Perturbative renormalisation for not-quite-connected bialgebras

TL;DR

This paper extends the Connes--Kreimer Hopf-algebraic approach to perturbative renormalisation to a broader class of filtered bialgebras, including those relevant in quantum field theory and combinatorics, enabling new interpretations of M"obius inversion.

Contribution

It generalizes the renormalisation framework from Hopf algebras to filtered bialgebras with group-like elements, broadening its applicability.

Findings

Applicable to pointed bialgebras with the coradical filtration

Enables interpretation of M"obius inversion as renormalisation

Works in both quantum field theory and combinatorics

Abstract

We observe that the Connes--Kreimer Hopf-algebraic approach to perturbative renormalisation works not just for Hopf algebras but more generally for filtered bialgebras $B$ with the property that $B_0$ is spanned by group-like elements (e.g. pointed bialgebras with the coradical filtration). Such bialgebras occur naturally both in Quantum Field Theory, where they have some attractive features, and elsewhere in Combinatorics, where they cover a comprehensive class of incidence bialgebras. In particular, the setting allows us to interpret M\"obius inversion as an instance of renormalisation.