[Connexion de Knizhnik–Zamolodchikov et formalité pour la bigèbre de Lie de Goldman–Turaev]
Soit Σ une variété orientable de dimension 2, de genre g et avec n composants de bord. L'espace a une structure de bigèbre de Lie de Goldman–Turaev définie par les intersections et les autointersections des courbes sur Σ. La bigèbre de Lie graduée associée (par rapport à la filtration naturelle) est décrite par l'espace des mots cycliques en . En genre zéro, l'isomorphisme entre ces deux bigèbres de Lie a été établi dans [13] en utilisant l'intégrale de Kontsevich, et dans [2] en utilisant les solutions du problème de Kashiwara–Vergne.
Dans cette note, nous donnons une démonstration élémentaire de cet isomorphisme sur . Notre démonstration utilise la connexion de Knizhnik–Zamolodchikov sur . Nous montrons que cet isomorphisme dépend naturellement de la structure complexe sur Σ. La preuve de l'isomorphisme pour le crochet de Lie est une version d'un résultat classique de Hitchin [9]. D'une manière surprenante, un argument similaire s'applique également au cocrochet.
De plus, nous montrons que l'isomorphisme de formalité construit dans cette note coïncide avec l'isomorphisme défini dans [2] si on choisit la solution du problème de Kashiwara–Vergne, qui correspond à l'associateur de Knizhnik–Zamolodchikov.
For an oriented 2-dimensional manifold Σ of genus g with n boundary components, the space carries the Goldman–Turaev Lie bialgebra structure defined in terms of intersections and self-intersections of curves. Its associated graded Lie bialgebra (under the natural filtration) is described by cyclic words in and carries the structure of a necklace Schedler Lie bialgebra. The isomorphism between these two structures in genus zero has been established in [13] using Kontsevich integrals and in [2] using solutions of the Kashiwara–Vergne problem.
In this note, we give an elementary proof of this isomorphism over . It uses the Knizhnik–Zamolodchikov connection on . We show that the isomorphism naturally depends on the complex structure on the surface. The proof of the isomorphism for Lie brackets is a version of the classical result by Hitchin [9]. Surprisingly, it turns out that a similar proof applies to cobrackets.
Furthermore, we show that the formality isomorphism constructed in this note coincides with the one defined in [2] if one uses the solution of the Kashiwara–Vergne problem corresponding to the Knizhnik–Zamolodchikov associator.
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@article{CRMATH_2017__355_11_1138_0,
author = {Alekseev, Anton and Naef, Florian},
title = {Goldman{\textendash}Turaev formality from the {Knizhnik{\textendash}Zamolodchikov} connection},
journal = {Comptes Rendus. Math\'ematique},
pages = {1138--1147},
publisher = {Elsevier},
volume = {355},
number = {11},
year = {2017},
doi = {10.1016/j.crma.2017.10.013},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2017.10.013/}
}
TY - JOUR AU - Alekseev, Anton AU - Naef, Florian TI - Goldman–Turaev formality from the Knizhnik–Zamolodchikov connection JO - Comptes Rendus. Mathématique PY - 2017 SP - 1138 EP - 1147 VL - 355 IS - 11 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2017.10.013/ DO - 10.1016/j.crma.2017.10.013 LA - en ID - CRMATH_2017__355_11_1138_0 ER -
%0 Journal Article %A Alekseev, Anton %A Naef, Florian %T Goldman–Turaev formality from the Knizhnik–Zamolodchikov connection %J Comptes Rendus. Mathématique %D 2017 %P 1138-1147 %V 355 %N 11 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2017.10.013/ %R 10.1016/j.crma.2017.10.013 %G en %F CRMATH_2017__355_11_1138_0
Alekseev, Anton; Naef, Florian. Goldman–Turaev formality from the Knizhnik–Zamolodchikov connection. Comptes Rendus. Mathématique, Tome 355 (2017) no. 11, pp. 1138-1147. doi : 10.1016/j.crma.2017.10.013. http://www.numdam.org/articles/10.1016/j.crma.2017.10.013/
[1] Drinfeld associators, braid groups and explicit solutions of the Kashiwara–Vergne equations, Publ. Math. Inst. Hautes Études Sci., Volume 112 (2010) no. 1, pp. 143-189
[2] The Goldman–Turaev Lie bialgebra in genus zero and the Kashiwara–Vergne problem (preprint) | arXiv
[3] The hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces, Duke Math. J., Volume 93 (1998) no. 3, pp. 575-595
[4] Necklace lie algebras and noncommutative symplectic geometry, Math. Z., Volume 240 (2002) no. 1, pp. 141-167
[5] On quasitriangular quasi-Hopf algebras and on a group that is closely connected with , Algebra Anal., Volume 2 (1990) no. 4, pp. 149-181
[6] Flat connections and polyubles, Theor. Math. Phys., Volume 95 (1993) no. 2, pp. 526-534
[7] Non-commutative symplectic geometry, quiver varieties, and operads, Math. Res. Lett., Volume 8 (2001) no. 3, pp. 377-400
[8] Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math., Volume 85 (1986), pp. 263-302
[9] , Springer, Netherlands (1997), pp. 69-112
[10] Harmonic Magnus expansion on the universal family of Riemann surfaces (preprint) | arXiv
[11] A regular homotopy version of the Goldman–Turaev Lie bialgebra, the Enomoto–Satoh traces and the divergence cocycle in the Kashiwara–Vergne problem, RIMS Kôkyûroku Bessatsu, Volume 1936 (2015), pp. 137-141 | arXiv
[12] Formal (non)commutative symplectic geometry, 1990–1992, Birkhäuser Boston, Boston, MA, USA (1993), pp. 173-187
[13] Formal descriptions of Turaev's loop operations, Quantum Topol. (2017) (in press, also available at) | arXiv
[14] A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver, Int. Math. Res. Not., Volume 2005 (2005) no. 12, pp. 725-760
[15] Intersections of loops in two-dimensional manifolds, Mat. Sb., Volume 106 (1978) no. 148, pp. 566-588 (English translation: Math. USSR Sb., 35, 1979, pp. 229-250)
[16] Double Poisson algebras, Trans. Amer. Math. Soc., Volume 360 (2008), pp. 5711-5799
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