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Symmetril Moulds, Generic Group Schemes, Resummation of MZVs

  • Claudia MalvenutoEmail author
  • Frédéric Patras
Conference paper
  • 34 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 314)

Abstract

The present article deals with various generating series and group schemes (not necessarily affine ones) associated with MZVs. Our developments are motivated by Ecalle’s mould calculus approach to the latter. We propose in particular a Hopf algebra–type encoding of symmetril moulds and introduce a new resummation process for MZVs.

Keywords

Multiple zeta values Mould calculus Quasi-shuffle 

Notes

Acknowledgements

The authors acknowledge support from ICMAT, Madrid, and from the grant CARMA, ANR-12-BS01-0017.

References

  1. 1.
    Aguiar, M., Mahajan, S.: Monoidal functors, species and Hopf algebras. CRM Monogr. Ser. 29 (2010)Google Scholar
  2. 2.
    Cartier, P.: A primer of Hopf algebras. Frontiers in Number Theory, Physics, and Geometry II, pp. 537–615. Springer, Berlin (2007)CrossRefGoogle Scholar
  3. 3.
    Cartier, P.: Fonctions polylogarithmes, nombres polyzêtas et groupes prounipotents, Séminaire Bourbaki, Mars 2001, 53ème année, 2000–2001, no 885Google Scholar
  4. 4.
    Cresson, J.: Calcul moulien. Annales de la Faculté des Sciences de Toulouse. Mathématiques 18(2), 307–395 (2009)Google Scholar
  5. 5.
    Ebrahimi-Fard, K., Guo, L.: Multiple zeta values and Rota-Baxter algebras. Integers 8(2), 1553–1732 (2008)Google Scholar
  6. 6.
    Ebrahimi-Fard, K., Patras, F.: La structure combinatoire du calcul intégral. Gazette des Mathématiciens 138 (2013)Google Scholar
  7. 7.
    Ecalle, J.: ARI/GARI, la dimorphie et l’arithmétique des multizêtas: un premier bilan. Journal de Théorie des Nombres Bordeaux 15(2), 411–478 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Foissy, L., Patras, F.: Natural endomorphisms of shuffle algebras. Int. J. Algebra Comput. 23(4), 989–1009 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Foissy, L., Patras, F., Thibon, J.-Y.: Deformations of shuffles and quasi-shuffles. Annales de l’Institut Fourier (Grenoble) 66(1), 209–237 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Furusho, H.: Double shuffle relation for associators. Ann. Math. 174, 341–360 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hoffman, M.E.: Quasi-shuffle products. J. Algebr. Comb. 11(1), 49–68 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ihara, K., Kaneko, M., Zagier, D.: Derivation and double shuffle relations for multiple zeta values. Compos. Math. 142, 307–338 (2006)Google Scholar
  13. 13.
    Patras, F.: Generic algebras and iterated Hochschild homology. J. Pure Appl. Algebra 162, 337–357 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Patras, F., Reutenauer, C.: On descent algebras and twisted bialgebras. Moscow Math. J. 4(1), 199–216 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Patras, F., Schocker, M.: Twisted descent algebras and the Solomon-Tits algebra. Adv. Math. 199(1), 151–184 (2006)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Patras, F., Schocker, M.: Trees, set compositions and the twisted descent algebra. J. Algebr. Comb. 28, 3–23 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Peskin, M.E., Schroeder, D.V.: An Introduction to Quantum Field Theory. Westview (1995)Google Scholar
  18. 18.
    Racinet, G.: Doubles mélanges des polylogarithmes multiples aux racines de l’unité. Publications Mathématiques de l’Institut des Hautes Études Scientifiques 95(1), 185–231 (2002)Google Scholar
  19. 19.
    Reutenauer, C.: Free Lie Algebras. Oxford University Press (1993)Google Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Dipartimento di MatematicaSapienza Università di RomaRomeItaly
  2. 2.Université Côte d’Azur UMR 7351 CNRSNice Cedex 02France

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