Skip to main content Accessibility help
×
Home
Hostname: page-component-684899dbb8-ndjvl Total loading time: 0.357 Render date: 2022-05-27T13:45:58.688Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

3 - Around associators

Published online by Cambridge University Press:  05 October 2014

Hidekazu Furusho
Affiliation:
Nagoya University
Fred Diamond
Affiliation:
King's College London
Payman L. Kassaei
Affiliation:
King's College London
Minhyong Kim
Affiliation:
University of Oxford
Get access

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
Publisher: Cambridge University Press
Print publication year: 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[An] André, Y.; Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses, 17, Société Mathématique de France, Paris, 2004.
[AlET] Alekseev, A., Enriquez, B. and Torossian, C.; 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] Alekseev, A., and Meinrenken, E.; On the Kashiwara-Vergne conjecture, Invent. Math. 164 (2006), no. 3, 615–634.Google Scholar
[AlT] Alekseev, A., and Torossian, C.; The Kashiwara-Vergne conjecture and Drinfeld's associators, Ann. of Math. (2) 175 (2012), no. 2, 415–463.Google Scholar
[Ba95] Alekseev, A., Bar-Natan, D.; On the Vassiliev knot invariants, Topology 34 (1995), no. 2, 423–472.Google Scholar
[Ba97] Alekseev, A.,; Non-associative tangles, Geometric topology (Athens, GA, 1993), 139–183, AMAPIP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, RI, 1997.
[BaD] Alekseev, A., and Dancso, Z.; Pentagon and hexagon equations following Furusho, Proc. Amer. Math. Soc. 140 (2012), no. 4, 1243–1250.Google Scholar
[Bel] Bely, G. V., Galois extensions of a maximal cyclotomic field, (Russian)Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 2, 267–276, 479.Google Scholar
[BeF] Besser, A. and Furusho, H.; The double shuffle relations for p-adic multiple zeta values, Primes and Knots, AMS Contemporary Math, Vol 416, 2006, 9–29.Google Scholar
[Br] Brown, F.; Mixed Tate Motives over Spec(Z), Annals of Math., 175, (2012) no. 2, 949–976.Google Scholar
[C] Cartier, P.; 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] Deligne, P.; 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] Deligne, P. and Goncharov, A.; Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. Ecole Norm. Sup. (4) 38 (2005), no. 1, 1–56.Google Scholar
[DeM] Deligne, P. and Milne, J.; Tannakian categories, in Hodge cycles, motives, and Shimura varieties (P., Deligne, J., Milne, A., Ogus, K.-Y., Shih editors), Lecture Notes in Mathematics 900, Springer-Verlag, 1982.
[Dr] Drinfel'd, V. G.; 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] Enriquez, B.; Quasi-reflection algebras and cyclotomic associators, Selecta Math. (N.S.) 13 (2007), no. 3, 391–463.Google Scholar
[EnF] Deligne, P. and Furusho, H.; Mixed Pentagon, octagon and Broadhurst duality equation, J. Pure Appl. Algebra, 216, (2012), no. 4, 982–995. no. 4,Google Scholar
[EtK] Etingof, P. and Kazhdan, D.; Quantization of Lie bialgebras. II, Selecta Math. 4 (1998), no. 2, 213–231.Google Scholar
[Eu] Euler, L., 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] Furusho, H.; The multiple zeta value algebra and the stable derivation algebra, Publ.Res.Inst.Math.Sci. 39, (2003), no. 4, 695–720.Google Scholar
[F04] Furusho, H.; 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] Furusho, H.; p-adic multiple zeta values II – tannakian interpretations, Amer. J. Math. 129, (2007), no 4, 11051144.Google Scholar
[F10] Furusho, H.; Pentagon and hexagon equations, Annals of Math. 171 (2010), no. 1, 545–556.Google Scholar
[F11a] Furusho, H.; Double shuffle relation for associators, Annals of Math. 174 (2011), no. 1, 341–360.Google Scholar
[F11b] Furusho, H.; Four groups related to associators, report of the Mathematische Arbeitstagung in Bonn, June 2011, arXiv:1108.3389.
[F12] Furusho, H.; 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] Furusho, H. and Jafari, A.; Regularization and generalized double shuffle relations for p-adic multiple zeta values, Compositio Math. 143, (2007), 1089–1107.Google Scholar
[G] Grothendieck, A.; 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] Ihara, K., Kaneko, M. and Zagier, D.; Derivation and double shuffle relations for multiple zeta values, Compos. Math. 142 (2006), no. 2, 307–338.Google Scholar
[Iy90] Ihara, Y.; 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] Ihara, Y.; 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.
[JS] Joyal, A. and Street, R.; Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20–78.Google Scholar
[KswV] Kashiwara, M. and Vergne, M.; The Campbell-Hausdorff formula and invariant hyperfunctions, Invent. Math. 47 (1978), no. 3, 249–272.Google Scholar
[KssT] Kassel, C. and Turaev, V.; Chord diagram invariants of tangles and graphs, Duke Math. J. 92 (1998), no. 3, 497–552.Google Scholar
[Kon] Kontsevich, M.; Vassiliev's knot invariants, I. M. Gelfand Seminar, 137–150, Adv. Soviet Math., 16, Part 2, Amer. Math. Soc., Providence, RI, 1993.
[LM96a] Le, T.Q.T. and Murakami, J.; The universal Vassiliev-Kontsevich invariant for framed oriented links, Compositio Math. 102 (1996), no. 1, 41–64.Google Scholar
[LM96b] Le, T.Q.T.; Kontsevich's integral for the Kauffman polynomial, Nagoya Math. J. 142 (1996), 39–65.Google Scholar
[P] Piunikhin, S.; Combinatorial expression for universal Vassiliev link invariant, Comm. Math. Phys. 168 (1995), no. 1, 1–22.Google Scholar
[R] Racinet, G.; Doubles melangés des polylogarithmes multiples aux racines de l'unité, Publ. Math. Inst. Hautes Etudes Sci. 95 (2002), 185–231.Google Scholar
[S] Schneps, L.; Double shuffle and Kashiwara-Vergne Lie algebra, J. Algebra 367 (2012), 54–74.Google Scholar
[SW] Ševera, P and Willwacher, T.; Equivalence of formalities of the little discs operad, Duke Math. J. 160 (2011), no. 1, 175–206.Google Scholar
[Ta] Tamarkin, D. E.; Formality of chain operad of little discs, Lett. Math. Phys. 66 (2003), no. 1–2, 65–72.Google Scholar
[Te] Terasoma, T.; Geometry of multiple zeta values, International Congress of Mathematicians. Vol. II, 627–635, Eur. Math. Soc., Zürich, 2006.
[Wi] Willwacher, T.; M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra, arXiv:1009.1654, preprint (2010).
[Wo] Wojtkowiak, Z.; 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] Zagier, D.; Evaluation of the multiple zeta values ζ(2, ..., 2, 3, 2, ..., 2), Annals of Math., 175, (2012), no. 2, 977–1000.Google Scholar
1
Cited by

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×