Représentations des groupes quantiques

Marc Rosso

Séminaire Bourbaki (1990-1991)

  • Volume: 33, page 443-483
  • ISSN: 0303-1179

How to cite

top

Rosso, Marc. "Représentations des groupes quantiques." Séminaire Bourbaki 33 (1990-1991): 443-483. <http://eudml.org/doc/110147>.

@article{Rosso1990-1991,
author = {Rosso, Marc},
journal = {Séminaire Bourbaki},
keywords = {representations of quantized universal enveloping algebras; universal -matrix; Yang-Baxter equation; rational conformal field theory; fusion rules; monodromy representations; compact quantum matrix groups},
language = {fre},
pages = {443-483},
publisher = {Société Mathématique de France},
title = {Représentations des groupes quantiques},
url = {http://eudml.org/doc/110147},
volume = {33},
year = {1990-1991},
}

TY - JOUR
AU - Rosso, Marc
TI - Représentations des groupes quantiques
JO - Séminaire Bourbaki
PY - 1990-1991
PB - Société Mathématique de France
VL - 33
SP - 443
EP - 483
LA - fre
KW - representations of quantized universal enveloping algebras; universal -matrix; Yang-Baxter equation; rational conformal field theory; fusion rules; monodromy representations; compact quantum matrix groups
UR - http://eudml.org/doc/110147
ER -

