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 $GL\left(n,Q\right)$–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.