Combinatorial Hopf algebras from renormalization

Christian Brouder (1) , Alessandra Frabetti (2) , Frederic Menous (3)

1 IMPMC - Institut de minéralogie et de physique des milieux condensés
2 ICJ - Institut Camille Jordan
3 Département de Mathématiques-Université de Paris XI

In this paper we describe the right-sided combinatorial Hopf structure of three Hopf algebras appearing in the context of renormalization in quantum field theory: the non-commutative version of the Faà di Bruno Hopf algebra, the non-commutative version of the charge renormalization Hopf algebra on planar binary trees for quantum electrodynamics, and the non-commutative version of the Pinter renormalization Hopf algebra on any bosonic field. We also describe two general ways to define the associative product in such Hopf algebras, the first one by recursion, and the second one by grafting and shuffling some decorated rooted trees.

Keywords: Hopf algebra; combinatorial; brace; tree; renormalization

Domains: Quantum Algebra [math.QA]; Mathematical Physics [math-ph]; Mathematical Physics [math-ph]

Christian Brouder, Alessandra Frabetti, Frederic Menous. Combinatorial Hopf algebras from renormalization. Journal of Algebraic Combinatorics, 2010, ⟨10.1007/s10801-010-0227-7⟩. ⟨hal-00417946v2⟩