We define a Hopf algebra of motivic iterated integrals on the line and prove an explicit formula for the coproduct in this Hopf algebra. We show that this formula encodes the group law of the automorphism group of a certain noncommutative variety. We relate the coproduct to the coproduct in the Hopf algebra of decorated rooted plane trivalent trees, which is a plane decorated version of the one defined by Connes and Kreimer [CK]. As an application, we derive explicit formulas for the coproduct in the motivic multiple polylogarithm Hopf algebra. These formulas play a key role in the mysterious correspondence between the structure of the motivic fundamental group of , where is the group of all th roots of unity, and modular varieties for (see [G6], [G7]). In Section 7 we discuss some general principles relating Feynman integrals and mixed motives. They are suggested by Section 4 and the Feynman integral approach for multiple polylogarithms on curves given in [G7]. The appendix contains background material.
"Galois symmetries of fundamental groupoids and noncommutative geometry." Duke Math. J. 128 (2) 209 - 284, 1 June 2005. https://doi.org/10.1215/S0012-7094-04-12822-2