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 B0 is spanned by grouplike 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.

From Möbius inversion to renormalisation

“…It follows that group-like elements are always of filtration degree zero. We make the following standing assumption (see [27]):…”

“…Recall that a filtered bialgebra B is connected if B 0 is spanned by the unit, and that any connected bialgebras is Hopf [42]. We shall call B not-quite-connected [27] in the situation where B 0 is spanned by group-like elements. A notion of antipode for not-quite-connected bialgebras was introduced recently by Carlier and Kock [6].…”

“…I do not know of any reference for this result. It may be new, but is in any case a straightforward abstraction of the theorems of Rota and Content-Lemay-Leroux already quoted, once the degree-zero condition(5) has been identified[27] 13. The proof ingredients go a long way back.…”

Decomposition Spaces and Restriction Species

“…This variation, which is subsumed in the class of decomposition spaces coming from operads [21,35], has some different features which have been exploited to good effect in various contexts [15,31,32,34]. In particular it is important that the cut locus expresses a type match between the roots of the crown forest and the leaves of the bottom tree, and that there is a grading [19] given by number of leaves minus number of roots.…”

Antipodes of monoidal decomposition spaces

“…Recent developments have shown the utility of avoiding this reduction, which destroys useful information. For example, the Faà di Bruno formula for general operads [3], [12] crucially exploits the finer structure of the zeroth graded piece of the incidence bialgebra, and in the bialgebra version [10] of BPHZ renormalisation in perturbative quantum field theory, the zeroth graded piece of the bialgebra of Feynman graphs contains the terms of the Lagrangian (not visible in the quotient Hopf algebra usually employed).…”