FREE LIE ALGEBRAS AND ADAMS OPERATIONS

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.