Right-handed Hopf algebras and the preLie forest formula

Abstract

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.