The Decomposition of Lie Powers

Details The Decomposition of Lie Powers The Decomposition of Lie Powers

2005-05-16

arxiv.org/abs/math/0505325v1

Mathematics

Representation Theory

32 pages

Scientific paper

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, $$ L^{p^mk}(V) = L^{p^m}(B_k) \oplus L^{p^{m-1}}(B_{pk}) \oplus ... \oplus L^p(B_{p^{m-1}k}) \oplus L^1(B_{p^mk}). $$ 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.

Affiliated with

Also associated with

No associations

Rating

The Decomposition of Lie Powers does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The Decomposition of Lie Powers, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Decomposition of Lie Powers will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-271133