Problems in the theory of quantum groups

Wang, Shuzhou

Banach Center Publications, Tome 38 (1997), p. 67-78 / Harvested from The Polish Digital Mathematics Library

Résumé

This is a collection of open problems in the theory of quantum groups. Emphasis is given to problems in the analytic aspects of the subject.

Détail

Publié le : 1997-01-01

Zbl 0872.17007

EUDML-ID : urn:eudml:doc:252201

Bibliographie

[000] [1] Andersen, H.H. Polo, P. and Wen, K.: Representations of quantum algebras, Invent. Math. 104 (1991), 1-59. | Zbl 0724.17012

[001] [2] Andruskiewitsch, N. and Enriquez, B.: Examples of compact matrix pseudogroups arising from twisting operation, Commun. Math. Phys. 149 (1992), 195-208. | Zbl 0756.17006

[002] [3] Baaj, S.: Représentation régulière du groupe quantique $E_μ(2)$ de Woronowicz, C. R. Acad. Sci. Paris t. 314, Serie I (1992), 1021-1026. | Zbl 0771.17011

[003] [4] Baaj, S.: Représentation régulière du groupe quantique des déplacements de Woronowicz, in Recent Advances in Operator Algebras, Astérisque 232 (1995), 11-48.

[004] [5] Baaj, S. and Skandalis, G.: Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres, Ann. Sci. Ec. Norm. Sup. 26 (1993), 425-488.

[005] [6] Banica, T.: Théorie des représentations du groupe quantique compact libre O(n), C. R. Acad. Sci. Paris t. 322, Serie I (1996), 241-244.

[006] [7] Banica, T.: Le groupe quantique compact libre U(n), Preprint, Unversity of Paris VII, 1996.

[007] [8] Biedenharn, L. C. and Lohe, M. A.: An extension of the Borel-Weil construction to the quantum group $U_q(n)$, Comm. Math. Phys. 146 (1992), 483-504. | Zbl 0757.17009

[008] [9] Boca, F.: Ergodic actions of compact matrix pseudogroups on C*-algebras, in Recent Advances in Operator Algebras, Astérisque 232 (1995), 93-109. | Zbl 0842.46039

[009] [10] Borel, A. et al (ed): Automorphic Forms, Representations, and L-Functions, Proc. Symp. Pure Math., Vol 33, Amer. Math. Soc., 1979.

[010] [11] Brzeziński, T. and Majid, S.: Quantum group gauge theory on quantum spaces, Commun. Math. Phys. 157 (1993), 591-638. | Zbl 0817.58003

[011] [12] Burghelea, D.: The cyclic homology of the groups rings, Comment. Math. Helv. 60 (1985), 354-365. | Zbl 0595.16022

[012] [13] Chari, V. and Pressley, A.: A Guide to Quantum Groups, Cambridge University Press, 1994. | Zbl 0839.17009

[013] [14] Connes, A.: Noncommutative Geometry, Academic Press, 1994. | Zbl 0818.46076

[014] [15] Connes, A.: Non-commutative differential geometry, Publ. Math. IHES 62 (1985), 41-144. | Zbl 0592.46056

[015] [16] Connes, A.: Noncommutative geometry and reality, J. Math. Phys. 36 No 11 (1996). | Zbl 0871.58008

[016] [17] Curtis, C. W. and Reiner, I.: Methods in Representation Theory, I, Wiley, 1981. | Zbl 0469.20001

[017] [18] Curtis, C. W. and Reiner, I.: Methods in Representation Theory, II, Wiley, 1987. | Zbl 0616.20001

[018] [19] Drinfeld, V. G.: Quantum groups, Proc. ICM-1986, Berkeley, Vol I, Amer. Math. Soc., Providence, R.I., 1987, pp798-820.

[019] [20] Dubois-Violette, M. and Launer, G.: The quantum group of a non-degenerate bilinear form, Phys. Lett. B 245 (1990), 175-177. | Zbl 1119.16307

[020] [21] Effros, E. and Ruan, Z.-J.: Discrete quantum groups, I. The Haar measure, Internat. J. Math. 5 (1994), 681-723. | Zbl 0824.17020

[021] [22] Enoch, M. and Schwartz, J. M.: Kac Algebras and Duality of Locally Compact Groups, Springer-Verlag, New York, 1992.

