Title : The Hopf algebra of Labelled diagrams

Abstract : In this talk, we start from the problem of normal ordering and show
how it is related to matrix coefficients of infinite matrices. Interesting
matrices are "naturally and combinatorially" provided by the exponential
formula. Using the Bell polynomial expansion of free exponentials in Bender's
product formula yields a summation on a suited category of Feynman Diagrams. It
turns out that these Feynman diagrams are a natural basis of Hopf algebra
compatible with the monomial evaluation. We show the different correspondences
with other Hopf algebras (Free Quasisymmetric Functions, Plane Decorated Trees
and Connes-Kreimer) and expicit the structure of its Sweedler's Dual.