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.

Abstract (VO)
Résumé

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.

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.

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

@article{CRMATH_2017__355_2_123_0,
     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},
     language = {en},
}

TY  - JOUR
AU  - Anton Alekseev
AU  - Nariya Kawazumi
AU  - Yusuke Kuno
AU  - Florian Naef
TI  - Higher genus Kashiwara–Vergne problems and the Goldman–Turaev Lie bialgebra
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 123
EP  - 127
VL  - 355
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crma.2016.12.007
LA  - en
ID  - CRMATH_2017__355_2_123_0
ER  -

%0 Journal Article
%A Anton Alekseev
%A Nariya Kawazumi
%A Yusuke Kuno
%A Florian Naef
%T Higher genus Kashiwara–Vergne problems and the Goldman–Turaev Lie bialgebra
%J Comptes Rendus. Mathématique
%D 2017
%P 123-127
%V 355
%N 2
%I Elsevier
%R 10.1016/j.crma.2016.12.007
%G en
%F CRMATH_2017__355_2_123_0

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/

Bibliographie
Cité par

[1] A. Alekseev; C. Torossian The Kashiwara–Vergne conjecture and Drinfeld's associators, Ann. Math., Volume 175 (2012) no. 2, pp. 415-463

[2] N. Enomoto; T. Satoh New series in the Johnson cokernels of the mapping class groups of surfaces, Algebraic Geom. Topol., Volume 14 (2014), pp. 627-669

[3] B. Enriquez Elliptic associators, Sel. Math., Volume 20 (2014), pp. 491-584

[4] W. Goldman Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math., Volume 85 (1986), pp. 263-302

[5] M. Kashiwara; M. Vergne The Campbell–Hausdorff formula and invariant hyperfunctions, Invent. Math., Volume 47 (1978), pp. 249-271

[6] N. Kawazumi; Y. Kuno Intersections of curves on surfaces and their applications to mapping class groups, Ann. Inst. Fourier (Grenoble), Volume 65 (2015), pp. 2711-2762

[7] N. Kawazumi; Y. Kuno The Goldman–Turaev Lie bialgebra and the Johnson homomorphisms (A. Papadopoulos, ed.), Handbook of Teichmuller Theory, Volume V, EMS Publishing House, Zurich, Switzerland, 2016, pp. 97-165

[8] G. Massuyeau Formal descriptions of Turaev's loop operations (preprint) | arXiv

[9] G. Massuyeau; V.G. Turaev Quasi-Poisson structures on representation spaces of surfaces, Int. Math. Res. Not., Volume 2014 (2014) no. 1, pp. 1-64

[10] F. Naef Poisson brackets in Kontsevich's “Lie World” (preprint) | arXiv

[11] L. Schneps, Talk at the conference “Homotopical Algebra, Operads and Grothendieck–Teichmüller Groups”, Nice, France, 9–12 September 2014.

[12] L. Schneps, E. Raphael, On linearised and elliptic versions of the Kashiwara–Vergne Lie algebra, in preparation.

[13] V.G. Turaev Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. Éc. Norm. Supér. (4), Volume 24 (1991), pp. 635-704

[14] M. van den Bergh Double Poisson algebras, Trans. Amer. Math. Soc., Volume 360 (2008), pp. 5711-5799

[15] T. Willwacher Configuration spaces of points and their rational homotopy theory, mini-course given in Les Diablerets (Switzerland) http://drorbn.net/dbnvp/LD16_Willwacher-1.php (on 31 August–1 September 2016. Video available at)

Cité par Sources :

Commentaires - Politique