The $\mathsf{Lie}$ Lie algebra

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 $H^k$(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 Sp-representations with multiplicities given by spaces of modular forms. Existing conjectures in the literature on the twisted homology of SL${}_3(\mathbb{Z})$ imply that this gives a full account of the rank 3 part of the abelianization in even degrees.