PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2013 | 11 | 3 | 401-422
Tytuł artykułu

Categorification of Hopf algebras of rooted trees

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec ℕ) whose semiring of functions is (a P-version of) the Connes-Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to ℤ and collapsing H 0). The monoidal structure is itself given by a polynomial functor, represented by three easily described set maps; we show that these maps are the same as those occurring in the polynomial representation of the free monad on P.
Wydawca
Czasopismo
Rocznik
Tom
11
Numer
3
Strony
401-422
Opis fizyczny
Daty
wydano
2013-03-01
online
2012-12-22
Twórcy
autor
  • Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra (Barcelona), Spain, kock@mat.uab.cat
Bibliografia
  • [1] Awodey S., Category Theory, Oxford Logic Guides, 49, Clarendon Press, Oxford University Press, New York, 2006 http://dx.doi.org/10.1093/acprof:oso/9780198568612.001.0001[Crossref]
  • [2] Baez J.C., Dolan J., From finite sets to Feynman diagrams, In: Mathematics Unlimited-2001 and Beyond, Springer, Berlin, 2001, 29–50
  • [3] Bergbauer C., Kreimer D., Hopf algebras in renormalization theory: locality and Dyson-Schwinger equations from Hochschild cohomology, In: Physics and Number Theory, IRMA Lect. Math. Theor. Phys., 10, European Mathematical Society, Zürich, 2006, 133–164 http://dx.doi.org/10.4171/028-1/4[Crossref][WoS]
  • [4] Brouder Ch., Runge-Kutta methods and renormalization, Eur. Phys. J. C Part. Fields, 2000, 12(3), 521–534 http://dx.doi.org/10.1007/s100529900235[Crossref]
  • [5] Butcher J.C., An algebraic theory of integration methods, Math. Comp., 1972, 26(117), 79–106 http://dx.doi.org/10.1090/S0025-5718-1972-0305608-0[Crossref]
  • [6] Carboni A., Lack S., Walters R.F.C., Introduction to extensive and distributive categories, J. Pure Appl. Algebra, 1993, 84(2), 145–158 http://dx.doi.org/10.1016/0022-4049(93)90035-R[Crossref]
  • [7] Chapoton F., Livernet M., Pre-Lie algebras and the rooted trees operad, Int. Math. Res. Not. IMRN, 2001, 8, 395–408 http://dx.doi.org/10.1155/S1073792801000198[Crossref]
  • [8] Connes A., Kreimer D., Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys., 1998, 199(1), 203–242 http://dx.doi.org/10.1007/s002200050499[Crossref]
  • [9] Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann-Hilbert problem I: The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys., 2000, 210(1), 249–273 http://dx.doi.org/10.1007/s002200050779[Crossref]
  • [10] Crane L., Yetter D.N., Examples of categorification, Cahiers Topologie Géom. Différentielle Catég., 1998, 39(1), 3–25
  • [11] Figueroa H., Gracia-Bondía J.M., Combinatorial Hopf algebras in quantum field theory I, Rev. Math. Phys., 2005, 17(8), 881–976 http://dx.doi.org/10.1142/S0129055X05002467[Crossref]
  • [12] Gálvez-Carrillo I., Kock J., Tonks A., Groupoids and Faà di Bruno formulae for Green functions in bialgebras of trees, preprint available at http://arxiv.org/abs/1207.6404 [WoS]
  • [13] Gambino N., Kock J., Polynomial functors and polynomial monads, Math. Proc. Cambridge Philos. Soc. (in press), DOI: 10.1017/S0305004112000394 [Crossref]
  • [14] Hartshorne R., Algebraic Geometry, Grad. Texts in Math., 52, Springer, New York-Heidelberg, 1977 [Crossref]
  • [15] Kock J., Polynomial functors and trees, Int. Math. Res. Not. IMRN, 2011, 3, 609–673
  • [16] Kock J., Data types with symmetries and polynomial functors over groupoids, In: Proceedings of the 28th Conference on the Mathematical Foundations of Programming Semantics, Bath, 2012, Electronic Notes in Theoretical Computer Science, 286, preprint available at http://arxiv.org/abs/1210.0828
  • [17] Kock J., Categorical formalisms of graphs and trees in quantum field theory (in preparation)
  • [18] Kock J., Joyal A., Batanin M., Mascari J.-F., Polynomial functors and opetopes, Adv. Math., 2010, 224(6), 2690–2737 http://dx.doi.org/10.1016/j.aim.2010.02.012[WoS][Crossref]
  • [19] Kreimer D., On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys., 1998, 2(2), 303–334
  • [20] Kreimer D., On overlapping divergences, Comm. Math. Phys., 1999, 204(3), 669–689 http://dx.doi.org/10.1007/s002200050661[Crossref]
  • [21] Lawvere F.W., Menni M., The Hopf algebra of Möbius intervals, Theory Appl. Categ., 2010, 24(10), 221–265
  • [22] Mac Lane S., Categories for the Working Mathematician, 2nd ed., Grad. Texts in Math., 5, Springer, New York, 1998
  • [23] Moerdijk I., On the Connes-Kreimer construction of Hopf algebras, In: Homotopy Methods in Algebraic Topology, Boulder, June 20–24, 1999, Contemp. Math., 271, American Mathematical Society, Providence, 2001, 311–321 http://dx.doi.org/10.1090/conm/271/04360[Crossref]
  • [24] Weber M., Polynomials in categories with pullbacks, preprint available at http://arxiv.org/abs/1106.1983
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-012-0152-1
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.