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.

中文翻译：

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Goldman-Turaev 形式意味着 Kashiwara-Vergne

令 $\Sigma$ 是一个具有非空边界的紧凑连通定向二维流形。在我们之前的工作中，我们已经证明，对于自由李代数的自同构 $F \in {\rm Aut}(L)$ 的广义（更高属）Kashiwara-Vergne 方程的解意味着 Goldman-Turaev 之间的同构李双代数 $\mathfrak{g}(\Sigma)$ 及其相关的分级 ${\rm gr}\, \mathfrak{g}(\Sigma)$。在本文中，我们证明了反过来：如果 $F$ 诱导同构 $\mathfrak{g}(\Sigma) \cong {\rm gr} \, \mathfrak{g}(\Sigma)$，那么它满足Kashiwara-Vergne 方程直到共轭。作为我们结果的应用，我们计算 Kirillov-Kostant-Souriau 双括号的一阶非交换泊松上同调。