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

TL;DR

This paper analyzes the structure of derivation algebras related to free Lie algebras, decomposes Johnson homomorphisms, and explores trace maps and twisted cohomology, revealing new algebraic insights.

Contribution

It provides the irreducible decomposition of Johnson homomorphism images and characterizes the abelianization of the derivation algebra of the Chen Lie algebra.

Findings

Decomposition of Johnson homomorphisms for free and metabelian groups

Identification of the abelianization of the derivation algebra via trace maps

Construction of a non-trivial twisted second cohomology class

Abstract

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. We determine the abelianization of the derivation algebra of the Chen Lie algebra as a Lie algebra, and show that the abelianizaton is given by the degree one part and the Morita's trace maps. We also consider twisted cohomology groups of the automorphism group of a free nilpotent group. We show that the trace map for the exterior product defines a non-trivial twisted second cohomology class of it.