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

TL;DR

This paper computes the abelianizations of various derivation Lie algebras related to free associative and Lie algebras, providing new insights and applications in algebraic topology and mapping class groups.

Contribution

It determines the abelianizations of derivation Lie algebras of free associative and Lie algebras, and applies these results to prove a vanishing theorem in topology.

Findings

Abelianizations of derivation Lie algebras are explicitly computed.

New proof of Harer's vanishing theorem using Kontsevich's theorem.

Results connect algebraic structures with topological properties of mapping class groups.

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.