On group actions on free Lie algebras

Johnson, Marianne (2007) On group actions on free Lie algebras. Doctoral thesis, University of Manchester.

We first study the module structure of the free Lie algebra $L(V)$ in characteristic zero under the action of the general linear group. Here we give a new, purely combinatorial, proof of Klyachko's celebrated theorem on Lie representations using the Kra\'{s}kiewicz-Weyman theorem. We then give a new factorisation of the Dynkin-Specht-Wever idempotent and use this to prove that $L_2(L_k(V))$ is a $KG$-module direct summand of $L_{2k}(V)$, for $G$ an arbitrary group, $K$ a field of characteristic $p \nmid k$ and $V$ a $KG$-module. For finite-dimensional modules $V$, this follows immediately from the Decomposition Theorem of Bryant and Schocker. We consider a small example of this theorem, namely the sixth Lie power over a field of characteristic $3$. Here we show explicitly that $L_6(V)$ decomposes into a direct sum of the modules $L_3(L_2(V))$ and $L_2(V) \otimes S_2(V) \otimes S_2(V)$, where $S_2(V)$ denotes the symmetric square of $V$. We give a description, up to isomorphism, of the modules $B_{p^mk}$ occurring in the Decomposition Theorem. Finally, we apply our knowledge of Lie powers to a group theoretic problem. We show that the torsion subgroup $t_c$ of the quotient $\gamma_c R/ [\gamma_c R, F]$ is bounded as follows, for $c=2p^m$ or $c=3p^m$, where $p$ is an arbitrary prime, $m\geq 0$: \begin{eqnarray*} 2t_{2p^m} = 0 \;\;\;\mbox{provided } G=F/R \mbox{ has no 2-torsion and no p-torsion,}\\ 3t_{3p^m} = 0 \;\;\;\mbox{provided } G=F/R \mbox{ has no 3-torsion and no p-torsion.} \end{eqnarray*} Thus, we have that $\gamma_6 R/ [\gamma_6 R, F]$ is torsion-free, provided that $G=F/R$ has no elements of order $2$ or $3$.