Rooted tree polylogarithms

Abstract

We study multiple polylogarithms indexed by rooted forests and show some properties of them from a viewpoint of the Connes-Kreimer Hopf algebra of rooted trees $H$. In particular, we show that almost all primitive elements in $H$ give relations among the polylogarithms. Moreover, we compare the algebraic structure of the polylogarithms with that of rooted tree maps.