A comodule-bialgebra structure for word-series substitution and mould composition

TL;DR

This paper introduces a new coproduct structure compatible with Hoffman's quasi-shuffle product, linking it to Ecalle's mould calculus and expanding the algebraic framework for word-series substitution.

Contribution

It defines an internal coproduct compatible with Hoffman's quasi-shuffle product and establishes a comodule-Hopf algebra structure relating to mould calculus.

Findings

Hoffman's quasi-shuffle Hopf algebra is a comodule-Hopf algebra over the new bialgebra

The relation between the coproduct and Ecalle's mould calculus is clarified

The algebraic structures facilitate advanced word-series substitution techniques

Abstract

An internal coproduct is described, which is compatible with Hoffman's quasi-shuffle product. Hoffman's quasi-shuffle Hopf algebra, with deconcatenation coproduct, is a comodule-Hopf algebra over the bialgebra thus defined. The relation with J. Ecalle's mould calculus, i.e. mould composition and contracting arborification, is precised.