Double shuffle relation for associators

Abstract

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 ${\rm GRT}_1$ into Racinet’s double shuffle group ${\rm DMR}_0$ 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.

  • [AET] Go to document 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.
    @article {AET, MRKEY = {2737979},
      AUTHOR = {Alekseev, A. and Enriquez, B. and Torossian, C.},
      TITLE = {Drinfeld associators, braid groups and explicit solutions of the {K}ashiwara-{V}ergne equations},
      JOURNAL = {Publ. Math. Inst. Hautes Études Sci.},
      FJOURNAL = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
      VOLUME = {112},
      YEAR = {2010},
      PAGES = {143--189},
      ISSN = {0073-8301},
      MRCLASS = {17B35 (16S30)},
      MRNUMBER = {2737979},
      ZBLNUMBER={pre05873393},
      DOI = {10.1007/s10240-010-0029-4},
      }
  • [AT] A. Alekseev and C. Torossian, The Kashiwara-Vergne conjecture and Drinfeld’s associators.
    @misc{AT,
      author = {Alekseev, A. and Torossian, C.},
      TITLE = {The {K}ashiwara-{V}ergne conjecture and {D}rinfeld's associators},
      NOTE={preprint},
      ARXIV={0802.4300},
      SORTYEAR={2012},
      }
  • [A] Y. André, Une Introduction aux Motifs (Motifs Purs, Motifs Mixtes, Périodes), Paris: Société Mathématique de France, 2004, vol. 17.
    @book {A, MRKEY = {2115000},
      AUTHOR = {Andr{é},
      Yves},
      TITLE = {Une Introduction aux Motifs (Motifs Purs, Motifs Mixtes, Périodes)},
      SERIES = {Panoramas et Synthèses [Panoramas and Syntheses]},
      VOLUME = {17},
      PUBLISHER = {Société Mathématique de France},
      ADDRESS = {Paris},
      YEAR = {2004},
      PAGES = {xii+261},
      ISBN = {2-85629-164-3},
      MRCLASS = {14F42 (11J91 14C25 19E15)},
      MRNUMBER = {2115000},
      MRREVIEWER = {Luca Barbieri Viale},
      }
  • [B] F. C. S. Brown, "Multiple zeta values and periods of moduli spaces $\overline{\mathfrak{M}}_{0,n}$," Ann. Sci. École Norm. Supér., vol. 42, iss. 3, pp. 371-489, 2009.
    @article {B, MRKEY = {2543329},
      AUTHOR = {Brown, Francis C. S.},
      TITLE = {Multiple zeta values and periods of moduli spaces {$\overline{\mathfrak{M}}\sb {0,n}$}},
      JOURNAL = {Ann. Sci. École Norm. Supér.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {42},
      YEAR = {2009},
      NUMBER = {3},
      PAGES = {371--489},
      ISSN = {0012-9593},
      MRCLASS = {32G20 (11G55 11M32 14F42 14H10 19F27 33B30)},
      MRNUMBER = {2543329},
      MRREVIEWER = {Wadim Zudilin},
      ZBLNUMBER = {05620646},
      }
  • [BF] 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.
    @incollection {BF, MRKEY = {2276133},
      AUTHOR = {Besser, Amnon and Furusho, Hidekazu},
      TITLE = {The double shuffle relations for {$p$}-adic multiple zeta values},
      BOOKTITLE = {Primes and Knots},
      SERIES = {Contemp. Math.},
      VOLUME = {416},
      PAGES = {9--29},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2006},
      MRCLASS = {11G55 (11M41 11S80)},
      MRNUMBER = {2276133},
      MRREVIEWER = {Jan Nekov{á}{\v{r}}},
      ZBLNUMBER = {1172.11043},
      }
  • [C] Go to document K. T. Chen, "Iterated path integrals," Bull. Amer. Math. Soc., vol. 83, iss. 5, pp. 831-879, 1977.
    @article {C, MRKEY = {0454968},
      AUTHOR = {Chen, Kuo Tsai},
      TITLE = {Iterated path integrals},
      JOURNAL = {Bull. Amer. Math. Soc.},
      FJOURNAL = {Bulletin of the American Mathematical Society},
      VOLUME = {83},
      YEAR = {1977},
      NUMBER = {5},
      PAGES = {831--879},
      ISSN = {0002-9904},
      MRCLASS = {55D35 (58A99)},
      MRNUMBER = {0454968},
      MRREVIEWER = {Jean-Michel Lemaire},
      DOI = {10.1090/S0002-9904-1977-14320-6},
      ZBLNUMBER = {0389.58001},
      }
  • [De] P. Deligne, "Le groupe fondamental de la droite projective moins trois points," in Galois Groups over ${\bf Q}$, New York, 1989, pp. 79-297.
    @inproceedings {De, MRKEY = {1012168},
      AUTHOR = {Deligne, P.},
      TITLE = {Le groupe fondamental de la droite projective moins trois points},
      BOOKTITLE = {Galois Groups over {${\bf Q}$}},
      VENUE={{B}erkeley, {CA},
      1987},
      SERIES = {Math. Sci. Res. Inst. Publ.},
      VOLUME = {16},
