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Double shuffle relation for associators. (English) Zbl 1321.11088

Summary: It is proved that Drinfel’d’s pentagon equation implies the generalized double shuffle relation. As a corollary, an embedding from the Grothendieck-Teichmüller group \(\text{GRT}_1\) into Racinet’s double shuffle group \(\text{DMR}_0\) is obtained, which settles the project of Deligne-Terasoma. It is also proved that the gamma factorization formula follows from the generalized double shuffle relation.

MSC:

11M32 Multiple Dirichlet series and zeta functions and multizeta values
14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
11G55 Polylogarithms and relations with \(K\)-theory
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
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References:

[1] A. Alekseev, B. Enriquez, and C. Torossian, ”Drinfeld associators, braid groups and explicit solutions of the Kashiwara-Vergne equations,” Publ. Math. Inst. Hautes Études Sci., vol. 112, pp. 143-189, 2010. · Zbl 1238.17008 · doi:10.1007/s10240-010-0029-4
[2] A. Alekseev and C. Torossian, The Kashiwara-Vergne conjecture and Drinfeld’s associators. · Zbl 1243.22009 · doi:10.4007/annals.2012.175.2.1
[3] Y. André, Une Introduction aux Motifs (Motifs Purs, Motifs Mixtes, Périodes), Paris: Société Mathématique de France, 2004, vol. 17. · Zbl 1060.14001
[4] F. C. S. Brown, ”Multiple zeta values and periods of moduli spaces \(\overline{\mathfrakM}_{0,n}\),” Ann. Sci. École Norm. Supér., vol. 42, iss. 3, pp. 371-489, 2009. · Zbl 1216.11079
[5] A. Besser and H. Furusho, ”The double shuffle relations for \(p\)-adic multiple zeta values,” in Primes and Knots, Providence, RI: Amer. Math. Soc., 2006, vol. 416, pp. 9-29. · Zbl 1172.11043
[6] K. T. Chen, ”Iterated path integrals,” Bull. Amer. Math. Soc., vol. 83, iss. 5, pp. 831-879, 1977. · Zbl 0389.58001 · doi:10.1090/S0002-9904-1977-14320-6
[7] P. Deligne, ”Le groupe fondamental de la droite projective moins trois points,” in Galois Groups over \({\mathbf Q}\), New York, 1989, pp. 79-297. · Zbl 0742.14022
[8] P. Deligne and A. B. Goncharov, ”Groupes fondamentaux motiviques de Tate mixte,” Ann. Sci. École Norm. Sup., vol. 38, iss. 1, pp. 1-56, 2005. · Zbl 1084.14024 · doi:10.1016/j.ansens.2004.11.001
[9] P. Deligne and T. Terasoma, Harmonic shuffle relation for associators.
[10] V. G. Drinfel\('\)d, ”On quasitriangular quasi-Hopf algebras and on a group that is closely connected with \({ Gal}(\overline{\mathbf Q}/{\mathbf Q})\),” Algebra i Analiz, vol. 2, iss. 4, pp. 149-181, 1990. · Zbl 0728.16021
[11] L. Euler, ”Meditationes circa singulare serierum genus,” Novi Comm. Acad. Sci. Petropol., vol. 20, pp. 140-186, 1776.
[12] H. Furusho, ”The multiple zeta value algebra and the stable derivation algebra,” Publ. Res. Inst. Math. Sci., vol. 39, iss. 4, pp. 695-720, 2003. · Zbl 1115.11055 · doi:10.2977/prims/1145476044
[13] H. Furusho, ”Pentagon and hexagon equations,” Ann. of Math., vol. 171, iss. 1, pp. 545-556, 2010. · Zbl 1257.17019 · doi:10.4007/annals.2010.171.545
[14] A. B. Goncharov, Periods and mixed motives, 2002.
[15] A. Grothendieck, Esquisse d’un programme (1984), reproduced in Geometric Galois Actions, 1, Cambridge: Cambridge Univ. Press, 1997, vol. 242. · Zbl 0901.14001
[16] M. E. Hoffman, ”The algebra of multiple harmonic series,” J. Algebra, vol. 194, iss. 2, pp. 477-495, 1997. · Zbl 0881.11067 · doi:10.1006/jabr.1997.7127
[17] Y. Ihara, ”On beta and gamma functions associated with the Grothendieck-Teichmüller groups,” in Aspects of Galois Theory, Cambridge, 1999, pp. 144-179. · Zbl 1046.14010
[18] K. Ihara, M. Kaneko, and D. Zagier, ”Derivation and double shuffle relations for multiple zeta values,” Compos. Math., vol. 142, iss. 2, pp. 307-338, 2006. · Zbl 1186.11053 · doi:10.1112/S0010437X0500182X
[19] T. T. Q. Le and J. Murakami, ”Kontsevich’s integral for the Kauffman polynomial,” Nagoya Math. J., vol. 142, pp. 39-65, 1996. · Zbl 0866.57008
[20] G. Racinet, ”Doubles mélanges des polylogarithmes multiples aux racines de l’unité,” Publ. Math. Inst. Hautes Études Sci., iss. 95, pp. 185-231, 2002. · Zbl 1050.11066 · doi:10.1007/s102400200004
[21] T. Terasoma, ”Geometry of multiple zeta values,” in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 627-635. · Zbl 1097.14009
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