Goldman–Turaev formality implies Kashiwara–Vergne

Abstract

Let $\Sigma$ be a compact connected oriented 2-dimensional manifold with non-empty boundary. In our previous work, we have shown that every solution of the higher genus Kashiwara–Vergne equations for an automorphism $F\in \mathrm{aut}(L)$ of a free Lie algebra induces an isomorphism between the Goldman–Turaev Lie bialgebra $\mathfrak{g}(\Sigma )$ and its associated graded gr $\mathfrak{g}(\Sigma )$. In this paper, we prove the converse: if $F$ induces an isomorphism $\mathfrak{g}(\Sigma )\cong \mathrm{gr}\mathfrak{g}(\Sigma )$, then it satisfies the Kashiwara–Vergne equations up to conjugation. As an application of our results, we compute the degree one non-commutative Poisson cohomology of the Kirillov–Kostant–Souriau double bracket. The main technical tool used in the paper is a novel characterization of conjugacy classes in the free Lie algebra in terms of cyclic words.