      PAGES = {79--297},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1989},
      MRCLASS = {14G25 (11G35 11M06 11R70 14F35 19E99 19F27)},
      MRNUMBER = {1012168},
      MRREVIEWER = {James Milne},
      ZBLNUMBER = {0742.14022},
      }
  • [DG] Go to document 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.
    @article {DG, MRKEY = {2136480},
      AUTHOR = {Deligne, Pierre and Goncharov, Alexander B.},
      TITLE = {Groupes fondamentaux motiviques de {T}ate mixte},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {38},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {1--56},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {11G55 (14F42 14G10 19F27)},
      MRNUMBER = {2136480},
      MRREVIEWER = {Tam{á}s Szamuely},
      DOI = {10.1016/j.ansens.2004.11.001},
      ZBLNUMBER = {1084.14024},
      }
  • [DT] Go to document P. Deligne and T. Terasoma, Harmonic shuffle relation for associators.
    @misc{DT,
      author={Deligne, Pierre and Terasoma, T.},
      TITLE = {Harmonic shuffle relation for associators},
      NOTE={preprint},
      URL={www2.lifl.fr/mzv2005/DOC/Terasoma/lille\_terasoma.pdf},
      }
  • [Dr] V. G. Drinfel$’$d, "On quasitriangular quasi-Hopf algebras and on a group that is closely connected with ${ Gal}(\overline{\bf Q}/{\bf Q})$," Algebra i Analiz, vol. 2, iss. 4, pp. 149-181, 1990.
    @article {Dr, MRKEY = {1080203},
      AUTHOR = {Drinfel$'$d, V. G.},
      TITLE = {On quasitriangular quasi-{H}opf algebras and on a group that is closely connected with {${\rm Gal}(\overline{\bf Q}/{\bf Q})$}},
      JOURNAL = {Algebra i Analiz},
      FJOURNAL = {Algebra i Analiz},
      VOLUME = {2},
      YEAR = {1990},
      NUMBER = {4},
      PAGES = {149--181},
      ISSN = {0234-0852},
      MRCLASS = {16W30 (17B37)},
      MRNUMBER = {1080203},
      MRREVIEWER = {Ivan Penkov},
      ZBLNUMBER={0728.16021},
      }
  • [E] L. Euler, "Meditationes circa singulare serierum genus," Novi Comm. Acad. Sci. Petropol., vol. 20, pp. 140-186, 1776.
    @article{E,
      author={Euler, L.},
      KEY={Eul},
      TITLE={Meditationes circa singulare serierum genus},
      JOURNAL={Novi Comm. Acad. Sci. Petropol.},
      VOLUME={20},
      YEAR={1776},
      PAGES={140--186},
      NOTE={and {\it Opera Omnia: Series} 1 {\bf 15} (1927), B. G. Teubner, Berline, pp. 217--267},
      }
  • [F03] Go to document 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.
    @article {F03, MRKEY = {2025460},
      AUTHOR = {Furusho, Hidekazu},
      TITLE = {The multiple zeta value algebra and the stable derivation algebra},
      JOURNAL = {Publ. Res. Inst. Math. Sci.},
      FJOURNAL = {Kyoto University. Research Institute for Mathematical Sciences. Publications},
      VOLUME = {39},
      YEAR = {2003},
      NUMBER = {4},
      PAGES = {695--720},
      ISSN = {0034-5318},
      CODEN = {KRMPBV},
      MRCLASS = {11M41 (14G32)},
      MRNUMBER = {2025460},
      URL = {http://projecteuclid.org/getRecord?id=euclid.prims/1145476044},
      ZBLNUMBER = {1115.11055},
      }
  • [F08] Go to document H. Furusho, "Pentagon and hexagon equations," Ann. of Math., vol. 171, iss. 1, pp. 545-556, 2010.
    @article {F08, MRKEY = {2630048},
      AUTHOR = {Furusho, Hidekazu},
      TITLE = {Pentagon and hexagon equations},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {171},
      YEAR = {2010},
      NUMBER = {1},
      PAGES = {545--556},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {16T20 (17B01 20E18)},
      MRNUMBER = {2630048},
      MRREVIEWER = {Pierre A. Lochak},
      DOI = {10.4007/annals.2010.171.545},
      ZBLNUMBER = {05712736},
      }
  • [G] A. B. Goncharov, Periods and mixed motives, 2002.
    @misc{G,
      author={Goncharov, A. B.},
      TITLE={Periods and mixed motives},
      ARXIV={math/0202154},
      YEAR={2002},
      NOTE={preprint},
      }
  • [Gr] A. Grothendieck, Esquisse d’un programme (1984), reproduced in Geometric Galois Actions, 1, Cambridge: Cambridge Univ. Press, 1997, vol. 242.
    @book {Gr, MRKEY = {1483107},
      KEY={Gro84},
      AUTHOR = {Grothendieck, Alexandre},
      TITLE = {{\rm Esquisse d'un programme (1984), reproduced in} Geometric {G}alois Actions, 1},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      VOLUME = {242},
      PAGES = {5--48},