References

top
  1. [A-G-S] L. Alvarez-Gaumé, C. Gomez, G. Sierra - Quantum group interpretation of some conformal field theories, prépublication CERN. 
  2. [A-P-W1] H.H. Andersen, P. Polo et Wen K. - Representations of quantum algebras, Invent. Math.104 (1991), 1-60. Zbl0724.17012MR1094046
  3. [A-P-W2] H.H. Andersen, P. Polo et Wen K. - Injective modules for quantum algebras, à paraître dans Amer. J. Math. Zbl0755.17005MR1165354
  4. [A-W] H.H. Andersen, Wen K. - Representations of quantum algebras : the mixed case, prépublication Aarhus Universitet1991, n° 5. MR1162431
  5. [An] N. Andruskiewitsh - Some exceptional compact matrix pseudogroups, prépublication École Polytechnique. 
  6. [Ar] D. Arnaudon - Periodic and flat irreducible representations of SU(3)q, Comm. Math. Phys.134 (1990), 523-537. Zbl0718.17015MR1086742
  7. [Ar-Ch1] D. Arnaudon, A. Chakrabarti - Periodic and partially periodic representations of SU(N)q, à paraître dans Comm. Math. Phys. Zbl0751.17006MR1121129
  8. [Ar-Ch2] D. Arnaudon, A. Chakrabarti - Flat periodic representations of Uq(G), prépublication École PolytechniqueA023-1190. 
  9. [B] O. Babelon - Extended conformal algebra of Yang-Baxter equation, Phys. Lett. B125 (1988), 523-529. MR972439
  10. [Ba1] R.J. Baxter - Partition function of the eight-vertex lattice model, Ann. Phys.70 (1972), 193-228. Zbl0236.60070MR290733
  11. [Ba2] R.J. Baxter - Exactly solved models in Statistical Mechanics, Academic Press, London, 1982. Zbl0723.60120MR690578
  12. [B-S] V.V. Bazhanov, Y.G. Stroganov - Chiral Potts models as a descendant of the six vertex model, J. Stat. Phys.51 (1990), 799-817. Zbl0718.60123MR1063182
  13. [B-M-L] A.A. Beilinson, R. Mac Pherson, G. Lusztig - A geometric setting for the quantum deformation of GLn, Duke Math. J.61 (1990). Zbl0713.17012MR1074310
  14. [Be] D. Bernard - Vertex operator representations of the quantum affine algebra Uq(Br(1)), Lett. Math. Phys.17 (1989), 239-245. Zbl0691.17017MR995803
  15. [B-W] J. Birman, H. Wenzl - Braids, link polynomials and a new algebra, Trans. AMS313 (1989), 249-273. Zbl0684.57004MR992598
  16. [Ca] P. Cartier - Développements récents sur les groupes de tresses ; applications à l'algèbre et à la topologie, Sém. Bourbaki, exp. n° 716, Astérisque189-190 (1991), 17-67. Zbl0737.57001MR1099871
  17. [C-P1] V. Chari, A. Pressley - Quantum affine algebras, à paraître dans Comm. Math. Phys. (1991). Zbl0739.17004MR1137064
  18. [C-P2] V. Chari, A. Pressley - Yangians and R-matrices, L'Enseignement Math.36 (1990), 267-302. Zbl0726.17013MR1096420
  19. [C-P3] V. Chari, A. Pressley - Fundamental representations of Yangians and rational R-matrices, à paraître dans J. für reine und angewandte Mathematik. Zbl0726.17014
  20. [Ch1] J. Cherednik - Quantum groups as hidden symmetries of classical representation theory, Proceed. of 17th Int. Conf. on diff. geom. methods in theoretical physics, World Scient. (1989), 47. MR1124414
  21. [Ch2] J. Cherednik - A new interpretation of Gelfand-Tzetlin bases, Duke Math. J.54 (1987), 563. Zbl0645.17006MR899405
  22. [C-G] E. Cremmer, J.-L. Gervais - The quantum group structure associated with non linearly extended Virasoro algebras, Comm. Math. Phys.134 (1990), 619. Zbl0726.17027MR1086746
  23. [DJMM1] E. Date, M. Jimbo, K. Miki, T. Miwa - New R-matrices associated with cyclic representations of Uq(A(2)2), prépublication RIMS706 (1990). MR1140680
  24. [DJMM2] E. Date, M. Jimbo, K. Miki, T. Miwa - R-matrix for cyclic representations of Uq(sl(3,C)) at q3 = 1, Phys. Lett. A148 (1990), 45-49. Zbl0749.17013MR1070333
  25. [DJMM3] E. Date, M. Jimbo, K. Miki, T. Miwa - Cyclic representations of Uq(sl(n + 1, C)) at qN = 1, à paraître dans Public. RIMS. Zbl0752.17011
