## 2007.28: The Decomposition of Lie Powers

2007.28:
R. M. Bryant and M. Schocker
(2006)
*The Decomposition of Lie Powers.*
Proceedings of the London Mathematical Society, 93 (1).
pp. 175-196.
ISSN 0024-6093

Full text available as:

PDF - Archive staff only - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader 257 Kb |

DOI: 10.1017/S0024611505015728

## 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 |
---|---|

Subjects: | MSC 2000 > 17 Nonassociative rings and algebras MSC 2000 > 20 Group theory and generalizations |

MIMS number: | 2007.28 |

Deposited By: | Miss Louise Stait |

Deposited On: | 23 March 2007 |

Download Statistics: last 4 weeks

Repository Staff Only: edit this item