I. Papistas
  Faculty of Sciences
  Department of Mathematics
  Aristotle University of Thessaloniki
  GR 541 24, Thessaloniki

Let G be a finite group, K a field, and V a finite-dimensional KG-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 K 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-dimensional submodule of the regular module D. In the third result, G is again the Klein four-group, K is any field of characteristic 2 with more than 2 elements, V is any faithful 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 $L(V \otimes V)$.
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