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QED Hopf algebras on planar binary trees. (English) Zbl 1056.16026
QED is the quantum field theory describing the dynamics of interacting electrons and photons. The interaction is represented by Feynman diagrams. A. Connes and D. Kreimer [Commun. Math. Phys. 210, No. 1, 249-273 (2000; Zbl 1032.81026)] and D. Kreimer [Adv. Theor. Math. Phys. 2, No. 2, 303-334 (1998; Zbl 1041.81087)] discussed the renormalization procedure and group in terms of a commutative Hopf algebra on the set of Feynman diagrams labeled by some indices.
In the paper under review, the authors replace Feynman diagrams by planar binary trees, and describe renormalization in terms of various noncommutative (and noncocommutative) Hopf algebras, and various coactions. The coactions replace the characters in the Connes-Kreimer approach. The main tool is the smash coproduct of Hopf algebras.

MSC:
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
81T17 Renormalization group methods applied to problems in quantum field theory
81T18 Feynman diagrams
81V10 Electromagnetic interaction; quantum electrodynamics
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