Goldman-Turaev formality implies Kashiwara-Vergne

Tuesday, 4 December, 2018

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Let Σ 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}(Σ) and its associated graded {\rm gr}\, \mathfrak{g}(Σ). In this paper, we prove the converse: if F induces an isomorphism \mathfrak{g}(Σ) \cong {\rm gr} \, \mathfrak{g}(Σ), 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.


Anton Alekseev
Nariya Kawazumi
Yusuke Kuno
Florian Naef