The Decomposition of Lie Powers

Let G be a group, F a field of prime characteristic p and V a finite-dimensional FGmodule. 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 Lr(V ) and T r(V ) be the rth homogeneous components of L(V ) and T (V ), respectively. Here Lr(V ) is called the rth Lie power of V . Our main result is that there are submodules B1, B2, . . . of L(V ) such that, for all r, Br is a direct summand of T r(V ) and, whenever m > 0 and k is not divisible by p, L mk(V ) = L m (Bk)⊕ L m−1(Bpk)⊕ · · · ⊕ L (Bpm−1k)⊕ L (Bpmk). 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.