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Goldman-Turaev formality implies Kashiwara-Vergne | Anton Alekseev
; Nariya Kawazumi
; Yusuke Kuno
; Florian Naef
; | Date: |
4 Dec 2018 | 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 $F
in {
m Aut}(L)$ 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 | Services: | Forum | Review | PDF | Favorites |
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