[022] [23] Enock, M. and Vainerman, L.: Deformation of a Kac algebra by an abelian subgroup, Preprint, Univ. Pierre et Marrie Curie, Aug, 1995. | Zbl 0876.46042

[023] [24] Faddeev, L. D. Reshetikhin, N. Y. and Takhtajan, L. A.: Quantization of Lie groups and Lie algebras, Algebra and Analysis 1 (1990), 193-225. | Zbl 0715.17015

[024] [25] Gelbart, S. S.: Automorphic Forms on Adele Groups, Annals of Mathematics Studies 83, Princeton University Press, 1975. | Zbl 0329.10018

[025] [26] Goodman, F. M. de la Harpe, P. and Jones, V. F. R.: Coxeter Graphs and Towers of Algebras, MSRI Publ. 14, Springer-Verlag, 1989. | Zbl 0698.46050

[026] [27] Hajac, P. M.: Strong Connections and $U_q(2)$-Yang-Mills Theory on Quantum Principal Bundles, Ph.D Thesis, University of California at Berkeley, May, 1994.

[027] [28] Kac, G.: Certain arithmetic properties of ring groups, Funct. Anal. Appl. 6 (1972), 158-160. | Zbl 0258.16007

[028] [29] Kac, G. and Palyutkin, V.: An example of a ring group generated by Lie groups, Ukrain. Math. J. 16 (1964), 99-105.

[029] [30] Kac, G. and Palyutkin, V.: Finite ring groups, Trans. Moscow Math. Soc. 15 (1966), 251-294.

[030] [31] Knapp, A. W.: Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986. | Zbl 0604.22001

[031] [32] Landstad, M. B.: Quantizations arising from abelian subgroups, Internat. J. Math. 5 (1994), 897-936. | Zbl 0853.17014

[032] [33] Levendorskii, S.: Twisted algebra of functions on compact quantum group and their representations, St. Petersburg Math. J. 3:2 (1992), 405-423. | Zbl 0791.17012

[033] [34] Levendorskii, S. and Soibelman, Y.: Algebra of functions on compact quantum groups, Schubert cells, and quantum tori, Comm. Math. Phys. 139, (1991), 141-170.

[034] [35] Lusztig, G.: Introduction to Quantum Groups, Progress in Mathematics Vol 110, Birkhauser, 1993. | Zbl 0788.17010

[035] [36] Lusztig, G.: Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math. 70 (1988), 237-249. | Zbl 0651.17007

[036] [37] Majid, S.: Physics for algebraists: non-commutative and noncocommutative Hopf algebras by a bycrossproduct construction, J. Algebra 130 (1990), 17-64. | Zbl 0694.16008

[037] [38] Manin, Y.: Quantum Groups and Noncommutative Geometry, Publications du C.R.M. 1561, Univ de Montreal, 1988. | Zbl 0724.17006

[038] [39] Masuda, T. Mimachi, K. Nakagami, Y. Noumi, M. Saburi, Y. and Ueno, K.: Unitary representations of the quantum group $SU_q(1,1)$: I-Structure of the dual space of $U_qsl(2)$, Lett. Math. Phys. 19 (1990), 187-194. | Zbl 0704.17007

[039] [40] Masuda, T. Mimachi, K. Nakagami, Y. Noumi, M. Saburi, Y. and Ueno, K.: Unitary representations of the quantum group $SU_q(1,1)$: II-Matrix elements of unitary representations and basic hypergeometric functions, Lett. Math. Phys. 19 (1990), 195-204. | Zbl 0704.17008

[040] [41] Masuda, T. and Nakagami, Y.: A von Neumann algebra framework for the duality of the quantum groups, Publ. RIMS 30:5 (1994), 799-850. | Zbl 0839.46055

[041] [42] Masuda, T. Nakagami, Y. and Watanabe, J.: Noncommutative geometry on the quantum SU(2) I: An algebraic viewpoint, K-theory 4 (1990), 157-180. | Zbl 0719.46042

[042] [43] Masuda, T. Nakagami, Y. and Woronowicz, S. L.: A C*-algebraic framework for quantum groups, to appear. | Zbl 1053.46050

[043] [44] Nakagami, Y.: Takesaki duality for the crossed product by quantum groups, in Quantum and Non-Commutative Analysis, H. Araki ed., Kluwer Academic Publishers, 1993, 263-281. | Zbl 0840.46045

[044] [45] Pal, A.: Induced representation and Frobenius reciprocity for compact quantum groups, Proc. Indian Acad. Sci. (Math. Sci.) 105:2 (1995), 157-167. | Zbl 0959.22004

[045] [46] Parshall, B. and Wang, J.: Quantum linear groups, Memoirs AMS 439, 1991.

[046] [47] Podleś, P.: Quantum spheres, Letters Math. Phys. 14 (1987), 193-202. | Zbl 0634.46054

[047] [48] Podleś, P.: Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups, Comm. Math. Phys. 170 (1995), 1-20. | Zbl 0853.46074

[048] [49] Podleś, P. and Woronowicz, S. L.: Quantum deformation of Lorentz group, Comm. Math. Phys. 130 (1990), 381-431. | Zbl 0703.22018

[049] [50] Pusz, W.: Irreducible unitary representations of quantum Lorentz group, Comm. Math. Phys. 152 (1993), 591-626. | Zbl 0810.17009

[050] [51] Pusz, W. and Woronowicz, S. L.: Unitary representations of quantum Lorentz group, Preprint, Warsaw University, 1993. | Zbl 0828.17015

[051] [52] Rieffel, M.: Noncommutative tori, Contemp. Math 105 (1990), 191-211.

