The Lie algebra

  • James Conant

    University of Tennessee, Knoxville, USA

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