Perturbative renormalisation for not-quite-connected bialgebras

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 Möbius inversion as an instance of renormalisation.

12 pages, comments are most welcome

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Cited by in corpus (10)

• Decomposition spaces, incidence algebras and Möbius inversion I: basic theory
• Decomposition spaces, incidence algebras and Möbius inversion II: completeness, length filtration, and finiteness
• The Hopf algebra structure of the $R^{\ast }$-operation
• Decomposition spaces and restriction species
• Polynomial functors and combinatorial Dyson-Schwinger equations
• Faà di Bruno for operads and internal algebras
• The geometry of characters of Hopf algebras
• The incidence comodule bialgebra of the Baez-Dolan construction
• Combinatorial Dyson-Schwinger equations and inductive data types
• Antipodes of monoidal decomposition spaces