Loop of formal diffeomorphisms and Faà di Bruno coloop bialgebra

Abstract

We consider a generalization of (pro)algebraic loops defined on general categories of algebras and the dual notion of a coloop bialgebra suitable to represent them as functors. We prove that the natural loop of formal diffeomorphisms with associative coefficients is proalgebraic, and we give the closed formulas of the codivisions on its coloop bialgebra. This result provides a generalization of the Lagrange inversion formula to series with non-commutative coefficients, and a loop-theoretic explanation to the existence of the non-commutative Faà di Bruno Hopf algebra. MSC: 20N05, 14L17, 18D35, 16T30Inv qnd Diff and which turn out to be extremely efficient in handling the combinatorial content of renormalization procedures [6,7,8,47,43].The toy model φ 3 theory used by Connes-Kreimer is a scalar field theory and leads to the commutative algebra A " C of amplitudes. However, interesting physical situations involve noncommutative algebras. In fact, Feynman amplitudes are complex numbers for single scalar fields, the coupling constants and the renormalization factors, but they are 4ˆ4 complex matrices for the fermionic or bosonic fields, and may be represented by higher order matrices for theories involving several interacting fields. In this case, forcing the final counterterms to be scalar, as imposed by the fact that the renormalization factors act on the (scalar) Lagrangian, prevents us from describing the renormalization in a functorial way, as shown by the results in [48], where the Hopf algebra does not represent a functorial group on A " M 4 pCq. In order to preserve this functoriality, there is a need to understand Dyson's formulas for sets of series InvpAq and DiffpAq also when A is not a commutative algebra. This is the motivation for the present work.