RIGHT-HANDED BIALGEBRAS AND THE PRELIE FOREST FORMULA

Frédéric Menous (1) , Frédéric Patras (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 LJAD - Laboratoire Jean Alexandre Dieudonné

Three equivalent methods allow to compute the antipode of the Hopf algebras of Feynman diagrams in perturbative quantum field theory: the Dyson-Salam formula, the Bogoliubov formula, and the Zimmermann forest formula. Whereas the first two hold generally for arbitrary 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 the formula 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.

Mots clés: Forest formula; Zimmerman forest formula; preLie algebra; preLie coalgebra; enveloping algebra; Hopf algebra; right-handed bialgebra; right-sided bialgebra

Domaines: Anneaux et algèbres [math.RA]; Combinatoire [math.CO]; Physique mathématique [math-ph]

Frédéric Menous, Frédéric Patras. RIGHT-HANDED BIALGEBRAS AND THE PRELIE FOREST FORMULA. 2015. ⟨hal-01232450v1⟩