Développements récents sur les groupes de tresses. Applications à la topologie et à l'algèbre
Séminaire Bourbaki (1989-1990)
- Volume: 32, page 17-67
- ISSN: 0303-1179
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topCartier, Pierre. "Développements récents sur les groupes de tresses. Applications à la topologie et à l'algèbre." Séminaire Bourbaki 32 (1989-1990): 17-67. <http://eudml.org/doc/110124>.
@article{Cartier1989-1990,
author = {Cartier, Pierre},
journal = {Séminaire Bourbaki},
keywords = {braid groups; number theory; statistical mechanics; quantum field theory; representations of braid groups; colored braids; cablings; link polynomials; quantum groups; modular Hopf algebras; triangulated Hopf algebras},
language = {fre},
pages = {17-67},
publisher = {Société Mathématique de France},
title = {Développements récents sur les groupes de tresses. Applications à la topologie et à l'algèbre},
url = {http://eudml.org/doc/110124},
volume = {32},
year = {1989-1990},
}
TY - JOUR
AU - Cartier, Pierre
TI - Développements récents sur les groupes de tresses. Applications à la topologie et à l'algèbre
JO - Séminaire Bourbaki
PY - 1989-1990
PB - Société Mathématique de France
VL - 32
SP - 17
EP - 67
LA - fre
KW - braid groups; number theory; statistical mechanics; quantum field theory; representations of braid groups; colored braids; cablings; link polynomials; quantum groups; modular Hopf algebras; triangulated Hopf algebras
UR - http://eudml.org/doc/110124
ER -
References
top- [1] E. Artin - Collected papers (édités par S. Lang et J. Tate), Addison-Wesley, 1965. [Voir en particulier la partie consacrée à la Topologie (p. 416-498) qui contient les articles célèbres : Theorie der Zöpfe (1925), Theory of braids (1947), Braids and permutations (1947).] MR176888
- [2] W. Burau - Über Zopfinvarianten, Hamburg Abh.9 (1932), p. 117-124. JFM58.0614.03
- [3] A.A. Markov - Über die freie Equivalenz der geschlossenen Zöpfe, Recueil Soc. Math. Moscou43 (1936), p. 73-78. Zbl0014.04202
- [4] R.H. Fox - Free differential calculus I : Derivations in the free group ring, Ann. of Math.57 (1953), p. 547-560. Zbl0050.25602MR53938
- [5] E. Fadell and L. Neuwirth - Configuration spaces, Math. Scand.10 (1962), p. 111-118. Zbl0136.44104MR141126
- [6] R.H. Fox and L. Neuwirth - The braid groups, Math. Scand.10 (1962), p. 119-126. Zbl0117.41101MR150755
- [7] K. Reidemeister - Knotentheorie, Erg. Math. Vol. 1, Springer (1932). Zbl0005.12001
- [8] R.H. Crowell and R.H. Fox - Introduction to knot theory, Ginn Co. (1963) (réimprimé dans GTM, vol. 57, Springer (1977)). Zbl0362.55001MR445489
- [9] L. Neuwirth - Knot groups, Annals Math. Studies56, Princeton Univ. Press (1965). Zbl0184.48903MR176462
- [10] W. Magnus, A. Karass, D. Solitar - Combinatorial group theory, Interscience/Wiley (1966). Zbl0138.25604
- [11] J. Birman - Braids, links and mapping class groups, Annals Math. Studies82, Princeton Univ. Press (1974). Zbl0305.57013MR375281
- [12] J. Rolfsen - Knots and links, Publish or Perish (1976). Zbl0339.55004MR515288
- [13] L. Kauffman - Formal knot theory, Math. Notes30, Princeton Univ. Press (1983). Zbl0537.57002MR712133
- [14] G. Burde and H. Zieschang - Knots, W. de Gruyter (1985). Zbl0568.57001MR808776
- [15] A. Gramain - Rapport sur la théorie classique des nœuds (1ère partie), Sém. Bourbaki, exp. n° 485, juin 1976, Springer, Lect. Notes Math.567 (1977), p. 222-237. Zbl0342.57009
- [16] A. Douady - Nœuds et structures de contact en dimension 3 [d'après D. Bennequin], Sém. Bourbaki, exp. n° 604 (février 1983), Astérisque105-106, 1983, p. 129-148. Pour terminer, mentionnons un ouvrage récent explorant la plupart des aspects liés aux groupes de tresses : Zbl0522.53034
