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29 March 2024
 
  » arxiv » 1812.1159

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Goldman-Turaev formality implies Kashiwara-Vergne
Anton Alekseev ; Nariya Kawazumi ; Yusuke Kuno ; Florian Naef ;
Date 4 Dec 2018
AbstractLet $Sigma$ 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 $F in { m Aut}(L)$ of a free Lie algebra implies an isomorphism between the Goldman-Turaev Lie bialgebra $mathfrak{g}(Sigma)$ and its associated graded ${ m gr}, mathfrak{g}(Sigma)$. In this paper, we prove the converse: if $F$ induces an isomorphism $mathfrak{g}(Sigma) cong { m gr} , mathfrak{g}(Sigma)$, 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.
Source arXiv, 1812.1159
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