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On the derivation
algebra of the free Lie algebra and trace maps
Naoya Enomoto and Takao Satoh |
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Algebraic & Geometric Topology 11 (2011) 2861–2901
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Abstract |
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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. |
Keywords
derivation, free Lie algebra, Chen Lie algebra, trace map,
Johnson homomorphism, automorphism group, free nilpotent
group
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Mathematical Subject Classification 2010
Primary: 17B40, 20C15
Secondary: 20F28
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References |
Publication
Received: 19 December 2010
Revised: 29 July 2011
Accepted: 14 September 2011
Published: 14 October 2011
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Authors |
| Naoya Enomoto | |
| Department of Mathematics, Graduate
School of Science Kyoto University Kitashirakawaoiwake-cho Sakyo-ku Kyoto city 606-8502 Japan |
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| Takao Satoh | |
| Department of Mathematics, Faculty
of Science Division II Tokyo University of Science 1-3 Kagurazaka Shinjyuku-ku Tokyo 162-8601 Japan |
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