- [17] J. Birman and A. Libgober (éditeurs) - Braids, Proceedings of a Research Conference, Contemporary Math. vol. 78, American Math. Soc. (1988). MR975074
- [1] S. Maclane - Natural associativity and commutativity, Rice Univ. Studies, 49 (1963), p. 28-46. Zbl0244.18008MR170925
- [2] S. Eilenberg and G.M. Kelly - Closed categories, in Proc. Conf. Categorical Algebra, Springer (1966). Zbl0192.10604MR225841
- [3] S. Maclane - Categories for the working mathematician, GTM vol. 5, Springer (1974). Zbl0705.18001MR1712872
- [4] G.M. Kelly and M.L. Laplaza - Coherence for compact closed categories, J. Pure Appl. Alg.19 (1980), p. 193-213. Zbl0447.18005MR593254
- [5] T. Tannaka - Über den Dualitätssatz der nicktkommutativen topologischen Gruppen, Tohoku Math. J.45 (1939), p. 1-12. JFM64.0362.01
- [6] M.G. Krein - A principle of duality for a bicompact group and a square block algebra, Dokl. Akad. Nauk SSSR69 (1949), p. 725-728. MR33809
- [7] M. Takesaki - A characterization of group algebras as a converse of Tannaka-Stinespring-Tatsuuma duality theorem, Amer. J. Math.91 (1969), p. 529-564. Zbl0182.18103MR244437
- [8] S. Doplicher and J. Roberts - Duals of compact Lie groups realized in the Cuntz algebras and their actions on C*-algebras, J. Funct. Anal.74 (1987), p. 96-120. Zbl0619.46053MR901232
- [9] S. Doplicher and J. Roberts - Endomorphisms of C*-algebras, cross products and duality for compact groups, Ann. of Math.130 (1989), p. 78-119. Zbl0702.46044MR1005608
- [10] S. Doplicher and J. Roberts - A new duality theory for compact groups, Invent. Math.98 (1989), p. 157-218. Pour une extension dans le cas des groupes quantiques : Zbl0691.22002MR1010160
- [11] P. Ghez, R. Lima and J. Roberts - W*-categories, Pacific J. Math.120 (1985), p. 79-109. Zbl0609.46033MR808930
- [12] S.L. Woronowicz - Duality in the C*-algebra theory, Proc. Int. Cong. Math., Varsovie1983, vol. 2, p. 1347-1350. Zbl0584.46046MR804783
- [13] S.L. Woronowicz - Compact matrix pseudo-groups, Comm. Math. Phys.111 (1987), p. 613-665. Zbl0627.58034MR901157
- [14] S.L. Woronowicz - Tannaka-Krein duality for compact matrix pseudo-groups. Twisted SU(N), Invent. Math.93 (1988), p. 35-76. Zbl0664.58044MR943923
- [15] C. Chevalley - Théorie des groupes de Lie, tome II : groupes algébriques, Hermann (1951). Zbl0186.33104MR51242
- [16] P. Cartier - Dualité de Tannaka et algèbres de Lie, C.R. Acad. Sci. Paris, 242 (1956), p. 322-325. Zbl0070.02506MR75536
- [17] S. Saavedra - Catégories tannakiennes, Lect. Notes Math. vol. 265, Springer (1972). MR338002
- [18] P. Deligne et J.S. Milne - Tannakian categories in Hodge cycles, Motives and Shimura varieties, Lect. Notes Math. vol. 900, Springer (1982). Zbl0477.14004MR654325
- [19] P. Deligne - Catégories tannakiennes, in Grothendieck's Festschrift, Prog. in Math., Birkhäuser (1990), à paraître. Zbl0727.14010MR1106898
- [1] V. Jones - Index for subfactors, Invent. Math.72 (1983), p. 1-25. Zbl0508.46040MR696688
- [2] V. Jones - A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc.12 (1985), p. 103-112. Zbl0564.57006MR766964
- [3] V. Jones - Braid groups, Hecke algebras and type II1 factors, in "Geometric methods in operator algebras", Proc. U.S.-Japan Symposium, Wiley, 1986, p. 242-273. Zbl0659.46054MR866500
- [4] V. Jones - On a certain value of the Kauffman polynomial, Comm. Math. Phys. Zbl0695.57003MR1022523
- [5] V. Jones - Hecke algebra representations of braid groups and link polynomials, Ann. of Math.126 (1987), p. 335-388. Zbl0631.57005MR908150
- [6] V. Jones - Notes on subfactors and statistical mechanics, in "Braid group, knot theory and statistical mechanics" (C.N. Yang et M.L. Ge, éditeurs), World Scientific, 1989, p. 1-25. Zbl0725.46038MR1062421
- [7] A. Ocneanu - Quantized groups, string atgebras and Galois theory for algebras, prépublication Penn. State Univ., 1985.
