Higher genus Kashiwara–Vergne problems and the Goldman–Turaev Lie bialgebra

Anton Alekseev; Nariya Kawazumi; Yusuke Kuno; Florian Naef

doi:10.1016/j.crma.2016.12.007

Lie algebras/Topology

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]

Présenté par : Michèle Vergne

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

Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 123-127.

Résumé
Abstract

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 $gr g^{(g, n + 1)}$ . 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 $g \geq 1$ , $n = 0$ 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.

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 $gr g^{(g, n + 1)}$ . For $g = 0$ , a similar result was obtained by G. Massuyeau using the Kontsevich integral. For $g \geq 1$ , $n = 0$ , our results imply that the obstruction to surjectivity of the Johnson homomorphism provided by the Turaev cobracket is equivalent to the Enomoto–Satoh obstruction.

Reçu le : 2016-12-02
Accepté le : 2016-12-20
Publié le : 2016-12-28

DOI : 10.1016/j.crma.2016.12.007

Affiliations des auteurs :

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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     author = {Anton Alekseev and Nariya Kawazumi and Yusuke Kuno and Florian Naef},
     title = {Higher genus {Kashiwara{\textendash}Vergne} problems and the {Goldman{\textendash}Turaev} {Lie} bialgebra},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {123--127},
     publisher = {Elsevier},
     volume = {355},
     number = {2},
     year = {2017},
     doi = {10.1016/j.crma.2016.12.007},
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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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