Table of Contents
Algebra
Volume 2013, Article ID 370618, 10 pages
http://dx.doi.org/10.1155/2013/370618
Research Article

Two Interacting Coordinate Hopf Algebras of Affine Groups of Formal Series on a Category

Université Paris 13, Sorbonne Paris Cité, LIPN, CNRS (UMR 7030), 93430 Villetaneuse, France

Received 19 March 2013; Revised 17 May 2013; Accepted 14 July 2013

Academic Editor: Stefaan Caenepeel

Copyright © 2013 Laurent Poinsot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. Warner, Topological Fields, vol. 157 of North-Holland Mathematical Studies, Elsevier, New York, NY, USA, 1989.
  2. J. Berstel and C. Reutenauer, Noncommutative Rational Series with Applications, vol. 137 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 2010.
  3. L. Poinsot, Contributions à l’Algèbre, à l’Analyse et à la Combinatoire des Endomorphismes sur les Espaces de Séries, (French Language), Habilitation à Diriger des Recherches en Mathématiques, Université Paris 13, Sorbonne Paris Cité, Villetaneuse, France, 2011, http://lipn.univ-paris13.fr/~poinsot/HDR/HDR.pdf.
  4. L. Poinsot, “Rigidity of the topoogical dual of spaces of formal series with respect to product topologies,” 2010, preprint, http://lipn.univ-paris13.fr/~poinsot/Articles/Top-dual-space-v8.pdf.
  5. S. Dascalescu, C. Nastasescu, and S. Raianu, Hopf Algebras, vol. 235 of Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2001.
  6. C. Kassel, Quantum Groups, vol. 155 of Graduate Texts in Mathematics, Springer, Berlin, Germany, 2010.
  7. M. Demazure and P. Gabriel, Groupes Algébriques: Volume 1, Elsevier, New York, NY, USA, 1970.
  8. G. Hochschild and G. D. Mostow, “Pro-affine algebraic groups,” American Journal of Mathematics, vol. 91, no. 4, pp. 1127–1140, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. J. S. Milne, Basic Theory of Affine Group Schemes, Version 1. 00, On-line course material, 2012.
  10. S. MacLane, Categories for the Working Mathematician, vol. 5 of Graduate Texts in Mathematics, Springer, Berlin, Germany, 1998.
  11. H. Figueroa, J. M. Gracia-Bondía, and J. C. Várilly, “Faà di Bruno Hopf algebras,” in Encyclopedia of Mathematics, M. Hazewinkel, Ed., Springer, Berlin, Germany.
  12. S. A. Joni and G. Rota, “Coalgebras and bialgebras in combinatorics,” Studies in Applied Mathematics, vol. 61, pp. 93–139, 1979. View at Google Scholar · View at Zentralblatt MATH
  13. W. R. Schmitt, “Incidence Hopf algebras,” Journal of Pure and Applied Algebra, vol. 96, no. 3, pp. 299–330, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. S. A. Jennings, “Substitution groups of formal power series,” Canadian Journal of Mathematics, vol. 6, pp. 325–340, 1954. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. L. Foissy, “Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations,” Advances in Mathematics, vol. 218, no. 1, pp. 136–162, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. R. P. Stanley, Enumerative Combinatorics, Cambridge University Press, Cambridge, UK, 1997.
  17. P. Ageron, “Logic without self-deductibility, logical universalis,” in Towards a General Theory of Logic, J. Y. Beziau, Ed., pp. 89–95, Birkhaüser, Basel, Switzerland, 2005. View at Google Scholar
  18. N. Bourbaki, Elements of Mathematics—Algebra, chapter 1–3, Springer, Berlin, Germany, 1989.
  19. A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, vol. 7 of Mathematical Surveys, American Mathematical Society, Providence, RI, USA, 1961.
  20. L. Poinsot, G. H. E. Duchamp, and C. Tollu, “Möbius inversion formula for monoids with zero,” Semigroup Forum, vol. 81, no. 3, pp. 446–460, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. M. Content, F. Lemay, and P. Leroux, “Catégories de Möbius et fonctorialités : un cadre général pour l’inversion de Möbius,” Journal of Combinatorial Theory A, vol. 28, pp. 169–190, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. P. Leroux, “Les catégories de Möbius,” Cahiers de Topologie et Géométrie Différentielle, vol. 16, pp. 280–282, 1975. View at Google Scholar
  23. F. W. Lawvere and M. Menni, “The Hopf algebra of Möbius intervals,” Theory and Applications of Categories, vol. 24, pp. 221–265, 2010. View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  24. I. Assem, D. Simon, and A. Skowronski, Elements of the Representation Theory of Associative Algebras, 1: Techniques of Representation Theory, Cambridge University Press, Cambridge, UK, 2006.
  25. B. Mitchell, “Rings with several objects,” Advances in Mathematics, vol. 8, no. 1, pp. 1–161, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  26. N. Bourbaki, Elements of Mathematics—General Topology, chapter 1–4, Springer, Berlin, Germany, 1998.
  27. L. Comtet, Advanced Combinatorics—The Art of Finite and infinite Expansions, Reidel, Dordrecht, The Netherlands, 1974.
  28. R. G. Heyneman and M. E. Sweedler, “Affine Hopf algebras, I,” Journal of Algebra, vol. 13, no. 2, pp. 192–241, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  29. R. K. Molnar, “Semi-direct products of Hopf algebras,” Journal of Algebra, vol. 47, no. 1, pp. 29–51, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  30. D. E. Radford, “The structure of Hopf algebras with a projection,” Journal of Algebra, vol. 92, no. 2, pp. 322–347, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  31. L. Poinsot and G. H. E. Duchamp, “A formal calculus on the Riordan near algebra,” Advances and Applications in Discrete Mathematics, vol. 6, no. 1, pp. 11–44, 2010. View at Google Scholar · View at Zentralblatt MATH
  32. D. Calaque, K. Ebrahimi-Fard, and D. Manchon, “Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution of B-series,” Advances in Applied Mathematics, vol. 47, no. 2, pp. 282–308, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  33. Ph. Chartier, E. Hairer, and G. Vilmart, “A substitution law for B-series vector fields,” Preprint INRIA 5498, INRIA, Paris, France, 2005. View at Google Scholar
  34. E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 2002.