- [8] A. Ocneanu - A polynomial invariant for knots : a combinatorial and an algebraic approach, à paraître.
- [9] H. Wenzl - Representations of Hecke algebras and subfactors, Thèse, Univ. of Pennsylvania, 1985.
- [10] H. Wenzl - Hecke algebras of type An and subfactors, Invent. Math.92 (1988), p. 349-383. Zbl0663.46055MR936086
- [11] H. Wenzl - Braid group representations and the quantum Yang-Baxter equation, à paraître. Zbl0735.57004
- [12] P. Freyd, D. Yetter, J. Hoste, W. Lickorish, K. Millett, A. Ocneanu - A new polynomial invariant of knots and links, Bull. Amer. Math. Soc.12 (1985), p. 239-246. Zbl0572.57002MR776477
- [13] A. Connes - Indice des sous-facteurs, algèbres de Hecke et théorie des nœuds, Sém. Bourbaki, exp. n° 647, juin 1985, Astérisque133-134 (1986), p. 289-308. Zbl0597.57005
- [14] F. Goodman, P. de la Harpe et V. Jones - Coxeter-Dynkin diagrams and towers of algebras, M.S.R.I. Publ. vol. 14, Springer (1989). Zbl0698.46050
- [15] P. de la Harpe, M. Kervaire et C. Weber - On the Jones polynomial, Ens. Math.32 (1986), p. 271-335. Zbl0622.57002MR874691
- [16] P. Vogel - Représentations et traces des algèbres de Hecke, polynôme de Jones-Conway, Ens. Math.34 (1988), p. 333-356. Zbl0686.57003MR979646
- [17] P.N. Hoefsmit - Representations of Hecke algebras of finite groups with BN pairs of classical type, Thèse, Univ. of British Columbia, 1974.
- [18] M. Atiyah - Topological quantum field theories, Publ. I.H.E.S.68 (1988), p. 175-186. Zbl0692.53053MR1001453
- [19] N. Reshetikhin and V. Turaev - Invariants of 3-manifolds via link polynomial and quantum groups, à paraître. Zbl0725.57007
- [20] E. Witten - Quantum field theory and the Jones polynomial, Comm. Math. Phys.121 (1989), p. 351-399. Zbl0667.57005MR990772
- [1] V. Drinfeld - Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl.32 (1985), p. 254-258. Zbl0588.17015MR802128
- [2] V. Drinfeld - Quantum groups, Proc. Int. Cong. Math., Berkeley1986, vol. 1, p. 798-820. Zbl0667.16003MR934283
- [3] J.-L. Verdier - Groupes quantiques [d'après V. Drinfeld], Sém. Bourbaki, exp. n° 685, juin 1987, Astérisque152-153 (1987), p. 305-319. Sur les représentations des groupes quantiques : Zbl0645.16006MR936861
- [4] J. Jimbo - A q-analogue of U(gl(N + 1)), Hecke algebras and the Yang-Baxter equation, Lett. Math. Phys.11 (1986), p. 247-252. Zbl0602.17005MR841713
- [5] A. Kirillov and N. Reshetikhin - Representations of the algebra Uq(sl(2)), q-orthogonal polynomials and invariants of links, LOMI preprint, Leningrad, 1988. MR977004
- [6] G. Lusztig - Quantum deformations of certain simple modules over enveloping algebras, Adv. Math.70 (1988), p. 237-249. Zbl0651.17007MR954661
- [7] M. Rosso - Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra, Comm. Math. Phys.117 (1988), p. 581-593. [Corrections dans la thèse de l'auteur.] Zbl0651.17008MR953821
- [8] V. Drinfeld - Sur les algèbres de Hopf presque cocommutatives, à paraître dans "Algebra and Analysis", Leningrad, 1989 (en russe).
- [9] V. Drinfeld - Quasi-algèbres de Hopf, Algebra and Analysis, vol. 1, n° 2 (1989), p. 30-46 (en russe). MR1091757
- [10] N. Reshetikhin - Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links, I, II, LOMI preprint, Léningrad, 1988.