  26. [DJMM4] E. Date, M. Jimbo, K. Miki, T. Miwa - Generalized chiral Potts models and minimal cyclic representations of Uq(gl(n, C)), à paraître dans Comm. Math. Phys. Zbl0737.17013MR1099258
  27. [DC-K] C. De Concini, V.G. Kac - Representations of quantum groups at roots of 1, Colloque Dixmier1989, 471-506, Progress in Math.92, Birkhäuser (1990). Zbl0738.17008MR1103601
  28. [DC-K-P] C. De Concini, V.G. Kač, C. Procesi - Quantum coadjoint action, prépublication Scuola Normale SuperiorePisa, n° 95, 1991. 
  29. [D1] V.G. Drinfeld - Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl.32 (1985), 254-258. Zbl0588.17015MR802128
  30. [D2] V.G. Drinfeld - Quantum groups, Proceedings ICM, Berkeley (1986), vol. 1, 798-820. MR934283
  31. [D3] V.G. Drinfeld - A new realization of Yangians and quantized affine algebras, Soviet Math. Dokl.36 (1988), 212-216. Zbl0667.16004MR914215
  32. [D4] V.G. Drinfeld - Degenerate affine Hecke algebras and Yangians, Funct. Anal. and Applic.20 (1986), 58-60. Zbl0599.20049MR831053
  33. [D5] V.G. Drinfeld - On almost cocommutative Hopf algebras, Leningrad Math. J.1 (1990), 321-342. Zbl0718.16035MR1025154
  34. [D6] V.G. Drinfeld - Quasi-Hopf algebras, Leningrad Math. J.1 (1990), 1419-1457. MR1047964
  35. [D-L] M. Dyer, G. Lusztig - Appendice à [L5]. 
  36. [Fa] L.D. Faddeev - Integrable models in (1 + 1)-dimensional quantum field theory, Les Houches1982, Elsevier Science Publishers (1984). MR782509
  37. [F-R-T] L.D. Faddeev, N.Y. Reshetikhin, L.A. Takhtajan - Quantization of Lie groups and Lie algebras, Leningrad Math. J.1 (1990), 193-226. Zbl0715.17015MR1015339
  38. [Fr] J. Fröhlich - Statistics of fields, the Yang-Baxter equation and the theory of knots and links, dans "Non Perturbative Quantum Field Theory", eds. G. 't Hooft et al., Plenum Press (1988). MR1008276
  39. [F-J] I. Frenkel, N. Jing - Vertex representations of quantum affine algebras, Proc. Natl. Acad. Sci. USA85 (1988), 9373-9377. Zbl0662.17006MR973376
  40. [F-G-P] P. Furlan, A. Ganchev, V. Petkova - Quantum groups and fusion rules multiplicities, Nucl. Phys. B343 (1990), 205-227. MR1070939
  41. [H] T. Hayashi - Q-analogues of Clifford and Weyl algebras. Spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys.127 (1990), 129-144. Zbl0701.17008MR1036118
  42. [J1] M. Jimbo - Quantum R-matrix for the generalized Toda system, Comm. Math. Phys.102 (1986), 537-547. Zbl0604.58013MR824090
  43. [J2] M. Jimbo - A q-difference analogue of U(G) and the Yang-Baxter equation, Lett. Math. Phys.10 (1985), 63-69. Zbl0587.17004MR797001
  44. [J3] M. Jimbo - A q-difference analogue of U(gl(N + 1)), Hecke algebra and the Yang-Baxter equation, Lett. Math. Phys.11 (1986), 247-252. Zbl0602.17005MR841713
  45. [Jg] N. Jing - Twisted vertex representations of quantum affine algebras, Invent. Math.102 (1990), 663-690. Zbl0692.17006MR1074490
  46. [Jo] V.F.R. Jones - Hecke algebra representations of braid groups and link polynomials, Ann. Math.126 (1987), 335-388. Zbl0631.57005MR908150
  47. [J-L1] A. Joseph, G. Letzter - Local finiteness of the adjoint action for quantized enveloping algebras, à paraître dans J. of Algebra. Zbl0779.17012
  48. [J-L2] A. Joseph, G. Letzter - Separation of variables for quantized enveloping algebras, prépublication. 
  49. [Ka] M. Kashiwara - Crystalizing the q-analogue of universal enveloping algebras, prépublication RIMS ; Bases cristallines, CRASParis311 (1990), 277. Zbl0724.17008MR1090425
  50. [K-R1] A.N. Kirillov, N.Y. Reshetikhin - Yangians and Bethe ansatz, Lett. Math. Phys.12 (1986), 199-208. Zbl0643.20027MR865758
  51. [K-R2] A.N. Kirillov, N.Y. Reshetikhin - Representations of the algebra Uq(sl(2)), q-orthogonal polynomials and invariant of links, dans "Infinite dimensional Lie algebras and groups", ed. V.G. Kač, World Scientific, Singapore (1989). Zbl0742.17018MR1026957
