1 June 2005 Galois symmetries of fundamental groupoids and noncommutative geometry
A. B. Goncharov
Duke Math. J. 128(2): 209-284 (1 June 2005). DOI: 10.1215/S0012-7094-04-12822-2


We define a Hopf algebra of motivic iterated integrals on the line and prove an explicit formula for the coproduct Δ 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 Δ 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 1 - 0 μ N , where μ N is the group of all N th roots of unity, and modular varieties for GL m (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.


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A. B. Goncharov. "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


Published: 1 June 2005
First available in Project Euclid: 2 June 2005

zbMATH: 1095.11036
MathSciNet: MR2140264
Digital Object Identifier: 10.1215/S0012-7094-04-12822-2

Primary: 11G55
Secondary: 11F67 , 11R32 , 20F34

Rights: Copyright © 2005 Duke University Press


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Vol.128 • No. 2 • 1 June 2005
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