- [11] N. Reshetikhin - Algèbres de Hopf quasi-triangulaires et invariants des nœuds, Algebra and Analysis, vol. 1, n° 2 (1989), p. 169-188 (en russe). Zbl0715.17016
- [12] N. Reshetikhin, L. Takhtadjan et L. Faddeev - Groupes de Lie quantiques et algèbres de Lie, Algebra and Analysis, vol. 1, n° 2 (1989), p. 178-205 (en russe). Zbl0715.17015
- [13] V. Turaev - The Yang-Baxter equation and invariants of links, Invent. Math.92 (1988), p. 527-553. Zbl0648.57003MR939474
- [14] V. Turaev - Algebra of loops on surfaces, algebra of knots, and quantization, LOMI preprint, Léningrad, 1988. Référence supplémentaire : MR1062423
- [15] C.M. Ringel - Hall algebras and quantum groups, à paraître (Bielefeld, RFA). Zbl0735.16009
- [1] V.I. Arnold - The cohomology ring of the colored braid group, Mat. Zametki5 (1969), p. 227-231. Zbl0277.55002MR242196
- [2] V.I. Arnold - Topological invariants of algebraic functions II, Funct. Anal. Appl.4 (1970), p. 91-98. Zbl0239.14012MR276244
- [3] E. Brieskorn - Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe, Invent. Math.12 (1971), p. 57-61. Zbl0204.56502MR293615
- [4] E. Brieskorn - Sur les groupes de tresses [d'après V.I. Arnold], Sém. Bourbaki, exp. n° 401, novembre 1971, Springer, Lect. Notes Math.317 (1973), p. 21-44. Zbl0277.55003MR422674
- [5] P. Cartier - Arrangements d'hyperplans : un chapitre de Géométrie combinatoire, Sém. Bourbaki, exp. n° 582, novembre 1980, Springer, Lect. Notes Math.901 (1982), p. 1-22. Zbl0483.51011
- [6] F.A. Garside - The braid groups and other groups, Quart. J. Math. Oxford20, (1969), p. 235-254. Zbl0194.03303MR248801
- [7] E. Brieskorn and K. Saito - Artin-Gruppen und Coxeter-gruppen, Invent. Math.17, (1972), p. 245-271. Zbl0243.20037MR323910
- [8] P. Deligne - Les immeubles des groupes de tresses généralisés, Invent. Math.17, (1972), p. 273-302. Zbl0238.20034MR422673
- [9] M. Falk and R. Randell - Pure braid groups and products of free groups, in "Braids", Contemporary Math., Amer. Math. Soc. (1988), vol. 78, p. 217-228. Zbl0669.20028MR975081
- [1] P. Cartier - Jacobiennes généralisées, monodromie unipotente et intégrales itérées, Sém. Bourbaki, exp. n° 687, novembre 1987, Astérisque161–162 (1988), p. 31-52. Voici quelques références complétant la bibliographie de cet exposé : Zbl0688.14036
- [2] A.A. Belavin and V. Drinfeld - Solution of the classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl.16 (1982), p. 1-29. Zbl0511.22011MR674005
- [3] T. Kohno - Homology of a local system on the complement of hyperplanes, Proc. Japan Acad. Sci.62, série A (1986), p. 144-147. Zbl0611.55005MR846350
- [4] T. Kohno - One-parameter family of linear representations of Artin's braid groups, Adv. Stud. in Pure Math.12 (1987), p. 189-200. Zbl0654.20037MR948243
- [5] T. Kohno - Linear representations of braid groups and classical Yang-Baxter equations, in "Braids", Contemporary Math., Amer. Math. Soc. (1988), vol. 78, p. 339-363. Zbl0661.20026MR975088
- [6] A.B. Givental and V.V. Shekhtman - Monodromy groups and Hecke algebras, Usp. Math. Nauk12 n° 4 (1987), p. 138- .
- [7] K. Aomoto - A construction of integrable differential system associated with braid groups, in "Braids", Contemporary Math., Amer. Math. Soc. (1988), vol. 78, p. 1-11. Zbl0671.58016MR975075
- [8] A.B. Givental - Twisted Picard-Lefschetz formulas, Funct. Anal. Appl.22 (1988), p. 10-18. Zbl0665.32011MR936695
- [9] A.A. Belavin, A.N. Polyakov and A.B. Zamolodchikov - Infinite dimensional symmetries in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984), p. 333-380. Zbl0661.17013MR757857
- [10] V.G. Knizhnik and A.B. Zamolodchikov - Current algebras and Wess-Zumino models in two dimensions, Nucl. Phys. B 247 (1984), p. 83-103. Zbl0661.17020MR853258
- [11] A. Tsuchiya and Y. Kanie - Vertex operators in conformal field theory on P1 and monodromy representations of braid group, Adv. Studies in Pure Math.12 (1988), p. 297-372. Zbl0661.17021MR972998
- [1] C.N. Yang and M.L. Ge (éditeurs) - Braid group, knot theory and statistical mechanics, Adv. Series Math. Phys. vol. 9, World Scientific (1989). Zbl0716.00010MR1062420
- [2] M.L. Ge and N.C. Song (éditeurs) - Conformal field theory and braid group (Nankai lectures in mathematical physics, 1988), World Scientific (1989).
