# Goldman–Turaev formality implies Kashiwara–Vergne

### Anton Alekseev

Université de Genève, Switzerland### Nariya Kawazumi

University of Tokyo, Japan### Yusuke Kuno

Tsuda University, Tokyo, Japan### Florian Naef

Trinity College, Dublin, Ireland

## 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.

## Cite this article

Anton Alekseev, Nariya Kawazumi, Yusuke Kuno, Florian Naef, Goldman–Turaev formality implies Kashiwara–Vergne. Quantum Topol. 11 (2020), no. 4, pp. 657–689

DOI 10.4171/QT/143