The Lie algebra
James Conant
University of Tennessee, Knoxville, USA
![The $\mathsf {Lie}$ Lie algebra cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-qt-volume-8-issue-4.png&w=3840&q=90)
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 (Out( 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 imply that this gives a full account of the rank 3 part of the abelianization in even degrees.
Cite this article
James Conant, The Lie algebra. Quantum Topol. 8 (2017), no. 4, pp. 667–714
DOI 10.4171/QT/99