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A three-parameter Hopf deformation of the algebra of Feynman-like diagrams*

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Journal of Russian Laser Research Aims and scope

Abstract

We construct a three-parameter deformation of the Hopf algebra LDIAG. This is the algebra that appears in an expansion in terms of Feynman-like diagrams of the product formula in a simplified version of quantum field theory. This new algebra is a true Hopf deformation which reduces to LDIAG for some parameter values and to the algebra of matrix quasi-symmetric functions (MQSym) for others, and thus relates LDIAG to other Hopf algebras of contemporary physics. Moreover, there is an onto linear mapping preserving products from our algebra to the algebra of Euler–Zagier sums.

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References

  1. A. I. Solomon, G. Duchamp, P. Blasiak, et al., “Hopf algebra structure of a model quantum field theory,” in: J. L. Birman, S. Catto, and B. Nicolescu (eds.), Proceedings of the 26th International Colloquium on Group Theoretical Methods in Physics (New York, 2007), Canopus Publishing (2009), p. 490.

  2. D. Kreimer, Knots and Feynman Diagrams, Cambridge Lecture Notes in Physics (2000), Vol. 13.

  3. C. M. Bender, D. C. Brody, and B. K. Meister, J. Math. Phys., 40, 12 (1999).

    Article  MathSciNet  ADS  Google Scholar 

  4. G. H. E. Duchamp, P. Blasiak, A. Horzela, et al., “Feynman graphs and related Hopf algebras,” SSPCM’05 (Myczkowce, Poland), J. Phys.: Conf. Ser., 30, 107 (2006) [arXiv:cs.SC/0510041].

    Article  Google Scholar 

  5. P. Cartier, “Fonctions polylogarithmes, nombres polyzeta et groupes pro-unipotents,” Séminaire Bourbaki, Asterisque n. 282 (2002).

  6. A. I. Solomon, P. Blasiak, G. Duchamp, et al., “Combinatorial physics, normal order, and model Feynman graphs,” in: B. Gruber, G. Marmo, and N. Yoshinaga (eds.), Proceedings of the Symposium ‘Symmetries in Science XIII’ (Bregenz, Austria, 2003), Kluwer Academic Publishers (2004), p. 527 [arXiv:quant-ph/0310174].

  7. A. I. Solomon, G. Duchamp, P. Blasiak, et al., “Normal order: Combinatorial graphs quantum theory and symmetries,” in: P. C. Argyres, T. J. Hodges, F. Mansouri, et al. (eds.), Proceedings of the 3rd International Symposium on Quantum Theory and Symmetries (Cincinnati, USA, 2003), World Scientific, Singapore (2004), p. 527 [arXiv:quant-ph/0402082].

    Google Scholar 

  8. A. I. Solomon, G. Duchamp, P. Blasiak, et al., “Partition functions and graphs: A combinatorial approach,” in: C. Burdik, O. Navratil, and S. Posta (eds.), Proceedings of the XI International Conference on Symmetry Methods in Physics (Prague, Czech Republic, June 2004), JINR Publishers, Dubna (2004) [arXiv:quant-ph/0409082].

    Google Scholar 

  9. A. Horzela, P. Blasiak, G. Duchamp, et al., “A product formula and combinatorial field theory,” in: C. Burdik, O. Navratil, and S. Posta (eds.), Proceedings of the XI International Conference on Symmetry Methods in Physics (Prague, Czech Republic, June 2004), JINR Publishers, Dubna (2004) [arXiv:quant-ph/0409152].

    Google Scholar 

  10. P. Blasiak, A. Horzela, K. A. Penson, et al., Phys. Lett. A, 338, 108 (2005).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. P. Blasiak, K. A. Penson, A. I. Solomon, et al., J. Math. Phys., 46, 052110 (2005).

    Article  MathSciNet  ADS  Google Scholar 

  12. G. Duchamp, A. I. Solomon, P. Blasiak, et al., “A multipurpose Hopf deformation of the algebra of Feynman-like diagrams,” in: J. L. Birman, S. Catto, and B. Nicolescu (eds.), Proceedings of the 26th International Colloquium on Group Theoretical Methods in Physics (New York, 2006), Canopus Publishing (2009), p. 511 [arXiv:cs.OH/0609107].

