Loop of formal diffeomorphisms and F{\`a}a di Bruno coloop bialgebra

TL;DR

This paper generalizes the concept of formal diffeomorphisms and their associated coloop bialgebras, providing a non-commutative Lagrange inversion formula and explaining the structure of the non-commutative Faà di Bruno Hopf algebra.

Contribution

It proves the proalgebraic nature of the loop of formal diffeomorphisms with associative coefficients and describes the codivisions on its coloop bialgebra, extending classical formulas to non-commutative settings.

Findings

The loop of formal diffeomorphisms is proalgebraic.

Full description of codivisions on the coloop bialgebra.

Generalization of Lagrange inversion to non-commutative series.

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. 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 F{\`a}a di Bruno Hopf algebra. MSC: 20N05, 14L17, 18D35, 16T30