Duke Mathematical Journal

Galois symmetries of fundamental groupoids and noncommutative geometry

A. B. Goncharov

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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

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

Dates
First available: 2 June 2005

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1117728416

Digital Object Identifier
doi:10.1215/S0012-7094-04-12822-2

Mathematical Reviews number (MathSciNet)
MR2140264

Zentralblatt MATH identifier
02201102

Subjects
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]

Citation

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.


Export citation

References

  • A. A. Beilinson and P. Deligne, Motivic polylogarithms and Zagier's conjecture, unpublished manuscript, version of 1992.
  • A. A. Beilinson, A. B. Goncharov, V. V. Schechtman, and A. N. Varchenko, ``Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of pairs of triangles on the plane'' in The Grothendieck Festschtrift, Vol. I, Progr. Math. 86, Birkhäuser, Boston, 1990, 135--172.
  • P. Belkale and P. Brosnan, Matroids, motives, and a conjecture of Kontsevich, Duke Math. J. 116 (2003), 147--188.
  • A. Borel, Cohomologie de $\rm SL\sbn$ et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 613--636.; Errata, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), 373. ;
  • D. J. Broadhurst, J. A. Gracey, and D. Kreimer, Beyond the triangle and uniqueness relations: Non-zeta counterterms at large $N$ from positive knots, Z. Phys. C 75 (1997), 559--574.
  • K. T. Chen, Iterated path integrals. Bull. Amer. Math. Soc. 83 (1977), 831--879.
  • A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys. 199 (1998), 203--242.
  • P. Deligne, ``Le groupe fondamental de la droite projective moins trois points'' in Galois Groups over $\mathbbQ$ (Berkeley, 1987), Math. Sci Res. Inst. Publ. 16, Springer, New York, 1989, 79--297.
  • --. --. --. --., ``Catégories tannakiennes'' in The Grothendieck Festschrift, Vol. II, Progr. Math. 87, Birkhäuser, Boston, 1990, 111--195.
  • P. Deligne and A. B. Goncharov, Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. École Norm Sup. (4), 38 (2005), 1--56.
  • V. G. Drinfeld, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $\Gal(\overline\mathbbQ/\mathbbQ)$ (in Russian), Algebra i Analiz 2, no. 4 (1990), 149--181.; English translation in Leningrad Math. J. 2 (1991), 829--860.
  • A. B. Goncharov, ``Polylogarithms and motivic Galois groups'' in Motives (Seattle, 1991), Proc. Sympos. Pure Math. 55, Part 2, Amer. Math. Soc., Providence, 1994, 43--96.
  • --. --. --. --., ``Polylogarithms in arithmetic and geometry'' in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 374--387.
  • --. --. --. --., ``Mixed elliptic motives'' in Galois representations in Arithmetic Algebraic Geometry, (Durham, 1996), London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press, Cambridge, 1998, 147--221.
  • --. --. --. --., Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998), 497--516.
  • --. --. --. --., Volumes of hyperbolic manifolds and mixed Tate motives, J. Amer. Math. Soc. 12 (1999), 569--618.
  • --. --. --. --., The dihedral Lie algebras and Galois symmetries of $\pi_1^(l)(\mathbbP^1 - (\0, \infty\\cup \mu_N)$, Duke Math. J. 110 (2001), 397--487.
  • --. --. --. --., ``Multiple $\zeta$-values, Galois groups, and geometry of modular varieties'' in European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math. 201, Birkhäuser, Basel, 2001, 361--392.
  • --------, Multiple $\zeta$-numbers, hyperlogarithms and mixed Tate motives, preprint, June 1993, Mathematical Sciences Research Institute, Berkeley, no. 058-93.
  • --------, Galois groups, geometry of modular varieties and graphs, Proceedings of Arbeitstagung, June 1999, preprint, no. MPIM1999-50f, http://www.mpim-bonn.mpg.de
  • --------, Multiple polylogarithms and mixed Tate motives, preprint.
  • --------, Periods and mixed motives, preprint.
  • D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys. 2 (1998), 303--334.
  • M. Levine, ``Tate motives and the vanishing conjectures for algebraic $K$-theory'' in Algebraic $K$-Theory and Algebraic Topology (Lake Louise, Canada, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 407, Kluwer, Dordrecht, 1993, 167--188.
  • J.-L. Loday, ``Dialgebras'' in Dialgebras and Related Operads, Lecture Notes in Math. 1763, Springer, Berlin, 2001, 7--66.
  • --. --. --. --., Arithmetree, J. Algebra 258 (2002), 275--309.
  • D. Zagier, ``Periods of modular forms, traces of Hecke operators, and multiple zeta values'' in Research into Automorphic Forms and $L$ Functions (Kyoto, 1992) (in Japanese), Sūrikaisekikenkyūsho Kōkyūroku 843, Kyoto Univ., Kyoto, 1993, 162--170.