Galois symmetries of fundamental groupoids and noncommutative geometry
The author introduces a Hopf algebra of motivic iterated integrals on the line, and proves an explicit formula for the coproduct \(\Delta\) which encodes the group law of the automorphism group of a certain noncommutative variety. He also relates the coproduct \(\Delta\) to the coproduct in the Hopf algebra of decorated rooted plane trivalent trees and give, as an application, explicit formulas for the coproduct in the motivic multiple polylogarithms Hopf algebra. These formulas play a key role in the striking correspondence between the motivic fundamental group of \(\mathbb{P}^1-(\{0,\infty\}^u\mu_N)\) (\(\mu_N\) being the group of all \(N\)th roots of unit) and modular varieties of \(\text{GL}_n\), which was previously investigated by the author. Finally, the author discusses some general principles relating Feynman integrals and mixed motives.
motivic multiple polylogarithm Hopf algebra
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