Algebraic aspects of rooted tree maps

Abstract

Based on the Connes–Kreimer Hopf algebra of rooted trees, rooted tree maps are defined as linear maps on the noncommutative polynomial algebra \(\mathbb {Q}\langle x,y\rangle \). It is known that they induce a large class of linear relations for multiple zeta values. In this paper, we show for any rooted tree f there exists a unique polynomial in \(\mathbb {Q}\langle x,y\rangle \) that gives the image of the rooted tree map \({\widetilde{f}}\) explicitly. We also characterize the antipode maps as the conjugation by the special map \(\tau \).