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Combinatorial Hopf algebras in quantum field theory. I. (English) Zbl 1090.16016

This survey on combinatorial Hopf algebras and its application in quantum field theory collects and expands for the most part a series of lectures delivered by the second-named author in the framework of the joint mathematical physics seminar of the Universités d’Artois and Lille 1.
It recalls in section 1 some facts on Hopf algebras. The results are illustrated by examples (universal enveloping algebras …). Section 2 gives, after a direct approach to Faà di Bruno Hopf algebras, a relation between Connes-Moscovi algebras and Faà di Bruno algebras (Thm 2.9). Section 3 deals with incidence bialgebras and Connes-Kreimer Hopf algebras on Feynman diagrams. They start by an example (Ginzburg-Landau \(\phi_4^4\) scalar model in Euclidean field theory). Then they give a simple derivation of the combinatorial part of Zimmermann’s cancellation-free method (sec. 3.2) and proof of the general cancellation-free formula for antipodes (sec. 3.4). In the last section the authors give 2 theorems of structure of commutative Hopf algebras (Thm 4.1 and Proposition 4.4).

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
81T18 Feynman diagrams
05E99 Algebraic combinatorics
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81T25 Quantum field theory on lattices
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