Book contents
- Frontmatter
- Contents
- List of contributors
- Preface
- 1 On the local structure of ordinary Hecke algebras at classical weight one points
- 2 Vector bundles on curves and p-adic Hodge theory
- 3 Around associators
- 4 The stable Bernstein center and test functions for Shimura varieties
- 5 Conditional results on the birational section conjecture over small number fields
- 6 Blocks for mod p representations of GL2(ℚp)
- 7 From étale P+-representations to G-equivariant sheaves on G/P
- 8 Intertwining of ramified and unramified zeros of Iwasawa modules
- References
3 - Around associators
Published online by Cambridge University Press: 05 October 2014
Book contents
- Frontmatter
- Contents
- List of contributors
- Preface
- 1 On the local structure of ordinary Hecke algebras at classical weight one points
- 2 Vector bundles on curves and p-adic Hodge theory
- 3 Around associators
- 4 The stable Bernstein center and test functions for Shimura varieties
- 5 Conditional results on the birational section conjecture over small number fields
- 6 Blocks for mod p representations of GL2(ℚp)
- 7 From étale P+-representations to G-equivariant sheaves on G/P
- 8 Intertwining of ramified and unramified zeros of Iwasawa modules
- References
Summary
Abstract
This is a concise exposition of recent developments around the study of associators. It is based on the author's talk at the Mathematische Arbeitstagung in Bonn, June 2011 (cf. [F11b]) and at the Automorphic Forms and Galois Representations Symposium in Durham, July 2011. The first section is a review of Drinfeld's definition [Dr] of associators and the results [F10, F11a] concerning the definition. The second section explains the four pro-unipotent algebraic groups related to associators; the motivic Galois group, the Grothendieck-Teichmüller group, the double shuffle group and the Kashiwara-Vergne group. Relationships, actually inclusions, between them are also discussed.
1. Associators
We recall the definition of associators [Dr] and explain our main results in [F10, F11a] concerning the defining equations of associators.
The notion of associators was introduced by Drinfeld in [Dr]. They describe monodromies of the KZ (Knizhnik–Zamolodchikov) equations. They are essential for the construction of quasi-triangular quasi-Hopf quantized universal enveloping algebras ([Dr]), for the quantization of Lie-bialgebras (Etingof–Kazhdan quantization [EtK]), for the proof of formality of chain operad of little discs by Tamarkin [Ta] (see also Ševera and Willwacher [SW]) and also for the combinatorial reconstruction of the universal Vassiliev knot invariant (the Kontsevich invariant [Kon, Ba95]) by Bar-Natan [Ba97], Cartier [C], Kassel and Turaev [KssT], Le and Murakami [LM96a] and Piunikhin [P].
- Type
- Chapter
- Information
- Automorphic Forms and Galois Representations , pp. 105 - 117Publisher: Cambridge University PressPrint publication year: 2014
References
[An] ; Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses, 17, Société Mathématique de France, Paris, 2004.
[AlET] , and ; Drinfeld associators, braid groups and explicit solutions of the Kashiwara-Vergne equations, Publ. Math. Inst. Hautes Études Sci. 112 (2010), 143–189.Google Scholar
[AlM] , and ; On the Kashiwara-Vergne conjecture, Invent. Math. 164 (2006), no. 3, 615–634.Google Scholar
[AlT] , and ; The Kashiwara-Vergne conjecture and Drinfeld's associators, Ann. of Math. (2) 175 (2012), no. 2, 415–463.Google Scholar
[Ba97] ,; Non-associative tangles, Geometric topology (Athens, GA, 1993), 139–183, AMAPIP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, RI, 1997.
[BaD] , and ; Pentagon and hexagon equations following Furusho, Proc. Amer. Math. Soc. 140 (2012), no. 4, 1243–1250.Google Scholar
[Bel] , Galois extensions of a maximal cyclotomic field, (Russian)Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 2, 267–276, 479.Google Scholar
[BeF] and ; The double shuffle relations for p-adic multiple zeta values, Primes and Knots, AMS Contemporary Math, Vol 416, 2006, 9–29.Google Scholar
[C] ; Construction combinatoire des invariants de Vassiliev-Kontsevich des nœuds, C. R. Acad. Sci. Paris Ser. I Math. 316 (1993), no. 11, 1205–1210.Google Scholar
[De] ; Le groupe fondamental de la droite projective moins trois points, Galois groups over Q (Berkeley, CA, 1987), 79–297, Math. S. Res. Inst. Publ., 16, Springer, New York-Berlin, 1989.
[DeG] and ; Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. Ecole Norm. Sup. (4) 38 (2005), no. 1, 1–56.Google Scholar
[DeM] and ; Tannakian categories, in Hodge cycles, motives, and Shimura varieties (, , , editors), Lecture Notes in Mathematics 900, Springer-Verlag, 1982.