  13. G. Duchamp, A. I. Solomon, K. A. Penson, et al., “One-parameter groups and combinatorial physics,” in: J. Govaerts, M. N. Hounkonnou, and A. Z. Msezane (eds.), Proceedings of the Third International Workshop on Contemporary Problems in Mathematical Physics (Porto-Novo, Benin, November 2003), World Scientific Publishing, Singapore (2004), p. 436 [arXiv:quantph/04011262].

    Chapter  Google Scholar 

  14. N. Bourbaki, Theory of Sets, Springer (1989).

  15. J. Berstel and C. Reutenauer, Rational Series and Their Languages, EATCS Monographs on Theoretical Computer Science, Springer (1988).

  16. N. Bourbaki, Elements of Mathematics. Commutative Algebra, Springer (1989), ch. III.

  17. G. H. E. Duchamp, J. -G. Luque, J. -C. Novelli, et al., “Hopf algebras of diagrams,” Contribution to the XIX International Conference on Formal Power Series & Algebraic Combinatorics (Tianjin, China, 2007) [arXiv:math.CO/0710.5661v1].

  18. M. Hazewinkel, “Hopf algebras of endomorphisms of Hopf algebras,” arXiv:math.QA/0410364.

  19. C. Reutenauer, Free Lie Algebras, Oxford University Press (1993).

  20. G. Duchamp, F. Hivert, and J. Y. Thibon, Int. J. Algebra Comput., 12, 671 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  21. L. Foissy (personal communication).

  22. L. Foissy, “Isomorphisme entre l’algèbre des fonctions quasi-symétriques libres et une algèbre de Hopf des arbres enracinés décorés plans” (personal communication).

  23. L. Foissy, “Les algèbres de Hopf des arbres enracinés decorés,” PhD Memoir, Reims University (2002).

  24. P. Cartier, “A primer of Hopf algebras,” IHES Preprint IHES/M/06/40 (2006).

  25. F. Bayen, M. Flato, C. Fronsdal, et al., Ann. Phys., 111, 61 (1978).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press (1994).

  27. M. Rosso, Quantum Groups and Quantum Shuffles, Inventiones Mathematicae (1998), Vol. 133.

  28. P. Ochsenschl¨ager, “Binomialkoeffitzenten und Shuffle-Zahlen,” Technischer Bericht, Fachbereich Informatik, T. H. Darmstadt (1981).

  29. G. Duchamp and J.-G. Luque, “Congruences Compatible with the Shuffle Product,” arXiv:math.CO/0607419.

  30. G. Duchamp, M. Flouret, E. Laugerotte, and J.-G. Luque, Theor. Comput. Sci., 267, 105 (2001) [arXiv:math.CO/0607412].

    Article  MATH  MathSciNet  Google Scholar 

  31. M. E. Hoffman, J. Algebraic Combin., 11, 49 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  32. G. H. E. Duchamp, G. Koshevoy, K. A. Penson, et al., “Geometric combinatorial twisting and shifting,” Séminaire Lotharingien (in preparation).

  33. N. Bourbaki, Elements of Mathematics. Commutative Algebra, Springer (1989), ch VI.

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Authors and Affiliations

  1. LIPN - UMR 7030, CNRS - Université Paris 13, Villetaneuse, F-93430, France

    G. H. E. Duchamp

  2. H. Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences, Eliasza Radzikowskiego Street 152, Kraków, PL 31342, Poland

    P. Blasiak & A. Horzela

  3. Laboratoire de Physique Théorique de la Matière Condensée, Université Pierre et Marie Curie, CNRS UMR 7600, Tour 24 - 2ième ét., 4 pl. Jussieu, F 75252, Paris Cedex 05, France

    K. A. Penson & A. I. Solomon

  4. The Open University, Physics and Astronomy Department, Milton Keynes, MK7 6AA, United Kingdom

    A. I. Solomon

Authors
  1. G. H. E. Duchamp
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  2. P. Blasiak
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  3. A. Horzela
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  4. K. A. Penson
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  5. A. I. Solomon
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Correspondence to A. I. Solomon.

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*Dedicated to Margarita Man’ko, colleague and friend.

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Duchamp, G.H.E., Blasiak, P., Horzela, A. et al. A three-parameter Hopf deformation of the algebra of Feynman-like diagrams*. J Russ Laser Res 31, 162–181 (2010). https://doi.org/10.1007/s10946-010-9135-5

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  • DOI: https://doi.org/10.1007/s10946-010-9135-5

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