  52. [K-R3] A.N. Kirillov, N.Y. Reshetikhin - q-Weyl group and a multiplicative formula for universal R-matrices, Comm. Math. Phys. (1991). Zbl0723.17014
  53. [K-K] H.T. Kœlink, T.H. Koornwinder - The Clebsch-Gordan coefficients for the quantum group SUμ(2) and q-Hahn polynomials, Nederl. Akad. Wetensch. Proc. Ser. A92 (1989), 443-456. Zbl0708.22007
  54. [Koh] T. Kohno - Monodromy representations of braid groups and Yang-Baxter equations, Ann. Inst. Fourier37 (1987), 139-160 ; Quantized universal enveloping algebras and braid groups, prépublication. Zbl0634.58040MR927394
  55. [Koo] T.H. Koornwinder - Representations of the twisted SU(2) quantum group and some q-hypergeometric orthogonal polynomials, Nederl. Akad. Wetensch. Proc. Ser. A92 (1989), 97-117. Zbl0681.22020MR993682
  56. [Ku-R] P.P. Kulish, N.Y. Reshetikhin - The quantum linear problem for the sine-Gordan equation and higher representations, J. Soviet Math.23 (1983), n° 4. 
  57. [L-S1] S.Z. Levendorskii, Y.S. Soibelman - Some applications of quantum Weyl group 1. The multiplicative formula for universal R-matrix for simple Lie algebras, Journ. Geom. and Phys.7 (1991), n° 4. MR1120927
  58. [L-S2] S.Z. Levendorskii, Y.S. Soibelman - Algebras of functions on compact quantum groups, Schubert cells and quantum tori, à paraître dans Comm. Math. Phys. Zbl0729.17011
  59. [L1] G. Lusztig - Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math.70 (1988), 237-249. Zbl0651.17007MR954661
  60. [L2] G. Lusztig - Modular representations and quantum groups, Contemp. Math.82 (1989), 59-77. Zbl0665.20022MR982278
  61. [L3] G. Lusztig - On quantum groups, J. Algebra131 (1990), 466-475. Zbl0698.16007MR1058558
  62. [L4] G. Lusztig - Finite dimensional Hopf algebras arising from quantized enveloping algebras, Journ. Amer. Math. Soc.3 (1990), 257-296. Zbl0695.16006MR1013053
  63. [L5] G. Lusztig - Quantum groups at roots of 1, Geom. Dedicata (1990). Zbl0714.17013MR1066560
  64. [L6] G. Lusztig - Canonical bases arising from quantized enveloping algebras, I : J. Amer. Math. Soc.3 (1990), 447-498, II : Progress of Theor. Physics (à paraître). Zbl0703.17008MR1035415
  65. [L7] G. Lusztig - Introduction to quantized enveloping algebras, prépublication. 
  66. [Ma] Y. Manin - Quantum groups and non-commutative geometry, CRM Université de Montréal (1989). Zbl0724.17006
  67. [MMNNU] T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi, K. UenoRepresentations of the quantum group SUq(2) and the little q-Jacobi polynomials, J. Funct. Analysis. Zbl0737.33012
  68. [M-R] G. Moore, N.Y. Reshetikhin - A comment on quantum group symmetry in conformal field theory, prépublication. 
  69. [Mu] J. Murakami - The Kauffman polynomial of links and representation theory, Osaka J. Math.24 (1987), 745-758. Zbl0666.57006MR927059
  70. [O] G.I. Olshanskii - Extension of the algebra U(G) for infinite dimensional classical Lie algebras Ç and the Yangians Y(gl(m)), Soviet Math. Dokl.36 (1988), 569-573. Zbl0662.22016MR936073
  71. [P-W] B. Parshall, J.P. Wang - Quantum linear groups, Memoirs of AMS, vol. 89, n° 439 (1991). Zbl0724.17011MR1048073
  72. [P-S] V. Pasquier, H. Saleur - Common structures between finite systems and conformal field theories through quantum groups, prépublication. 
  73. [P-W] P. Podles, S.L. Woronowicz - Quantum deformations of Lorentz group, Comm. Math. Phys.130 (1990), 381-431. Zbl0703.22018MR1059324
  74. [Re1] N.Y. Reshetikhin - Quantized enveloping algebras, the Yang-Baxter equation and invariants of links I et II, prépublications LOMI E-4-87 et E-17-87. 
  75. [Re2] N.Y. Reshetikhin - Quasitriangular Hopf algebras and invariants of tangles, Leningrad Math. J.1 (1990), 491-513. Zbl0715.17016MR1025161
