The combinatorics of Bogoliubov's recursion in renormalization

Abstract

We describe various combinatorial aspects of the Birkhoff-Connes-Kreimer factorization in perturbative renormalisation. The analog of Bogoliubov's preparation map on the Lie algebra of Feynman graphs is identified with the pre-Lie Magnus expansion. Our results apply to any connected filtered Hopf algebra, based on the pro-nilpotency of the Lie algebra of infinitesimal characters.Comment: improved version, 20 pages, CIRM 2006 workshop "Renormalization and Galois Theory", Org. F. Fauvet, J.-P. Rami