## Duke Mathematical Journal

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

#### Abstract

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.

#### Article information

Source
Duke Math. J., Volume 162, Number 5 (2013), 965-1002.

Dates
First available in Project Euclid: 29 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1364562916

Digital Object Identifier
doi:10.1215/00127094-2140560

Mathematical Reviews number (MathSciNet)
MR3047471

Zentralblatt MATH identifier
1308.17021

#### Citation

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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