Duke Mathematical Journal

Galois symmetries of fundamental groupoids and noncommutative geometry

A. B. Goncharov

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We define a Hopf algebra of motivic iterated integrals on the line and prove an explicit formula for the coproduct $\Delta$ 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 $\Delta$ 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 $\mathbb{P}^1 - (\{0, \infty\}\cup \mu_N)$, where $\mu_N$ 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.

Article information

Duke Math. J. Volume 128, Number 2 (2005), 209-284.

First available: 2 June 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G55: Polylogarithms and relations with $K$-theory
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11R32: Galois theory 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx]


Goncharov, A. B. Galois symmetries of fundamental groupoids and noncommutative geometry. Duke Mathematical Journal 128 (2005), no. 2, 209--284. doi:10.1215/S0012-7094-04-12822-2. http://projecteuclid.org/euclid.dmj/1117728416.

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