Algebraic & Geometric Topology

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

Naoya Enomoto and Takao Satoh

Full-text: Open access

Abstract

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.

Article information

Source
Algebr. Geom. Topol., Volume 11, Number 5 (2011), 2861-2901.

Dates
Received: 19 December 2010
Revised: 29 July 2011
Accepted: 14 September 2011
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715308

Digital Object Identifier
doi:10.2140/agt.2011.11.2861

Mathematical Reviews number (MathSciNet)
MR2846914

Zentralblatt MATH identifier
1259.17018

Subjects
Primary: 17B40: Automorphisms, derivations, other operators 20C15: Ordinary representations and characters
Secondary: 20F28: Automorphism groups of groups [See also 20E36]

Keywords
derivation free Lie algebra Chen Lie algebra trace map Johnson homomorphism automorphism group free nilpotent group

Citation

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


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