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Galois symmetries of fundamental groupoids and noncommutative geometry. (English) Zbl 1095.11036
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.

MSC:
 11G55 Polylogarithms and relations with $$K$$-theory 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11R32 Galois theory 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 19E15 Algebraic cycles and motivic cohomology ($$K$$-theoretic aspects) 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
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References:
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