Algebraic aspects of rooted tree maps

TL;DR

This paper explores algebraic properties of rooted tree maps derived from the Connes--Kreimer Hopf algebra, relating them to harmonic algebra and characterizing antipode maps through conjugation.

Contribution

It introduces new algebraic insights into rooted tree maps, connecting them with harmonic algebra and providing a characterization of antipode maps.

Findings

Rooted tree maps induce linear relations for multiple zeta values.

Antipode maps are characterized as conjugation by a specific map τ.

The paper relates rooted tree maps to harmonic algebra structures.

Abstract

Based on the Connes--Kreimer Hopf algebra of rooted trees, the rooted tree maps are defined as linear maps on noncommutative polynomial algebra in two indeterminates. It is known that they induce a large class of linear relations for multiple zeta values. In this paper, we investigate some basic algebraic properties of rooted tree maps by relating to the harmonic algebra. We also characterize the antipode maps as the conjugation by the special map $\tau$.