Loop of formal diffeomorphisms and Fàa di Bruno coloop bialgebra

Alessandra Frabetti (1) , Ivan P Shestakov

1 PSPM - Probabilités, statistique, physique mathématique

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. Our main result is the proof that the natural loop of formal diffeomorphisms with associative coefficients is proalgebraic, and give a full description 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 Fàa di Bruno Hopf algebra. MSC: 20N05, 14L17, 18D35, 16T30

Mots clés: Non-associative algebras; Formal diffeomorphisms; Loops; Algebraic groups; Cogroups; Coloops; Formal series

Domaines: Théorie des groupes [math.GR]

Alessandra Frabetti, Ivan P Shestakov. Loop of formal diffeomorphisms and Fàa di Bruno coloop bialgebra. 2018. ⟨hal-01849454v2⟩