The Goldman–Turaev Lie bialgebra in genus zero and the Kashiwara–Vergne problem

Abstract

In this paper, we describe a surprising link between the theory of the Goldman–Turaev Lie bialgebra on surfaces of genus zero and the Kashiwara–Vergne (KV) problem in Lie theory. Let Σ be an oriented 2-dimensional manifold with non-empty boundary and K a field of characteristic zero. The Goldman–Turaev Lie bialgebra is defined by the Goldman bracket {−,−} and Turaev cobracket δ on the K-span of homotopy classes of free loops on Σ.Applying an expansion θ:Kπ→K〈x1,…,xn〉 yields an algebraic description of the operations {−,−} and δ in terms of non-commutative variables x1,…,xn. If Σ is a surface of genus g=0 the lowest degree parts {−,−}−1 and δ−1 are canonically defined (and independent of θ). They define a Lie bialgebra structure on the space of cyclic words which was introduced and studied by Schedler [31]. It was conjectured by the second and the third authors that one can define an expansion θ such that {−,−}={−,−}−1 and δ=δ−1. The main result of this paper states that for surfaces of genus zero constructing such an expansion is essentially equivalent to the KV problem. In [24], Massuyeau constructed such expansions using the Kontsevich integral.In order to prove this result, we show that the Turaev cobracket δ can be constructed in terms of the double bracket (upgrading the Goldman bracket) and the non-commutative divergence cocycle which plays the central role in the KV theory. Among other things, this observation gives a new topological interpretation of the KV problem and allows to extend it to surfaces with arbitrary number of boundary components (and of arbitrary genus, see [2]).