Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T12:13:41.289Z Has data issue: false hasContentIssue false

THE DECOMPOSITION OF LIE POWERS

Published online by Cambridge University Press:  09 June 2006

R. M. BRYANT
Affiliation:
School of Mathematics, University of Manchester, P.O. Box 88, Manchester, M60 1QD, United [email protected]
M. SCHOCKER
Affiliation:
Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, United [email protected]
Get access

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 $r$th homogeneous components of $L(V)$ and $T(V)$, respectively. Here $L^r (V)$ is called the $r$th 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 $m \geqslant 0$ and $k$ is not divisible by $p$, the module $L^{p^mk} (V)$ is the direct sum of $L^{p^m} (B_k)$, $L^{p^{m - 1}} (B_{pk})$, ..., $L^1 (B_{p^mk})$. 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.

Type
Research Article
Copyright
2006 London Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)