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 $\mathbb{K}$ a field of characteristic zero. The Goldman-Turaev Lie bialgebra is defined by the Goldman bracket $\{ -,- \}$ and Turaev cobracket $δ$ on the $\mathbb{K}$-span of homotopy classes of free loops on $Σ$. Applying an expansion $θ: \mathbb{K}π\to \mathbb{K}\langle x_1, \dots, x_n \rangle$ yields an algebraic description of the operations $\{ -,- \}$ and $δ$ in terms of non-commutative variables $x_1, \dots, x_n$. 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 T. Schedler. 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. G. 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 [C. R. Acad. Sci. Paris, Ser. I 355 (2017), 123--127]).