Annales de l’Institut Henri Poincaré D

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Volume 5, Issue 1, 2018, pp. 103–125
DOI: 10.4171/AIHPD/49

Published online: 2018-02-01

Right-handed Hopf algebras and the preLie forest formula

Frédéric Menous[1] and Frédéric Patras[2]

(1) Université Paris-Sud, Orsay, France
(2) Université de Nice, France

Three equivalent methods allow to compute the antipode of the Hopf algebras of Feynman diagrams in perturbative quantum field theory (QFT): the Dyson–Salam formula, the Bogoliubov formula, and the Zimmermann forest formula. Whereas the first two hold generally for arbitrary connected graded Hopf algebras, the third one requires extra structure properties of the underlying Hopf algebra but has the nice property to reduce drastically the number of terms in the expression of the antipode (it is optimal in that sense).

The present article is concerned with the forest formula: we show that it generalizes to arbitrary right-handed polynomial Hopf algebras. These Hopf algebras are dual to the enveloping algebras of preLie algebras – a structure common to many combinatorial Hopf algebras which is carried in particular by the Hopf algebras of Feynman diagrams.

Keywords: Forest formula, Zimmermann forest formula, preLie algebra, enveloping algebra, Hopf algebra, right-sided bialgebra

Menous Frédéric, Patras Frédéric: Right-handed Hopf algebras and the preLie forest formula. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 5 (2018), 103-125. doi: 10.4171/AIHPD/49