DEFORMATION THEORY OF HOPF ALGEBRAS APPEARING IN COMBINATORIAL PHYSICS

G H E DUCHAMP^{†}, P BLASIAK^{‡}, A HORZELA^{‡}, 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 three-parameter deformation. It turns out that the two first parameters (deformation of the structure of algebra) are of different nature : the first is due to the deformation of the (tensor) space by a commutation factor and the second to a perturbation of the comultiplication. The very general tools for deformation are detailed (coloured products, dual laws, diagonal deformation, Swedler’s duals). As a byproduct, one gets an unexpected deformation of the Euler-Zagier sums.

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