Abelianizations of derivation Lie algebras of free associative algebra and free Lie algebra

Details Abelianizations of derivation Lie algebras of free associative algebra and free Lie algebra Abelianizations of derivation Lie algebras of free associative algebra and free Lie algebra

2011-07-19

arxiv.org/abs/1107.3686v1

Mathematics

Algebraic Topology

29 pages, 17 figures

Scientific paper

We determine the abelianizations of the following three kinds of graded Lie algebras in a certain stable range: derivations of the free associative algebra, derivations of the free Lie algebra and symplectic derivations of the free associative algebra. As an application of the last case, and by making use of a theorem of Kontsevich, we obtain a new proof of the vanishing theorem of Harer concerning the top rational cohomology group of the mapping class group with respect to its virtual cohomological dimension.

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