  76. [Re-S] N.Y. Reshetikhin, M. Semenov-Tian-Shanski - Central extensions of quantum current groups, Lett. Math. Phys.19 (1990), 133-143. Zbl0692.22011MR1039522
  77. [Ri] C.M. Ringel - Hall algebras and quantum groups, Invent. Math.101 (1990), 583-592. Zbl0735.16009MR1062796
  78. [R-A] P. Roche, D. Arnaudon - Irreducible representations of the quantum analogue of SU(2), Lett. Math. Phys.17 (1989), 295-300. Zbl0694.17005MR1001085
  79. [Ro1] M. Rosso - Finite dimensional representations of the quantum analogue of the enveloping algebra of a complex simple Lie algebra, Comm. Math. Phys.117 (1988), 581-593. Zbl0651.17008MR953821
  80. [Ro2] M. Rosso - Analogue of PBW theorem and the universal R-matrix for Uhsl(N + 1), Comm. Math. Phys.124 (1989), 307-318. Zbl0694.17006MR1012870
  81. [Ro3] M. Rosso - Analogues de la forme de Killing et du théorème d'Harish-Chandra pour les groupes quantiques, Annales Scient. de l'École Norm. Sup.23 (1990), 445-467. Zbl0721.17012MR1055444
  82. [Ro4] M. Rosso - Algèbres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul différentiel non commutatif, Duke Math. J.61 (1990), 11-40. Zbl0721.17013MR1068378
  83. [Ro5] M. Rosso - Groupes quantiques, représentations linéaires et applications, Thèse Université Paris7 (1990). 
  84. [Sk] E.K. Sklyanin - On an algebra generated by quadratic relations, Uspekhi Mat. Nauk40 (1985), n° 2. 
  85. [So1] Y. Soibelman - Algebra of functions on quantum groups SU(n) and Schubert cells, Dokl. Akad. Nauk SSSR307 (1989), 41-45. MR1017083
  86. [So2] Y. Soibelman - Algebra of functions on a compact quantum group and its representations, Algebra and Analysis2 (1990), n° 1. Zbl0708.46029MR1049910
  87. [So3] Y. Soibelman - Gelfand-Naimark states and quantum Weyl group for SU(n), Funct. Anal. and Applic.24 (1990), n° 3. MR1082047
  88. [So-V] Y. Soibelman, L. Vaksman - Algebra of functions on quantum group SU(2), Funct. Anal. and Applic.22 (1988), n° 3, 1-14. Zbl0661.43001MR961757
  89. [T] M. Takeuchi - The q-bracket product and quantum enveloping algebras of classical type, J. of Math. Soc. of Japan42 (1990), 605-630. Zbl0726.17012MR1069847
  90. [Ta] T. Tanisaki - Harish-Chandra isomorphisms for quantum algebras, prépublication. 
  91. [T-K] A. Tsuchiya, Y. Kanie - Vertex operators in two dimensional conformal field theory on P1 and monodromy representations of braid groups, Adv. Stud. in Pure Math., Vol. 16, Academic Press, Boston (1988), 297-372 et ibidem, Vol. 19 (1989), 675-682. Zbl0661.17021
  92. [Tu] V.G. Turaev - The Yang-Baxter equation and invariants of links, Invent. Math.92 (1988), 527-553. Zbl0648.57003MR939474
  93. [Ve] J.-L. Verdier - Groupes quantiques (d'après V. G. Drinfel'd), Sém. Bourbaki, exp. n° 685, Astérisque152-153 (1987), 305-319. Zbl0645.16006MR936861
  94. [We] H. Wenzl - Quantum groups and subfactors of type B, C and D, Comm. Math. Phys.133 (1990), 383-432. Zbl0744.17021MR1090432
  95. [W1] S.L. Woronowicz - Twisted SU(2) group. An example of non-commutative differential calculus, Publ. RIMS23 (1987), 117-181. Zbl0676.46050MR890482
  96. [W2] S.L. Woronowicz - Compact matrix pseudogroups, Comm. Math. Phys.111 (1987), 613-655. Zbl0627.58034MR901157
  97. [W3] S.L. Woronowicz - Tannaka-Krein duality for compact matrix pseudogroups, Twisted SU(N) group, Invent. Math.93 (1988), 35-76. Zbl0664.58044MR943923
  98. [Ym] H. Yamane - A Poincaré-Birkhoff-Witt theorem for quantized enveloping algebras of type AN, Publ. RIMS25 (1989), 503-520. Zbl0694.17007MR1018513
  99. [Yg] C.N. Yang - Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett.19 (1967), 1312-1315. Zbl0152.46301MR261870

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.