[Dr] .; On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(Q/Q), Leningrad Math. J. 2 (1991), no. 4, 829–860.Google Scholar
[En] ; Quasi-reflection algebras and cyclotomic associators, Selecta Math. (N.S.) 13 (2007), no. 3, 391–463.Google Scholar
[EnF] and ; Mixed Pentagon, octagon and Broadhurst duality equation, J. Pure Appl. Algebra, 216, (2012), no. 4, 982–995. no. 4,Google Scholar
[EtK] and ; Quantization of Lie bialgebras. II, Selecta Math. 4 (1998), no. 2, 213–231.Google Scholar
[Eu] , Meditationes circa singulare serierum genus, Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 140–186 and Opera Omnia: Series 1, Volume 15, pp. 217–267 (also available from www.math.dartmouth.edu/euler/).
[F03] ; The multiple zeta value algebra and the stable derivation algebra, Publ.Res.Inst.Math.Sci. 39, (2003), no. 4, 695–720.Google Scholar
[F04] ; p-adic multiple zeta values I – p-adic multiple polylogarithms and the p-adic KZ equation, Invent. Math. 155, (2004), no. 2, 253–286.Google Scholar
[F07] ; p-adic multiple zeta values II – tannakian interpretations, Amer. J. Math. 129, (2007), no 4, 1105–1144.Google Scholar
[F11a] ; Double shuffle relation for associators, Annals of Math. 174 (2011), no. 1, 341–360.Google Scholar
[F11b] ; Four groups related to associators, report of the Mathematische Arbeitstagung in Bonn, June 2011, arXiv:1108.3389.
[F12] ; Geometric interpretation of double shuffle relation for multiple L-values, Galois-Teichmüller theory and Arithmetic Geometry, Advanced Studies in Pure Math 63 (2012), 163–187.Google Scholar
[FJ] and ; Regularization and generalized double shuffle relations for p-adic multiple zeta values, Compositio Math. 143, (2007), 1089–1107.Google Scholar
[G] ; Esquisse d'un programme, 1983, available on pp. 243–283. London Math. Soc. LNS 242, Geometric Galois actions, 1, 5–48, Cambridge University Press.
[IkKZ] , and ; Derivation and double shuffle relations for multiple zeta values, Compos. Math. 142 (2006), no. 2, 307–338.Google Scholar
[Iy90] ; Braids, Galois groups, and some arithmetic functions, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 99–120, Math. Soc. Japan, Tokyo, 1991.
[Iy94] ; On the embedding of Gal(Q/Q) into GT, London Math. Soc. LNS 200, The Grothendieck theory of dessins d'enfants (Luminy, 1993), 289–321, Cambridge University Press, Cambridge, 1994.
[KswV] and ; The Campbell-Hausdorff formula and invariant hyperfunctions, Invent. Math. 47 (1978), no. 3, 249–272.Google Scholar
[KssT] and ; Chord diagram invariants of tangles and graphs, Duke Math. J. 92 (1998), no. 3, 497–552.Google Scholar
[Kon] ; Vassiliev's knot invariants, I. M. Gelfand Seminar, 137–150, Adv. Soviet Math., 16, Part 2, Amer. Math. Soc., Providence, RI, 1993.
[LM96a] and ; The universal Vassiliev-Kontsevich invariant for framed oriented links, Compositio Math. 102 (1996), no. 1, 41–64.Google Scholar
[LM96b] ; Kontsevich's integral for the Kauffman polynomial, Nagoya Math. J. 142 (1996), 39–65.Google Scholar
[P] ; Combinatorial expression for universal Vassiliev link invariant, Comm. Math. Phys. 168 (1995), no. 1, 1–22.Google Scholar
[R] ; Doubles melangés des polylogarithmes multiples aux racines de l'unité, Publ. Math. Inst. Hautes Etudes Sci. 95 (2002), 185–231.Google Scholar
[SW] and ; Equivalence of formalities of the little discs operad, Duke Math. J. 160 (2011), no. 1, 175–206.Google Scholar
[Ta] ; Formality of chain operad of little discs, Lett. Math. Phys. 66 (2003), no. 1–2, 65–72.Google Scholar
[Te] ; Geometry of multiple zeta values, International Congress of Mathematicians. Vol. II, 627–635, Eur. Math. Soc., Zürich, 2006.
[Wi] ; M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra, arXiv:1009.1654, preprint (2010).
[Wo] ; Monodromy of iterated integrals and non-abelian unipotent periods, Geometric Galois actions, 2, 219–289, London Math. Soc. LNS 243, Cambridge University Press, Cambridge, 1997.
[Z] ; Evaluation of the multiple zeta values ζ(2, ..., 2, 3, 2, ..., 2), Annals of Math., 175, (2012), no. 2, 977–1000.Google Scholar
- 2
- Cited by