Combinatorial Deformations: Twisting and Perturbations

The framework used to prove the multiplicative law deformation of the algebra of Feynman-Bender diagrams is a twisted shifted dual law (in fact, twice). We give here a clear interpretation of its two parameters. The crossing parameter is a deformation of the tensor structure whereas the superposition parameters is a perturbation of the shuffle coproduct of Hoffman type which, in turn, can be interpreted as the diagonal restriction of a superproduct. Here, we systematically detail these constructions. LIPN UMR 7030 CNRS Université Paris 13 F-93430 Villetaneuse, France Laboratoire de Physique Théorique de la Matière Condensée Université Pierre et Marie Curie, CNRS UMR 7600 Tour 24 2ième étage, 4 place Jussieu, F 75252 Paris cedex 05 Central Institute of Economics and Mathematics (CEMI) Russian Academy of Sciences