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

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 $A^{\text{tmop}st}$ associated to an algebra $A$. We prove then that for any connected Hopf algebra $H=k{1}_H\oplus H^{\prime }$, there exists a canonical injective morphism from $H$ to $H^{\prime \text{tmop}st}$. This morphism induces an action of $\mathcal{C}(A^{\text{tmop}st},A)$ 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 $\mathcal{C}(A^{\text{tmop}st},A)$.