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2006.271: Free Lie algebras and Adams operations

2006.271: R. M. Bryant (2003) Free Lie algebras and Adams operations. Journal of the London Mathematical Society, 68 (2). pp. 355-370. ISSN 0024-6107

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DOI: 10.1112/S0024610703004484

Abstract

Let $G$ be a group and $K$ a field. For any finite-dimensional $KG$-module $V$ and any positive integer $n$, let $L^n(V)$ denote the $n$th homogeneous component of the free Lie $K$-algebra generated by (a basis of) $V$. Then $L^n(V)$ can be considered as a $KG$-module, called the $n$th Lie power of $V$. The paper is concerned with identifying this module up to isomorphism. A simple formula is obtained which expresses $L^n(V)$ in terms of certain linear functions on the Green ring. When $n$ is not divisible by the characteristic of $K$ these linear functions are Adams operations. Some results are also obtained which clarify the relationship between Adams operations defined by means of exterior powers and symmetric powers and operations introduced by Benson. Some of these results are put into a more general setting in an appendix by Stephen Donkin.

Item Type:Article
Subjects:MSC 2000 > 17 Nonassociative rings and algebras
MSC 2000 > 20 Group theory and generalizations
MIMS number:2006.271
Deposited By:Miss Louise Stait
Deposited On:09 August 2006

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