Formulas for Birkhoff-(Rota-Baxter) decompositions related to connected bialgebra

Abstract

In recent years, The BPHZ algorithm for renormalization in quantum field theory has been interpreted, after dimensional regularization, as the Birkhoff-(Rota-Baxter) decomposition (BRB) of characters on the Hopf algebra of Feynmann graphs, with values in a Rota-Baxter algebra. We give in this paper formulas for the BRB decomposition in the group $\mathcal{C}(H, A)$ of characters on a connected Hopf algebra $H$, with values in a Rota-Baxter (commutative) algebra $A$. To do so we first define the stuffle (or quasi-shuffle) Hopf algebra $A^{\tmop{st}}$ associated to an algebra $A$. We prove then that for any connected Hopf algebra $H = k 1_H \oplus H'$, there exists a canonical injective morphism from $H$ to $H'^{\tmop{st}}$. This morphism induces an action of $\mathcal{C}(A^{\tmop{st}}, A)$ on $\mathcal{C}(H, A)$ so that the BRB decomposition in $\mathcal{C}(H, A)$ is determined by the action of a unique (universal) element of $\mathcal{C}(A^{\tmop{st}}, A)$.