COMBINATORIAL PHYSICS: SCHÜTZENBERGER FACTORIZATION AND

NON-COMMUTATIVE DIFFERENTIAL EQUATIONS.

G H E DUCHAMP^{†}, P BLASIAK^{‡}, A HORZELA^{‡}, HOANG NGOC MINH^{†}, K A PENSON^{♢} AND A I SOLOMON^{♢,♯}

Abstract The original Hopf algebra inherent to the pre-Feynman diagrams (LDIAG) of the Quantum Field Theory of Partitions (QFTP) admits a threeparameter deformation. It turns out that the two first parameters (deformation of the structure of algebra) are of different natures : the first is due to the deformation of the (tensor) space by a commutation factor and the second to a perturbation of the comultiplication. Details for these tools used for deformation will be indicated (coloured products, dual laws, diagonal deformation, Sweedler’s duals). As a byproduct, one gets an unexpected deformation of the Euler-Zagier sums. It will be shown also how these sums arise from a noncommutative fuchsian differential equation which produces a group-like element. We indicate a way of regularizing them through Schützenberger factorization. The talk is based on the two papers [1, 2].

References

[1] G. H. E. Duchamp, C. Tollu, K. A. Penson, G. Koshevoy, Combinatorial Deformations of Algebras: Twisting and Perturbations, Sminaire Lotharingien de Combinatoire, to appear.

      arXiv:0903.2101v2

[2] G. H.E. Duchamp, Hoang Ngoc Minh, Allan I. Solomon, Silvia Goodenough, An interface between physics and number theory, Proceedings of Group 28 (Group-Theoretical methods in Physics), Newcastle 2010, to appear.

      arXiv:1011.0523v1

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

‡ H.Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences, ul. Eliasza-Radzikowskiego 152, PL 31342 Kraków, Poland

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

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