2008.46: On group actions on free Lie algebras

2008.46: Marianne Johnson (2007) On group actions on free Lie algebras. PhD thesis, University of Manchester.

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Abstract

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 $3$.

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