The Lie Lie algebra

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 $H^k(\mathrm{O}\mathrm{u}\mathrm{t}(F_r);\mathbb{Q})$ 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 $\mathrm{S}\mathrm{P}$-representations with multiplicities given by spaces of modular forms. Existing conjectures in the literature on the twisted homology of ${\mathrm{S}\mathrm{L}}_3(\mathbb{Z})$ imply that this gives a full account of the rank $3$ part of the abelianization in even degrees.