Ordered forests, permutations and iterated integrals

TL;DR

This paper constructs an explicit algebraic isomorphism connecting heap-ordered trees and quasi-symmetric functions, providing a universal method to lift characters in the context of rough path theory.

Contribution

It introduces a new explicit Hopf algebra isomorphism that lifts measure-indexed characters from the Connes-Kreimer algebra to the shuffle algebra, enhancing the algebraic framework in rough path theory.

Findings

Established a Hopf algebra isomorphism between heap-ordered trees and quasi-symmetric functions.

Provided a universal lifting method for measure-indexed characters in rough path analysis.

Connected algebraic structures with applications in iterated integrals and permutations.

Abstract

We construct an explicit Hopf algebra isomorphism from the algebra of heap-ordered trees to that of quasi-symmetric functions, generated by formal permutations, which is a lift of the natural projection of the Connes-Kreimer algebra of decorated rooted trees onto the shuffle algebra. This isomorphism gives a universal way of lifting measure-indexed characters of the Connes-Kreimer algebra into measure-indexed characters of the shuffle algebra, already introduced in \cite{Unterberger} in the framework of rough path theory as the so-called Fourier normal ordering algorithm.