The Decomposition of Lie Powers

Let G be a group, F a field of prime characteristic p and Va finite-dimensional FG-module. Let L(V) denote the free Liealgebra on V regarded as an FG-submodule of the free associativealgebra (or tensor algebra) T(V). For each positive integerr, let L^r (V) and T^r (V) be the rth homogeneous components ofL(V) and T(V), respectively. Here L^r (V) is called the rth Liepower of V. Our main result is that there are submodules B₁,B₂, ... of L(V) such that, for all r, B_r is a direct summandof T^r(V) and, whenever m 0 and k is not divisible by p, themodule is the direct sum of , . Thus every Lie power is a direct sum of Lie powers of p-powerdegree. The approach builds on an analysis of T^r (V) as a bimodulefor G and the Solomon descent algebra. 2000 Mathematics SubjectClassification 17B01 (primary), 20C07, 20C20 (secondary).