Translator Disclaimer
2011 On the derivation algebra of the free Lie algebra and trace maps
Naoya Enomoto, Takao Satoh
Algebr. Geom. Topol. 11(5): 2861-2901 (2011). DOI: 10.2140/agt.2011.11.2861

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.

Citation

Download Citation

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

Information

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

zbMATH: 1259.17018
MathSciNet: MR2846914
Digital Object Identifier: 10.2140/agt.2011.11.2861

Subjects:
Primary: 17B40, 20C15
Secondary: 20F28

Rights: Copyright © 2011 Mathematical Sciences Publishers

JOURNAL ARTICLE
41 PAGES


SHARE
Vol.11 • No. 5 • 2011
MSP
Back to Top