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Comptes Rendus Mathématique
Volume 355, n° 2
pages 123-127 (février 2017)
Doi : 10.1016/j.crma.2016.12.007
Received : 2 December 2016 ;  accepted : 20 December 2016
Higher genus Kashiwara–Vergne problems and the Goldman–Turaev Lie bialgebra
Problèmes de Kashiwara–Vergne en genre supérieur et la bigèbre de Lie de Goldman–Turaev
 

Anton Alekseev a , Nariya Kawazumi b , Yusuke Kuno c , Florian Naef a
a Department of Mathematics, University of Geneva, 2–4, rue du Lièvre, 1211 Genève, Switzerland 
b Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan 
c Department of Mathematics, Tsuda College, 2-1-1 Tsuda-machi, Kodaira-shi, Tokyo 187-8577, Japan 

Abstract

We define a family   of Kashiwara–Vergne problems associated with compact connected oriented 2-manifolds of genus g with   boundary components. The problem   is the classical Kashiwara–Vergne problem from Lie theory. We show the existence of solutions to   for arbitrary g and n . The key point is the solution to   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  . In more detail, we show that every solution to   induces a Lie bialgebra isomorphism between   and its associated graded  . For  , a similar result was obtained by G. Massuyeau using the Kontsevich integral. For  ,  , our results imply that the obstruction to surjectivity of the Johnson homomorphism provided by the Turaev cobracket is equivalent to the Enomoto–Satoh obstruction.

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Résumé

Nous définissons une famille   de problèmes de Kashiwara–Vergne associés aux variétés compactes, connexes et orientées de dimension 2, de genre g avec   composantes du bord. Le problème   est un problème classique de la théorie de Lie. Nous montrons l'existence de solutions de   pour tous g et n . Le point crucial est la résolution de  , 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  . Nous montrons que chaque solution de   induit un isomorphisme de bigèbres de Lie entre   et sa graduée associée  . Dans le cas où  , un résultat similaire a été obtenu par G. Massuyeau en utilisant l'intégrale de Kontsevich. Dans le cas de  ,   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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