The Lie Lie algebra

TL;DR

This paper advances the understanding of Kontsevich's Lie algebra by computing the rank 3 abelianization, revealing connections to modular forms and implications for the homology of SL(3,Z).

Contribution

It provides a partial computation of the rank 3 abelianization of Kontsevich's Lie algebra, linking it to modular forms and conjectures in twisted homology.

Findings

Identified many irreducible SP-representations in the rank 3 abelianization.

Connected the abelianization to spaces of modular forms.

Implications for the twisted homology of SL(3,Z) in even degrees.

Abstract

We study the abelianization of Kontsevich's Lie algebra associated with the Lie operad and some related problems. Calculating the abelianization is a long-standing unsolved problem, which is important in at least two different contexts: constructing cohomology classes in $H^k(\mathrm{Out}(F_r);\mathbb Q)$ and related groups as well as studying the higher order Johnson homomorphism of surfaces with boundary. The abelianization carries a grading by "rank," with previous work of Morita and Conant-Kassabov-Vogtmann computing it up to rank $2$. This paper presents a partial computation of the rank $3$ part of the abelianization, finding lots of irreducible $\mathrm{SP}$-representations with multiplicities given by spaces of modular forms. Existing conjectures in the literature on the twisted homology of $\mathrm{SL}_3(\mathbb Z)$ imply that this gives a full account of the rank $3$ part of the abelianization in even degrees.