Représentations des groupes quantiques

Rosso, Marc

Séminaire Bourbaki : volume 1990/91, exposés 730-744, Astérisque, no. 201-202-203 (1991), Exposé no. 744, 41 p.

MR Zbl

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     author = {Rosso, Marc},
     title = {Repr\'esentations des groupes quantiques},
     booktitle = {S\'eminaire Bourbaki : volume 1990/91, expos\'es 730-744},
     series = {Ast\'erisque},
     note = {talk:744},
     pages = {443--483},
     year = {1991},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {201-202-203},
     mrnumber = {1157851},
     zbl = {0752.17016},
     language = {fr},
     url = {https://www.numdam.org/item/SB_1990-1991__33__443_0/}
}

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Rosso, Marc. Représentations des groupes quantiques, dans Séminaire Bourbaki : volume 1990/91, exposés 730-744, Astérisque, no. 201-202-203 (1991), Exposé no. 744, 41 p.. https://www.numdam.org/item/SB_1990-1991__33__443_0/

Bibliographie
Cité par

[A-G-S] L. Alvarez-Gaumé, C. Gomez, G. Sierra - Quantum group interpretation of some conformal field theories, prépublication CERN.

[A-P-W1] H. H. Andersen, P. Polo et Wen K. - Representations of quantum algebras, Invent. Math. 104 (1991), 1-60. | Zbl | MR

[A-P-W2] H. H. Andersen, P. Polo et Wen K. - Injective modules for quantum algebras, à paraître dans Amer. J. Math. | Zbl | MR

[A-W] H. H. Andersen, Wen K. - Representations of quantum algebras : the mixed case, prépublication Aarhus Universitet 1991, n° 5. | MR

[An] N. Andruskiewitsh - Some exceptional compact matrix pseudogroups, prépublication École Polytechnique.

[Ar] D. Arnaudon - Periodic and flat irreducible representations of SU(3)q, Comm. Math. Phys. 134 (1990), 523-537. | Zbl | MR

[Ar-Ch1] D. Arnaudon, A. Chakrabarti - Periodic and partially periodic representations of SU(N)q, à paraître dans Comm. Math. Phys. | Zbl | MR

[Ar-Ch2] D. Arnaudon, A. Chakrabarti - Flat periodic representations of Uq(G), prépublication École Polytechnique A023-1190.

[B] O. Babelon - Extended conformal algebra of Yang-Baxter equation, Phys. Lett. B125 (1988), 523-529. | MR

[Ba1] R. J. Baxter - Partition function of the eight-vertex lattice model, Ann. Phys. 70 (1972), 193-228. | Zbl | MR

[Ba2] R. J. Baxter - Exactly solved models in Statistical Mechanics, Academic Press, London, 1982. | Zbl | MR

[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. | Zbl | MR

[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). | Zbl | MR

[Be] D. Bernard - Vertex operator representations of the quantum affine algebra Uq(Br(1)), Lett. Math. Phys. 17 (1989), 239-245. | Zbl | MR

[B-W] J. Birman, H. Wenzl - Braids, link polynomials and a new algebra, Trans. AMS 313 (1989), 249-273. | Zbl | MR

[Ca] P. Cartier - Développements récents sur les groupes de tresses ; applications à l'algèbre et à la topologie, Sém. Bourbaki, exp. n° 716, Astérisque 189-190 (1991), 17-67. | Zbl | MR | Numdam

[C-P1] V. Chari, A. Pressley - Quantum affine algebras, à paraître dans Comm. Math. Phys. (1991). | Zbl | MR

[C-P2] V. Chari, A. Pressley - Yangians and R-matrices, L'Enseignement Math. 36 (1990), 267-302. | Zbl | MR

[C-P3] V. Chari, A. Pressley - Fundamental representations of Yangians and rational R-matrices, à paraître dans J. für reine und angewandte Mathematik. | Zbl

[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. | MR

[Ch2] J. Cherednik - A new interpretation of Gelfand-Tzetlin bases, Duke Math. J. 54 (1987), 563. | Zbl | MR

[C-G] E. Cremmer, J.-L. Gervais - The quantum group structure associated with non linearly extended Virasoro algebras, Comm. Math. Phys. 134 (1990), 619. | Zbl | MR

[DJMM1] E. Date, M. Jimbo, K. Miki, T. Miwa - New R-matrices associated with cyclic representations of Uq(A(2)2), prépublication RIMS 706 (1990). | MR

[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. | Zbl | MR

[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. | Zbl

[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. | Zbl | MR

[DC-K] C. De Concini, V. G. Kac - Representations of quantum groups at roots of 1, Colloque Dixmier 1989, 471-506, Progress in Math. 92, Birkhäuser (1990). | Zbl | MR

[DC-K-P] C. De Concini, V. G. Kač, C. Procesi - Quantum coadjoint action, prépublication Scuola Normale Superiore Pisa, n° 95, 1991.

