The Decomposition of Lie Powers

Abstract

Let G be a group, F a field of prime characteristic p and V a finite-dimensional FG-module. Let L(V) denote the free Lie algebra on V regarded as an FG-submodule of the free associative algebra (or tensor algebra) T(V). For each positive integer r, let L^r(V) and T^r(V) be the rth homogeneous components of L(V) and T(V), respectively. Here L^r(V) is called the rth Lie power of V. Our main result is that there are submodules B_1, B_2, ... of L(V) such that, for all r, B_r is a direct summand of T^r(V) and, whenever m \geq 0 and k is not divisible by p, $$L^{p^{m}k}(V)=L^{p^m}(B_k)\oplus L^{p^{m-1}}(B_{pk})\oplus ...\oplus L^p(B_{p^{m-1}k})\oplus L^1(B_{p^{m}k}).$$ Thus every Lie power is a direct sum of Lie powers of p-power degree. The approach builds on an analysis of T^r(V) as a bimodule for G and the Solomon descent algebra.

32 pages