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

TL;DR

This paper provides explicit formulas for Birkhoff-(Rota-Baxter) decompositions of characters on connected Hopf algebras, linking them to a universal element in a quasi-shuffle algebra, with applications to quantum field theory renormalization.

Contribution

It introduces formulas for BRB decomposition in the group of characters on connected Hopf algebras using a universal element derived from a quasi-shuffle algebra.

Findings

Established a canonical injective morphism from a connected Hopf algebra to a quasi-shuffle algebra.

Demonstrated that BRB decomposition is governed by a universal element in the quasi-shuffle algebra.

Linked the algebraic structures to the BPHZ renormalization process in quantum field theory.

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 $\mathcal{C}(A^{\tmop{st}}, A)$.