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      COMBINATORIAL HOPF ALGEBRAS IN QUANTUM FIELD THEORY I

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      Reviews in Mathematical Physics
      World Scientific Pub Co Pte Lt

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          Abstract

          This paper stands at the interface between combinatorial Hopf algebra theory and renormalization theory. Its plan is as follows: Sec. 1.1 is the introduction, and contains an elementary invitation to the subject as well. The rest of Sec. 1 is devoted to the basics of Hopf algebra theory and examples in ascending level of complexity. Section 2 turns around the all-important Faà di Bruno Hopf algebra. Section 2.1 contains a first, direct approach to it. Section 2.2 gives applications of the Faà di Bruno algebra to quantum field theory and Lagrange reversion. Section 2.3 rederives the related Connes–Moscovici algebras. In Sec. 3, we turn to the Connes–Kreimer Hopf algebras of Feynman graphs and, more generally, to incidence bialgebras. In Sec. 3.1, we describe the first. Then in Sec. 3.2, we give a simple derivation of (the properly combinatorial part of) Zimmermann's cancellation-free method, in its original diagrammatic form. In Sec. 3.3, general incidence algebras are introduced, and the Faà di Bruno bialgebras are described as incidence bialgebras. In Sec. 3.4, deeper lore on Rota's incidence algebras allows us to reinterpret Connes–Kreimer algebras in terms of distributive lattices. Next, the general algebraic-combinatorial proof of the cancellation-free formula for antipodes is ascertained. The structure results for commutative Hopf algebras are found in Sec. 4. An outlook section very briefly reviews the coalgebraic aspects of quantization and the Rota–Baxter map in renormalization.

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                Author and article information

                Journal
                Reviews in Mathematical Physics
                Rev. Math. Phys.
                World Scientific Pub Co Pte Lt
                0129-055X
                1793-6659
                November 21 2011
                September 2005
                November 21 2011
                September 2005
                : 17
                : 08
                : 881-976
                Affiliations
                [1 ]Departamento de Matemáticas, Universidad de Costa Rica, San Pedro 2060, Costa Rica
                [2 ]Departamento de Física Teórica I, Universidad Complutense, Madrid 28040, Spain
                Article
                10.1142/S0129055X05002467
                5672ae70-c5db-4ca9-b439-a8f2bd40bd7a
                © 2005
                History

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