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

@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)

Takashi Ichikawa Higher genus polylogarithms on families of Riemann surfaces, Nuclear Physics. B, Volume 1013 (2025), p. 16 (Id/No 116836) | DOI:10.1016/j.nuclphysb.2025.116836 | Zbl:8020259
Sergei A. Merkulov From gravity to string topology, Letters in Mathematical Physics, Volume 113 (2023) no. 3, p. 23 (Id/No 62) | DOI:10.1007/s11005-023-01686-8 | Zbl:1535.18034
Matteo Felder; Florian Naef; Thomas Willwacher Stable cohomology of graph complexes, Selecta Mathematica. New Series, Volume 29 (2023) no. 2, p. 72 (Id/No 23) | DOI:10.1007/s00029-023-00830-5 | Zbl:1518.81051
Sergei Merkulov Grothendieck-Teichmüller group, operads and graph complexes: a survey, Integrability, Quantization, and Geometry, Volume 103.2 (2021), p. 383 | DOI:10.1090/pspum/103.2/01863
Florian Naef; Yuting Qin The elliptic Kashiwara-Vergne Lie algebra in low weights, Journal of Lie Theory, Volume 31 (2021) no. 2, pp. 583-598 | Zbl:1482.17017
Richard Hain Hodge theory of the Turaev cobracket and the Kashiwara-Vergne problem, Journal of the European Mathematical Society (JEMS), Volume 23 (2021) no. 12, pp. 3889-3933 | DOI:10.4171/jems/1088 | Zbl:1483.14015
Élise Raphael; Leila Schneps On the elliptic Kashiwara-Vergne Lie algebra, Galois covers, Grothendieck-Teichmüller theory and dessins d'enfants. Interactions between geometry, topology, number theory and algebra. Proceedings from the London Mathematical Society Midlands regional meeting and workshop, Leicester, UK, June 4–7, 2018, Cham: Springer, 2020, pp. 227-240 | DOI:10.1007/978-3-030-51795-3_11 | Zbl:1483.17007
Richard Hain Hodge theory of the Goldman bracket, Geometry Topology, Volume 24 (2020) no. 4, pp. 1841-1906 | DOI:10.2140/gt.2020.24.1841 | Zbl:1470.14017
Johan Leray Shifted double Lie-Rinehart algebras, Theory and Applications of Categories, Volume 35 (2020), pp. 594-621 | Zbl:1447.18003
Anton Alekseev; Nariya Kawazumi; Yusuke Kuno; Florian Naef The Goldman-Turaev Lie bialgebra in genus zero and the Kashiwara-Vergne problem, Advances in Mathematics, Volume 326 (2018), pp. 1-53 | DOI:10.1016/j.aim.2017.12.005 | Zbl:1422.57053
Shinji Fukuhara; Nariya Kawazumi; Yusuke Kuno Self-intersections of curves on a surface and Bernoulli numbers, Osaka Journal of Mathematics, Volume 55 (2018) no. 4, pp. 761-768 | Zbl:1440.11026
Nariya Kawazumi The mapping class group orbits in the framings of compact surfaces, The Quarterly Journal of Mathematics, Volume 69 (2018) no. 4, pp. 1287-1302 | DOI:10.1093/qmath/hay024 | Zbl:1432.57029

Cité par 12 documents. Sources : Crossref, zbMATH

Commentaires - Politique

Il n'y a aucun commentaire pour cet article. Soyez le premier à écrire un commentaire !

Publier un nouveau commentaire:

Publier une nouvelle réponse: