Comptes Rendus
Lie algebras/Topology
Higher genus Kashiwara–Vergne problems and the Goldman–Turaev Lie bialgebra
Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 123-127.

We define a family KV(g,n+1) of Kashiwara–Vergne problems associated with compact connected oriented 2-manifolds of genus g with n+1 boundary components. The problem KV(0,3) is the classical Kashiwara–Vergne problem from Lie theory. We show the existence of solutions to KV(g,n+1) for arbitrary g and n. The key point is the solution to KV(1,1) based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman–Turaev Lie bialgebra g(g,n+1). In more detail, we show that every solution to KV(g,n+1) induces a Lie bialgebra isomorphism between g(g,n+1) and its associated graded grg(g,n+1). For g=0, a similar result was obtained by G. Massuyeau using the Kontsevich integral. For g1, n=0, our results imply that the obstruction to surjectivity of the Johnson homomorphism provided by the Turaev cobracket is equivalent to the Enomoto–Satoh obstruction.

Nous définissons une famille KV(g,n+1) de problèmes de Kashiwara–Vergne associés aux variétés compactes, connexes et orientées de dimension 2, de genre g avec n+1 composantes du bord. Le problème KV(0,3) est un problème classique de la théorie de Lie. Nous montrons l'existence de solutions de KV(g,n+1) pour tous g et n. Le point crucial est la résolution de KV(1,1), qui est basée sur les résultats de B. Enriquez sur les associateurs elliptiques. Notre construction est motivée par la question de formalité de la bigèbre de Lie de Goldman–Turaev g(g,n+1). Nous montrons que chaque solution de KV(g,n+1) induit un isomorphisme de bigèbres de Lie entre g(g,n+1) et sa graduée associée grg(g,n+1). Dans le cas où g=0, un résultat similaire a été obtenu par G. Massuyeau en utilisant l'intégrale de Kontsevich. Dans le cas de g1, n=0 nos résultats impliquent que l'obstacle à la surjectivité de l'homomorphisme de Johnson définie par le co-crochet de Turaev est équivalent à l'obstacle de Enomoto–Satoh.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2016.12.007

Anton Alekseev 1; Nariya Kawazumi 2; Yusuke Kuno 3; Florian Naef 1

1 Department of Mathematics, University of Geneva, 2–4, rue du Lièvre, 1211 Genève, Switzerland
2 Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
3 Department of Mathematics, Tsuda College, 2-1-1 Tsuda-machi, Kodaira-shi, Tokyo 187-8577, Japan
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Anton Alekseev; Nariya Kawazumi; Yusuke Kuno; Florian Naef. Higher genus Kashiwara–Vergne problems and the Goldman–Turaev Lie bialgebra. Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 123-127. doi : 10.1016/j.crma.2016.12.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.12.007/

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