Duke Mathematical Journal

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

Shigeyuki Morita, Takuya Sakasai, and Masaaki Suzuki

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We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra, and symplectic derivations of the free associative algebra. In each case, we consider both the whole derivation Lie algebra and its ideal consisting of derivations with positive degrees. 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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Duke Math. J., Volume 162, Number 5 (2013), 965-1002.

First available in Project Euclid: 29 March 2013

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Primary: 17B56: Cohomology of Lie (super)algebras 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20]
Secondary: 17B65: Infinite-dimensional Lie (super)algebras [See also 22E65] 20J06: Cohomology of groups


Morita, Shigeyuki; Sakasai, Takuya; Suzuki, Masaaki. Abelianizations of derivation Lie algebras of the free associative algebra and the free Lie algebra. Duke Math. J. 162 (2013), no. 5, 965--1002. doi:10.1215/00127094-2140560. https://projecteuclid.org/euclid.dmj/1364562916

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