The Decomposition of Lie Powers

2005/05/16 by R. M. Bryant, Bryant, R. M., M. Schocker +1
Mathematics · #17B01 (Primary) #20C07 #20C20 (Secondary) #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:17B01 #msc:20C07 #msc:20C20

paper · pdf · doi:10.48550/arxiv.math/0505325

32 pages

arxiv created 2005/05/16 · arxiv updated 2009/12/01

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 B₁, B₂, ... of L(V) such that, for all r, B_r is a direct summand of T^r(V) and, whenever m ≥ 0 and k is not divisible by p, L^pmk(V) = L^pm(B_k) ⊕ L^p^m-1(B_pk) ⊕ ... ⊕ L^p(B_p^m-1k) ⊕ L¹(B_p^mk). 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.

Related

• Factorisation of Lie Resolvents
• Modular Lie powers and the Solomon descent algebra
• Asymptotic behaviour of Lie powers and Lie modules
• Lie powers of the natural module for GL(2,K)
• Blocks of representations of Lie algebras