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

Abstract

2861On the derivation algebra of the free Lie algebra and trace maps NAOYA ENOMOTO TAKAO SATOHWe 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.

A survey of the Johnson homomorphisms of the automorphism groups of free groups and related topics

“…Using Bryant and Gupta's generators and some relations among them, we showed Proposition 9.6 (Enomoto and Satoh, [16]). For k ≥ 3, n ≥ k − 1 and l ≥ 3,…”

“…We remark that recently we [16] showed that the multiplicity of the Morita obstruction The goal of this subsection is to give an upper bound on Coker(τ k,Q ) as a GL(n, Q)-module. In order to do this, we consider the Johnson homomorphism…”

“…In our recent paper [16], which is a joint work with Naoya Enomoto, we gave a combinatorial description of the GL(n, Q)-irreducible decomposition of C Q n (k). We remark that, as a GL(n, Q)-module, C Q n (k) is isomorphic to the invariant part a n (k) :…”

“…In [16], we determined the abelianization of the derivation algebra Der + (gr(L M n )) as a Lie algebra using Morita's trace maps. That is, Theorem 5.3 (Enomoto and Satoh, [16]). For n ≥ 4, we have…”

“…In [16], we prove that the trace map Tr [1 k ] defines a non-trivial twisted second cohomology class of the automorphism group Aut N n,k of a free nilpotent group N n,k := F n /Γ n (k + 1) with coefficients in Λ k H Q for any k ≥ 2 and n ≥ k. Here we give a brief strategy to prove it.…”