[D1] V. G. Drinfeld - Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32 (1985), 254-258. | Zbl | MR

[D2] V. G. Drinfeld - Quantum groups, Proceedings ICM, Berkeley (1986), vol. 1, 798-820. | MR

[D3] V. G. Drinfeld - A new realization of Yangians and quantized affine algebras, Soviet Math. Dokl. 36 (1988), 212-216. | Zbl | MR

[D4] V. G. Drinfeld - Degenerate affine Hecke algebras and Yangians, Funct. Anal. and Applic. 20 (1986), 58-60. | Zbl | MR

[D5] V. G. Drinfeld - On almost cocommutative Hopf algebras, Leningrad Math. J. 1 (1990), 321-342. | Zbl | MR

[D6] V. G. Drinfeld - Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), 1419-1457. | MR

[D-L] M. Dyer, G. Lusztig - Appendice à [L5].

[Fa] L. D. Faddeev - Integrable models in (1 + 1)-dimensional quantum field theory, Les Houches 1982, Elsevier Science Publishers (1984). | MR

[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. | Zbl | MR

[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). | MR

[F-J] I. Frenkel, N. Jing - Vertex representations of quantum affine algebras, Proc. Natl. Acad. Sci. USA 85 (1988), 9373-9377. | Zbl | MR

[F-G-P] P. Furlan, A. Ganchev, V. Petkova - Quantum groups and fusion rules multiplicities, Nucl. Phys. B343 (1990), 205-227. | MR

[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. | Zbl | MR

[J1] M. Jimbo - Quantum R-matrix for the generalized Toda system, Comm. Math. Phys. 102 (1986), 537-547. | Zbl | MR

[J2] M. Jimbo - A q-difference analogue of U(G) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69. | Zbl | MR

[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. | Zbl | MR

[Jg] N. Jing - Twisted vertex representations of quantum affine algebras, Invent. Math. 102 (1990), 663-690. | Zbl | MR

[Jo] V. F. R. Jones - Hecke algebra representations of braid groups and link polynomials, Ann. Math. 126 (1987), 335-388. | Zbl | MR

[J-L1] A. Joseph, G. Letzter - Local finiteness of the adjoint action for quantized enveloping algebras, à paraître dans J. of Algebra. | Zbl

[J-L2] A. Joseph, G. Letzter - Separation of variables for quantized enveloping algebras, prépublication.

[Ka] M. Kashiwara - Crystalizing the q-analogue of universal enveloping algebras, prépublication RIMS ; Bases cristallines, CRAS Paris 311 (1990), 277. | Zbl | MR

[K-R1] A. N. Kirillov, N. Y. Reshetikhin - Yangians and Bethe ansatz, Lett. Math. Phys. 12 (1986), 199-208. | Zbl | MR

[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). | Zbl | MR

[K-R3] A. N. Kirillov, N. Y. Reshetikhin - q-Weyl group and a multiplicative formula for universal R-matrices, Comm. Math. Phys. (1991). | Zbl

[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. | Zbl

[Koh] T. Kohno - Monodromy representations of braid groups and Yang-Baxter equations, Ann. Inst. Fourier 37 (1987), 139-160 ; Quantized universal enveloping algebras and braid groups, prépublication. | Zbl | MR | Numdam

[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. | Zbl | MR

[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.

[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. | MR

[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. | Zbl

[L1] G. Lusztig - Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math. 70 (1988), 237-249. | Zbl | MR

[L2] G. Lusztig - Modular representations and quantum groups, Contemp. Math. 82 (1989), 59-77. | Zbl | MR

[L3] G. Lusztig - On quantum groups, J. Algebra 131 (1990), 466-475. | Zbl | MR

[L4] G. Lusztig - Finite dimensional Hopf algebras arising from quantized enveloping algebras, Journ. Amer. Math. Soc. 3 (1990), 257-296. | Zbl | MR

[L5] G. Lusztig - Quantum groups at roots of 1, Geom. Dedicata (1990). | Zbl | MR

[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). | Zbl | MR

[L7] G. Lusztig - Introduction to quantized enveloping algebras, prépublication.