- [3] L. Kauffman - Statistical mechanics and the Jones polynomial, in "Braids", Contemporary Math., Amer. Math. Soc. (1988), vol. 78, p. 263-297. Zbl0664.57002MR975085
- [4] L. Kauffman - New invariants in the theory of knots, Astérisque163-164 (1988), p. 137-219. Zbl0673.57007MR999974
- [5] L. Kauffman - Knots, abstract tensors and the Yang-Baxter equation, prépublication IHES/M/89/24 (avril 1989). MR1146944
- [6] R.J. Baxter - Exactly solved models in statistical mechanics, Acad. Press (1982). Zbl0723.60120MR690578
- [7] F.Y. Wu - The Potts model, Rev. Mod. Phys. vol. 54, n° 1, Janvier 1982. MR641370
- [8] R. Penrose - Applications of negative dimensional tensors, in Comb. Math. and its Appl. (Welsch éditeur), Acad. Press (1971). Zbl0216.43502MR281657
- [9] L. Kauffman - State models and the Jones polynomial, Topology26 (1987), p. 395-407. Zbl0622.57004MR899057
- [10] V. Jones - On knot invariants related to statistical mechanics models, Pac. J. Math.138 (1989), p.. Zbl0695.46029
- [11] M. Rosso - Groupes quantiques et modèles à vertex de V. Jones en théorie des nœuds, C.R. Acad. Sci. Paris, Série I, vol. 307 (1988), p. 207-210. Zbl0651.17009
- [1] V. Dotsenko and V. Fateev - Conformal algebra and multipoint correlation functions in 2D statistical mechanics, Nucl. Phys. B240 (1984), p. 312-. MR762194
- [2] J. Fröhlich - Statistics of fields, the Yang-Baxter equation and the theory of braids and links, Cargèse lectures1987, G.'t Hooft (éditeur), Plenum Press (1988). MR1008276
- [3] J. Fröhlich - Statistics and monodromy in two- and three-dimensional quantum field theory, in "Differential geometrical methods in mathematical physics", K. Bleuler et M.Werner (éditeurs), Dordrecht (1988). MR981379
- [4] K.H. Rehren and B. Shroer - Einstein causality and Artin braids, prépublication, Freie Universität Berlin, 1988.
- [5] J. Fröhlich and C. King - Two-dimensional conformal field theory and three-dimensional topology, prépublication ETH/TH/89-9, Zürich, 1989. MR1028434
- [6] G. Felder, J. Fröhlich and G. Keller - Braid matrices and structure constants for minimal conformal models, Comm. Math. Phys.124 (1989), p. 647-664. Zbl0696.17009MR1014118
- [7] J. Fröhlich, G. Gabbiani and P.A. Marchetti - Braid statistics in three-dimensional local quantum theory, in the Proceedings of the Banff Summer School in Theoretical physics (août 1989) on "Physics, Geometry and Topology", à paraître. Zbl0732.53076
- [1] A. Grothendieck - Esquisse d'un programme, Notes miméographiées, Univ. Montpellier, 1982.
- [2] R. Coleman - Dilogarithms, regulators and p-adic L-functions, Invent. Math.69 (1982), p. 171-208. Zbl0516.12017MR674400
- [3] Y. Ihara - Profinite braid groups, Galois representations and complex multiplication, Ann. of Math.123 (1986), p. 3-106. Zbl0595.12003MR825839
- [4] Y. Ihara - On Galois representations arising from towers of coverings of P1{0,1,∞, Invent. Math.86 (1986), p. 427-459.
- [5] B.H. Matzat - Konstruktive Galois Theorie, Lect. Notes Math.1284, Springer (1987). Zbl0634.12011MR1004467
- [6] T. Kohno and T. Oda - The lower central series of the pure braid group of an algebraic curve, Adv. Studies in Pure Math.12 (1988). Zbl0642.20035MR948244
- [7] P. Deligne - Le groupe fondamental de la droite projective moins trois points, in "Galois groups over Q", M.S.R.I. Publ. vol. 16, Springer, 1989. Zbl0742.14022MR1012168
Citations in EuDML Documents
top- Yvette Kosmann-Schwarzbach, Quasi-bigèbres jacobiennes
- Christian Kassel, L'ordre de Dehornoy sur les tresses
- Christian Kassel, Invariants des nœuds, catégories tensorielles et groupes quantiques
- Marc Rosso, Représentations des groupes quantiques
- Vaughan Jones, Fusion en algèbres de von Neumann et groupes de lacets
- Pierre Cartier, Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents
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