#### Volume 11, issue 5 (2011)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
On the derivation algebra of the free Lie algebra and trace maps

### Naoya Enomoto and Takao Satoh

Algebraic & Geometric Topology 11 (2011) 2861–2901
##### Abstract

We mainly study the derivation algebra of the free Lie algebra and the Chen Lie algebra generated by the abelianization $H$ of a free group, and trace maps. To begin with, we give the irreducible decomposition of the derivation algebra as a $GL\left(n,Q\right)$–module via the Schur–Weyl duality and some tensor product theorems for $GL\left(n,Q\right)$. Using them, we calculate the irreducible decomposition of the images of the Johnson homomorphisms of the automorphism group of a free group and a free metabelian group.

Next, we consider some applications of trace maps: Morita’s trace map and the trace map for the exterior product of $H$. First, we determine the abelianization of the derivation algebra of the Chen Lie algebra as a Lie algebra, and show that the abelianization is given by the degree one part and Morita’s trace maps. Second, we consider twisted cohomology groups of the automorphism group of a free nilpotent group. In particular, we show that the trace map for the exterior product of $H$ defines a nontrivial twisted second cohomology class of it.

##### Keywords
derivation, free Lie algebra, Chen Lie algebra, trace map, Johnson homomorphism, automorphism group, free nilpotent group
##### Mathematical Subject Classification 2010
Primary: 17B40, 20C15
Secondary: 20F28