Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 11, Number 5 (2011), 2861-2901.
On the derivation algebra of the free Lie algebra and trace maps
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.
Algebr. Geom. Topol., Volume 11, Number 5 (2011), 2861-2901.
Received: 19 December 2010
Revised: 29 July 2011
Accepted: 14 September 2011
First available in Project Euclid: 19 December 2017
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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