# The Decomposition of Lie Powers

Bryant, R. M. and Schocker, M. (2006) The Decomposition of Lie Powers. Proceedings of the London Mathematical Society, 93 (1). pp. 175-196. ISSN 0024-6093

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## 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.

Item Type: Article MSC 2010, the AMS's Mathematics Subject Classification > 17 Nonassociative rings and algebrasMSC 2010, the AMS's Mathematics Subject Classification > 20 Group theory and generalizations Ms Lucy van Russelt 23 Mar 2007 20 Oct 2017 14:12 http://eprints.maths.manchester.ac.uk/id/eprint/721

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