The Decomposition of Lie Powers

الأصل · EN

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 Lʳ(V) and Tʳ(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 Tʳ(V) and, whenever m ≥ 0 and k is not divisible by p, Lpᵐk(V) = Lᵖᵐ(Bₖ) ⊕ Lᵖᵐ⁻¹(Bpk) ⊕... ⊕ Lᵖ(Bₚᵐ⁻¹ₖ) ⊕ L¹(Bpᵐk). Thus every Lie power is a direct sum of Lie powers of p-power degree. The approach builds on an analysis of Tʳ(V) as a bimodule for G and the Solomon descent algebra.