We mainly study the derivation algebra of the free Lie algebra and the Chen Lie algebra generated by the abelianization of a free group, and trace maps. To begin with, we give the irreducible decomposition of the derivation algebra as a –module via the Schur–Weyl duality and some tensor product theorems for . 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 . 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 defines a nontrivial twisted second cohomology class of it.
Naoya Enomoto. Takao Satoh. "On the derivation algebra of the free Lie algebra and trace maps." Algebr. Geom. Topol. 11 (5) 2861 - 2901, 2011. https://doi.org/10.2140/agt.2011.11.2861