# Goldman-Turaev formality implies Kashiwara-Vergne

## TL;DR

This paper proves that Goldman-Turaev formality implies solutions to Kashiwara-Vergne equations, establishing a converse to previous results and applying it to compute non-commutative Poisson cohomology.

## Contribution

It demonstrates that Goldman-Turaev formality implies Kashiwara-Vergne solutions, providing a new characterization of conjugacy classes in free Lie algebras.

## Key findings

- Proves the converse implication of Goldman-Turaev formality and Kashiwara-Vergne equations.
- Computes degree one non-commutative Poisson cohomology of the Kirillov-Kostant-Souriau double bracket.
- Introduces a novel characterization of conjugacy classes in free Lie algebras using cyclic words.

## Abstract

Let $\Sigma$ be a compact connected oriented 2-dimensional manifold with non-empty boundary. In our previous work, we have shown that the solution of generalized (higher genus) Kashiwara-Vergne equations for an automorphism $F \in {\rm Aut}(L)$ of a free Lie algebra implies an isomorphism between the Goldman-Turaev Lie bialgebra $\mathfrak{g}(\Sigma)$ and its associated graded ${\rm gr}\, \mathfrak{g}(\Sigma)$. In this paper, we prove the converse: if $F$ induces an isomorphism $\mathfrak{g}(\Sigma) \cong {\rm gr} \, \mathfrak{g}(\Sigma)$, 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.