      NOTE = {with an English translation on pp. 243--283},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1997},
      MRCLASS = {14H10 (14F20 14H30)},
      MRNUMBER = {1483107},
      MRREVIEWER = {Hiroaki Nakamura},
      ZBLNUMBER = {0901.14001},
      }
  • [H] Go to document M. E. Hoffman, "The algebra of multiple harmonic series," J. Algebra, vol. 194, iss. 2, pp. 477-495, 1997.
    @article {H, MRKEY = {1467164},
      AUTHOR = {Hoffman, Michael E.},
      TITLE = {The algebra of multiple harmonic series},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {194},
      YEAR = {1997},
      NUMBER = {2},
      PAGES = {477--495},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {11M41 (05E05)},
      MRNUMBER = {1467164},
      DOI = {10.1006/jabr.1997.7127},
      ZBLNUMBER = {0881.11067},
      }
  • [I] Y. Ihara, "On beta and gamma functions associated with the Grothendieck-Teichmüller groups," in Aspects of Galois Theory, Cambridge, 1999, pp. 144-179.
    @inproceedings {I, MRKEY = {1708605},
      AUTHOR = {Ihara, Yasutaka},
      TITLE = {On beta and gamma functions associated with the {G}rothendieck-{T}eichmüller groups},
      BOOKTITLE = {Aspects of {G}alois Theory},
      VENUE={{G}ainesville, {FL},
      1996},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      VOLUME = {256},
      PAGES = {144--179},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1999},
      MRCLASS = {14G32 (11S80)},
      MRNUMBER = {1708605},
      MRREVIEWER = {N{ú}ria Vila},
      ZBLNUMBER={1046.14010},
      }
  • [IKZ] Go to document 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.
    @article {IKZ, MRKEY = {2218898},
      AUTHOR = {Ihara, Kentaro and Kaneko, Masanobu and Zagier, Don},
      TITLE = {Derivation and double shuffle relations for multiple zeta values},
      JOURNAL = {Compos. Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {142},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {307--338},
      ISSN = {0010-437X},
      MRCLASS = {11M41},
      MRNUMBER = {2218898},
      MRREVIEWER = {David Bradley},
      DOI = {10.1112/S0010437X0500182X},
      ZBLNUMBER = {1186.11053},
      }
  • [LM] Go to document T. T. Q. Le and J. Murakami, "Kontsevich’s integral for the Kauffman polynomial," Nagoya Math. J., vol. 142, pp. 39-65, 1996.
    @article {LM, MRKEY = {1399467},
      AUTHOR = {Le, Thang Tu Quoc and Murakami, Jun},
      TITLE = {Kontsevich's integral for the {K}auffman polynomial},
      JOURNAL = {Nagoya Math. J.},
      FJOURNAL = {Nagoya Mathematical Journal},
      VOLUME = {142},
      YEAR = {1996},
      PAGES = {39--65},
      ISSN = {0027-7630},
      CODEN = {NGMJA2},
      MRCLASS = {57M25 (11M99)},
      MRNUMBER = {1399467},
      MRREVIEWER = {Sergei K. Lando},
      URL = {http://projecteuclid.org/getRecord?id=euclid.nmj/1118772043},
      ZBLNUMBER = {0866.57008},
      }
  • [R] Go to document G. Racinet, "Doubles mélanges des polylogarithmes multiples aux racines de l’unité," Publ. Math. Inst. Hautes Études Sci., iss. 95, pp. 185-231, 2002.
    @article {R, MRKEY = {1953193},
      AUTHOR = {Racinet, Georges},
      TITLE = {Doubles mélanges des polylogarithmes multiples aux racines de l'unité},
      JOURNAL = {Publ. Math. Inst. Hautes Études Sci.},
      FJOURNAL = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
      NUMBER = {95},
      YEAR = {2002},
      PAGES = {185--231},
      ISSN = {0073-8301},
      MRCLASS = {11G55 (11M41)},
      MRNUMBER = {1953193},
      MRREVIEWER = {Jan Nekov{á}{\v{r}}},
      DOI = {10.1007/s102400200004},
      ZBLNUMBER = {1050.11066},
      }
  • [T] T. Terasoma, "Geometry of multiple zeta values," in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 627-635.
    @incollection {T, MRKEY = {2275614},
      AUTHOR = {Terasoma, Tomohide},
      TITLE = {Geometry of multiple zeta values},
      BOOKTITLE = {International {C}ongress of {M}athematicians. {V}ol. {\rm II}},
      PAGES = {627--635},
      PUBLISHER = {Eur. Math. Soc., Zürich},
      YEAR = {2006},
      MRCLASS = {14C30 (11M41 14F42)},
      MRNUMBER = {2275614},
      MRREVIEWER = {Jan Nekov{á}{\v{r}}},
      ZBLNUMBER = {1097.14009},
      }

Authors

Hidekazu Furusho

Graduate School of Mathematics
Nagoya University
Chikusa-ku
Furo-cho
Nagoya 464-8602
Japan