The \(\mathsf{Lie}\) Lie algebra (Q1691975)

Summary: 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 \textit{S. Morita} [Geom. Topol. Monogr. 2, 349--406 (1999; Zbl 0959.57018)] and Conant-Kassabov-Vogtmann [the author et al., J. Topol. 6, No. 1, 119--153 (2013; Zbl 1345.17015)] 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 \(\mathrm{SL}_3(\mathbb{Z})\) imply that this gives a full account of the rank 3 part of the abelianization in even degrees.