Volume 11, issue 5 (2011)

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

Naoya Enomoto and Takao Satoh

Algebraic & Geometric Topology 11 (2011) 2861–2901

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(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.

derivation, free Lie algebra, Chen Lie algebra, trace map, Johnson homomorphism, automorphism group, free nilpotent group
Mathematical Subject Classification 2010
Primary: 17B40, 20C15
Secondary: 20F28
Received: 19 December 2010
Revised: 29 July 2011
Accepted: 14 September 2011
Published: 14 October 2011
Naoya Enomoto
Department of Mathematics, Graduate School of Science
Kyoto University
Kyoto city 606-8502
Takao Satoh
Department of Mathematics, Faculty of Science Division II
Tokyo University of Science
1-3 Kagurazaka
Tokyo 162-8601