Figueroa, Héctor; Gracia-Bondía, José M. Combinatorial Hopf algebras in quantum field theory. I. (English) Zbl 1090.16016 Rev. Math. Phys. 17, No. 8, 881-976 (2005). 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). Reviewer: Jérôme Petit (Montpellier) Cited in 47 Documents 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 Keywords:Hopf algebras; combinatorics; renormalization; noncommutative geometry; Feynman diagrams; antipodes PDFBibTeX XMLCite \textit{H. Figueroa} and \textit{J. M. Gracia-Bondía}, Rev. Math. Phys. 17, No. 8, 881--976 (2005; Zbl 1090.16016) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Irregular triangle of multinomial coefficients of integer partitions read by rows (in Abramowitz and Stegun ordering) giving the coefficients of the cycle index polynomials for the symmetric groups S_n. Irregular triangle of multinomial coefficients, read by rows (version 1). Irregular triangle of coefficients of a partition transform for direct Lagrange inversion of an o.g.f., complementary to A134685 for an e.g.f.; Normalized by the factorials, these are signed, refined face polynomials of the associahedra Irregular triangle read by rows: coefficients C(j,k) of a partition transform for direct Lagrange inversion. Coefficients of the Faber partition polynomials. 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