[Ma] Y. Manin - Quantum groups and non-commutative geometry, CRM Université de Montréal (1989). | Zbl

[MMNNU] T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi, K. Ueno Representations of the quantum group SUq(2) and the little q-Jacobi polynomials, J. Funct. Analysis. | Zbl

[M-R] G. Moore, N. Y. Reshetikhin - A comment on quantum group symmetry in conformal field theory, prépublication.

[Mu] J. Murakami - The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), 745-758. | Zbl | MR

[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. | Zbl | MR

[P-W] B. Parshall, J. P. Wang - Quantum linear groups, Memoirs of AMS, vol. 89, n° 439 (1991). | Zbl | MR

[P-S] V. Pasquier, H. Saleur - Common structures between finite systems and conformal field theories through quantum groups, prépublication.

[P-W] P. Podles, S. L. Woronowicz - Quantum deformations of Lorentz group, Comm. Math. Phys. 130 (1990), 381-431. | Zbl | MR

[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.

[Re2] N. Y. Reshetikhin - Quasitriangular Hopf algebras and invariants of tangles, Leningrad Math. J. 1 (1990), 491-513. | Zbl | MR

[Re-S] N. Y. Reshetikhin, M. Semenov-Tian-Shanski - Central extensions of quantum current groups, Lett. Math. Phys. 19 (1990), 133-143. | Zbl | MR

[Ri] C. M. Ringel - Hall algebras and quantum groups, Invent. Math. 101 (1990), 583-592. | Zbl | MR

[R-A] P. Roche, D. Arnaudon - Irreducible representations of the quantum analogue of SU(2), Lett. Math. Phys. 17 (1989), 295-300. | Zbl | MR

[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. | Zbl | MR

[Ro2] M. Rosso - Analogue of PBW theorem and the universal R-matrix for Uhsl(N + 1), Comm. Math. Phys. 124 (1989), 307-318. | Zbl | MR

[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. | Zbl | MR | Numdam

[Ro4] M. Rosso - Algèbres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul différentiel non commutatif, Duke Math. J. 61 (1990), 11-40. | Zbl | MR

[Ro5] M. Rosso - Groupes quantiques, représentations linéaires et applications, Thèse Université Paris 7 (1990).

[Sk] E. K. Sklyanin - On an algebra generated by quadratic relations, Uspekhi Mat. Nauk 40 (1985), n° 2.

[So1] Y. Soibelman - Algebra of functions on quantum groups SU(n) and Schubert cells, Dokl. Akad. Nauk SSSR 307 (1989), 41-45. | MR

[So2] Y. Soibelman - Algebra of functions on a compact quantum group and its representations, Algebra and Analysis 2 (1990), n° 1. | Zbl | MR

[So3] Y. Soibelman - Gelfand-Naimark states and quantum Weyl group for SU(n), Funct. Anal. and Applic. 24 (1990), n° 3. | MR

[So-V] Y. Soibelman, L. Vaksman - Algebra of functions on quantum group SU(2), Funct. Anal. and Applic. 22 (1988), n° 3, 1-14. | Zbl | MR

[T] M. Takeuchi - The q-bracket product and quantum enveloping algebras of classical type, J. of Math. Soc. of Japan 42 (1990), 605-630. | Zbl | MR

[Ta] T. Tanisaki - Harish-Chandra isomorphisms for quantum algebras, prépublication.

[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. | Zbl

[Tu] V. G. Turaev - The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), 527-553. | Zbl | MR

[Ve] J.-L. Verdier - Groupes quantiques (d'après V. G. Drinfel'd), Sém. Bourbaki, exp. n° 685, Astérisque 152-153 (1987), 305-319. | Zbl | MR | Numdam

[We] H. Wenzl - Quantum groups and subfactors of type B, C and D, Comm. Math. Phys. 133 (1990), 383-432. | Zbl | MR

[W1] S. L. Woronowicz - Twisted SU(2) group. An example of non-commutative differential calculus, Publ. RIMS 23 (1987), 117-181. | Zbl | MR

[W2] S. L. Woronowicz - Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613-655. | Zbl | MR

[W3] S. L. Woronowicz - Tannaka-Krein duality for compact matrix pseudogroups, Twisted SU(N) group, Invent. Math. 93 (1988), 35-76. | Zbl | MR

[Ym] H. Yamane - A Poincaré-Birkhoff-Witt theorem for quantized enveloping algebras of type AN, Publ. RIMS 25 (1989), 503-520. | Zbl | MR

[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. | Zbl | MR