Drinfeld associators, Braid groups and explicit solutions of the Kashiwara–Vergne equations
Publications Mathématiques de l'IHÉS, Volume 112 (2010), p. 143-189

The Kashiwara–Vergne (KV) conjecture states the existence of solutions of a pair of equations related with the Campbell–Baker–Hausdorff series. It was solved by Meinrenken and the first author over ℝ, and in a formal version, by two of the authors over a field of characteristic 0. In this paper, we give a simple and explicit formula for a map from the set of Drinfeld associators to the set of solutions of the formal KV equations. Both sets are torsors under the actions of prounipotent groups, and we show that this map is a morphism of torsors. When specialized to the KZ associator, our construction yields a solution over ℝ of the original KV conjecture.

@article{PMIHES_2010__112__143_0,
     author = {Alekseev, A. and Enriquez, Benjamin and Torossian, C.},
     title = {Drinfeld associators, Braid groups and explicit solutions of the Kashiwara--Vergne equations},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     publisher = {Springer-Verlag},
     volume = {112},
     year = {2010},
     pages = {143-189},
     doi = {10.1007/s10240-010-0029-4},
     zbl = {1238.17008},
     mrnumber = {2737979},
     language = {en},
     url = {http://www.numdam.org/item/PMIHES_2010__112__143_0}
}
Alekseev, A.; Enriquez, B.; Torossian, C. Drinfeld associators, Braid groups and explicit solutions of the Kashiwara–Vergne equations. Publications Mathématiques de l'IHÉS, Volume 112 (2010) pp. 143-189. doi : 10.1007/s10240-010-0029-4. http://www.numdam.org/item/PMIHES_2010__112__143_0/

[AM1] A. Alekseev, E. Meinrenken, Poisson geometry and the Kashiwara–Vergne conjecture, C. R. Math. Acad. Sci. Paris 335 (2002), p. 723-728 | MR 1951805 | Zbl 1057.17004

[AM2] A. Alekseev, E. Meinrenken, On the Kashiwara–Vergne conjecture, Invent. Math. 164 (2006), p. 615-634 | MR 2221133 | Zbl 1096.22007

[AT1] A. Alekseev, C. Torossian, On triviality of the Kashiwara-Vergne problem for quadratic Lie algebras, C. R. Math. Acad. Sci. Paris 347 (2009), p. 1231-1236 | MR 2561029 | Zbl 1228.17015

[AT2] A. Alekseev and C. Torossian, The Kashiwara–Vergne conjecture and Drinfeld’s associators, arXiv:0802.4300 . | MR 2877064 | Zbl 1243.22009

[AL] A. Amitsur, J. Levitsky, Minimal identities for algebras, Proc. Am. Math. Soc. 1 (1950), p. 449-463 | MR 36751 | Zbl 0040.01101

[AST] M. Andler, S. Sahi, C. Torossian, Convolution of invariant distributions: proof of the Kashiwara–Vergne conjecture, Lett. Math. Phys. 69 (2004), p. 177-203 | MR 2104443 | Zbl 1059.22008

[B] D. Bar-Natan, On associators and the Grothendieck–Teichmüller group. I, Sel. Math. (N.S.) 4 (1998), p. 183-212 | MR 1669949 | Zbl 0974.16028

[Bo] M. Boyarchenko, Drinfeld associators and the Campbell–Hausdorff formula, Notes available at http://www.math.uchicago.edu/~mitya/associators-ch.pdf .

[Bk] N. Bourbaki, Élements de mathématique. XXVI. Groupes et algèbres de Lie. Chapitre 1: Algèbres de Lie, Actualités Sci. Ind. 1285 (1960), Hermann, Paris | MR 132805 | Zbl 0199.35203

[DT] P. Deligne and T. Terasoma, Harmonic shuffle relation for associators, www2.lifl.fr/mzv2005/DOC/Terasoma/lille_terasoma.pdf .

[Dr] V. Drinfeld, On quasitriangular quasi-Hopf algebras and a group closely connected with Gal( ¯/) , Leningr. Math. J. 2 (1991), p. 829-860 | MR 1080203 | Zbl 0728.16021

[E] B. Enriquez, On the Drinfeld generators of 𝔤𝔯𝔱 1 (𝐤) and Γ-functions for associators, Math. Res. Lett. 13 (2006), p. 231-243 | MR 2231114 | Zbl 1109.17004

[EG] B. Enriquez and F. Gavarini, A formula for the logarithm of the KZ associator, SIGMA, 2 (2006), paper 080, in memory of V. Kuznetsov. | MR 2264896 | Zbl 1188.17004

[HM] N. Habegger, G. Masbaum, The Kontsevich integral and Milnor’s invariants, Topology 39 (2000), p. 1253-1289 | MR 1783857 | Zbl 0964.57011

[Ih] Y. Ihara, On Beta and Gamma Functions Associated with the Grothendieck-Teichmüller Group, London Math. Soc. Lecture Note Ser. 256 (1999), Cambridge Univ. Press, Cambridge | MR 1708605 | Zbl 1046.14009

[JS] A. Joyal, R. Street, Braided tensor categories, Adv. Math. 102 (1993), p. 20-78 | MR 1250465 | Zbl 0817.18007

[KV] M. Kashiwara, M. Vergne, The Campbell–Hausdorff formula and invariant hyperfunctions, Invent. Math. 47 (1978), p. 249-272 | MR 492078 | Zbl 0404.22012

[LS] P. Lochak and L. Schneps, Every element of GT ^ is a twist, Preprint.

[Mag] W. Magnus, Über Automorphismen von Fundamentalgruppen berandeter Flächen, Math. Ann. 109 (1934), p. 617-646 | JFM 60.0091.01 | MR 1512913

[R] F. Rouvière, Démonstration de la conjecture de Kashiwara–Vergne pour l’algèbre 𝔰𝔩 2 , C. R. Acad. Sci. Paris, Sér. I, Math. 292 (1981), p. 657-660 | MR 618878 | Zbl 0467.22011

[Su] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), p. 269-331 | Numdam | MR 646078 | Zbl 0374.57002

[T1] C. Torossian, Sur la conjecture combinatoire de Kashiwara-Vergne, J. Lie Theory 12 (2002), p. 597-616 | MR 1923789 | Zbl 1012.17002

[T2] C. Torossian, La conjecture de Kashiwara-Vergne [d’après Alekseev-Meinrenken], in: Séminaire Bourbaki (volume 2006/07), exposé no. 980, Astérisque 317 (2008), | MR 2487742 | Zbl 1175.22008

[V] M. Vergne, Le centre de l’algèbre enveloppante et la formule de Campbell–Hausdorff, C. R. Acad. Sci. Paris, Sér. I, Math. 329 (1999), p. 767-772 | MR 1724537 | Zbl 0989.17007