## Algebraic & Geometric Topology

### On the derivation algebra of the free Lie algebra and trace maps

#### 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 $GL(n,Q)$–module via the Schur–Weyl duality and some tensor product theorems for $GL(n,Q)$. 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 11, Number 5 (2011), 2861-2901.

Dates
Revised: 29 July 2011
Accepted: 14 September 2011
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715308

Digital Object Identifier
doi:10.2140/agt.2011.11.2861

Mathematical Reviews number (MathSciNet)
MR2846914

Zentralblatt MATH identifier
1259.17018

#### Citation

Enomoto, Naoya; Satoh, Takao. On the derivation algebra of the free Lie algebra and trace maps. Algebr. Geom. Topol. 11 (2011), no. 5, 2861--2901. doi:10.2140/agt.2011.11.2861. https://projecteuclid.org/euclid.agt/1513715308

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