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Communications in Mathematical Physics

, Volume 327, Issue 1, pp 151–179 | Cite as

General Dyson–Schwinger Equations and Systems

  • Loïc FoissyEmail author
Article

Abstract

We classify combinatorial Dyson–Schwinger equations giving a Hopf subalgebra of the Hopf algebra of Feynman graphs of the considered Quantum Field Theory. We first treat single equations with an arbitrary (eventually infinite) number of insertion operators. We distinguish two cases; in the first one, the Hopf subalgebra generated by the solution is isomorphic to the Faà di Bruno Hopf algebra or to the Hopf algebra of symmetric functions; in the second case, we obtain the dual of the enveloping algebra of a particular associative algebra (seen as a Lie algebra). We also treat systems with an arbitrary finite number of equations, with an arbitrary number of insertion operators, with at least one of degree 1 in each equation.

Keywords

Hopf Algebra Rooted Tree Formal Series Feynman Graph Insertion Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Abe, E.: Hopf algebras, Cambridge Tracts in Mathematics, vol. 74. Cambridge University Press, Cambridge (1980) [Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka]Google Scholar
  2. 2.
    Bergbauer, C., Kreimer, D.: Hopf algebras in renormalization theory: locality and Dyson–Schwinger equations from Hochschild cohomology. IRMA Lect. Math. Theor. Phys., vol. 10, Eur. Math. Soc., Zürich (2006). arXiv:hep-th/0506190
  3. 3.
    Chapoton, F., Livernet, M.: Pre-Lie algebras and the rooted trees operad. Int. Math. Res. Notices 8, 395–408 (2001). arXiv:math/0002069 Google Scholar
  4. 4.
    Connes, A., Kreimer, D.: Hopf algebras, Renormalization and Noncommutative geometry. Commun. Math. Phys 199(1), 203–242 (1998). arXiv:hep-th/9808042 Google Scholar
  5. 5.
    Foissy, L.: Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson–Schwinger equations. Adv. Math. 218, 136–162 (2008). arXiv:0707.1204 Google Scholar
  6. 6.
    Foissy, L.: Classification of systems of Dyson–Schwinger equations of the Hopf algebra of decorated rooted trees. Adv. Math. 224(5), 2094–2150 (2010). arXiv:1101.5231 Google Scholar
  7. 7.
    Foissy, L.: Hopf subalgebras of rooted trees from Dyson–Schwinger equations. Motives, quantum field theory, and pseudodifferential operators. Clay Math. Proc., vol. 12, pp. 189–210. Amer. Math. Soc., Providence (2010)Google Scholar
  8. 8.
    Foissy, L.: Lie algebras associated to a system Dyson–Schwinger equations. Adv. Math. 226(6), 4702–4730 (2011). arXiv:1101.5231 Google Scholar
  9. 9.
    Gan W.L., Schedler T.: The necklace Lie coalgebra and renormalization algebras. J. Noncommut. Geom. 2(2), 195–214 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Grossman, R.L., Larson, R.G.: Hopf-algebraic structure of families of trees. J. Algebra 126(1), 184–210 (1989). arXiv:0711.3877 Google Scholar
  11. 11.
    Grossman R.L., Larson R.G.: Hopf-algebraic structure of combinatorial objects and differential operators. Israel J. Math. 72(1–2), 109–117 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Hoffman M.E.: Combinatorics of rooted trees and Hopf algebras. Trans. Am. Math. Soc. 355(9), 3795–3811 (2003)CrossRefzbMATHGoogle Scholar
  13. 13.
    Kreimer, D.: On overlapping divergences. Commun. Math. Phys. 204(3), 669–689 (1999). arXiv:hep-th/9810022 Google Scholar
  14. 14.
    Kreimer, D.: Anatomy of a gauge theory. Ann. Phys. 321(12), 2757–2781 (2006). arXiv:hep-th/0509135 Google Scholar
  15. 15.
    Milnor J.W., Moore J.C.: On the structure of Hopf algebras. Ann. Math. (2) 81, 211–264 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Oudom, J.-Mi., Guin, D.: Sur l’algèbre enveloppante d’une algèbre pré-Lie. C. R. Math. Acad. Sci. Paris 340(5), 331–336 (2005). arXiv:math/0404457
  17. 17.
    Panaite, F.: Relating the Connes–Kreimer and Grossman–Larson Hopf algebras built on rooted trees. Lett. Math. Phys. 51(3), 211–219 (2000). arXiv:math/0003074 Google Scholar
  18. 18.
    Tanasa, A., Kreimer, D.: Combinatorial Dyson–Schwinger equations in noncommutative field theory. J. Noncommut. Geom. 7(2013), 255 (2009). arXiv:0907.2182 Google Scholar
  19. 19.
    Yeats, K.A.: Growth estimates for Dyson–Schwinger equations. ProQuest LLC, Ann Arbor (2008). arXiv:0810.2249

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Laboratoire de MathématiquesUniversité de ReimsReims Cedex 2France

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