[052] [53] Rieffel, M.: Some solvable quantum groups, in Operator Algebras and Topology, W. B. Arveson, A. S. Mischenko, M. Putinar, M. A. Rieffel and S. Stratila, ed., Proc. Conf. Operator Algebras and their Connection with Topology and Ergodic Theory, 2nd Conference, Romania, 1989, Pitman Research Notes in Mathematics 270, Longman, Burnt Mill, England, 1992, pp146-159. | Zbl 0804.17010

[053] [54] Rieffel, M.: Compact quantum groups associated with toral subgroups, Contemp. Math. 145 (1993), 465-491. | Zbl 0795.17017

[054] [55] Rieffel, M.: Non-compact quantum groups associated with abelian subgroups, Comm. Math. Phys. 171 (1995), 181-201. | Zbl 0857.17014

[055] [56] Rosso, M.: Comparaison des groupes SU(2) quantiques de Drinfeld et Woronowicz, C. R. Acad. Sci. Paris 304 (1987), 323-326.

[056] [57] Rosso, M.: Finite dimensional representations of the quantum analog of the enveloping algebra of a complex semisimple Lie algebra, Comm. Math. Phys. 117 (1988), 581-593. | Zbl 0651.17008

[057] [58] Rosso, M.: Algèbres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul différentiel non-commutatif, Duke Math. J. 61 (1990), 11-40. | Zbl 0721.17013

[058] [59] Serre, J.-P.: Arbres, amalgames, $SL_2$, Soc. Math. France, Asterisque 46 (1977).

[059] [60] Skandalis, G.: Operator algebras and duality, Proc. ICM-1990, Kyoto, Vol II, Springer, New York, 1991, pp997-1009. | Zbl 0819.46054

[060] [61] Soibelman, Y.: Algebra of functions on a compact quantum group and its representations, Leningrad Math. J. 2:1 (1990), 161-178.

[061] [62] Vaksman, L. and Soibelman, Y.: The algebra of functions on quantum SU(2), Funct. Anal. ego Pril. 223, (1988), 1-14. | Zbl 0661.43001

[062] [63] Varadarajan, V. S.: Lie Groups, Lie Algebras, and their Representations, Graduate Text in Math, no. 102, Springer, 1984. | Zbl 0955.22500

[063] [64] Van Daele, A.: Quantum deformation of the Heisenberg group, Proc. Satellite Conf. of ICM-90, Current Topics in Operator Algebras, 314-325. | Zbl 0811.46074

[064] [65] Van Daele, A.: Discrete quantum groups, J. Alg. 180 (1996), 431-444. | Zbl 0864.17012

[065] [66] Van Daele, A. and Wang, S. Z.: Universal quantum groups, International J. Math 7:2 (1996), 255-264. | Zbl 0870.17011

[066] [67] Wang, S. Z.: General Constructions of Compact Quantum Groups, Ph.D Thesis, University of California at Berkeley, March, 1993.

[067] [68] Wang, S. Z.: Free products of compact quantum groups, Comm. Math. Phys. 167 (1995), 671-692. | Zbl 0838.46057

[068] [69] Wang, S. Z.: Tensor products and crossed products of compact quantum groups, Proc. London Math. Soc. 71 (1995), 695-720. | Zbl 0837.46052

[069] [70] Wang, S. Z.: Krein duality for compact quantum groups, J. Math. Phys. 38 No. 1 (1997). %Preprint, Spring, 1992.

[070] [71] Wang, S. Z.: Deformations of compact quantum groups via Rieffel's quantization, Commun. Math. Phys. 178 (1996), 747-764. | Zbl 0876.17021

[071] [72] Wang, S. Z.: New classes of compact quantum groups, Lecture notes for talks at the University of Amsterdam and the University of Warsaw, January and March, 1995.

[072] [73] S. Z. Wang, Classification of quantum groups $SU_q(n)$, to appear in J. London Math. Soc.

[073] [74] S. Z. Wang, Imprimitivity theory for compact quantum groups, In preparation.

[074] [75] S. Z. Wang, Rieffel type discrete deformations of finite quantum groups, In preparation.

[075] [76] S. Z. Wang, Actions of universal quantum groups on operator algebras, In preparation.

[076] [77] S. Z. Wang, Quantum symmetry groups of finite spaces, In preparation.

[077] [78] Wassermann, A.: Ergodic actions of compact groups on operator algebras III: Classification for SU(2), Invent. Math. 93 (1988), 309-355. | Zbl 0692.46058

[078] [79] Weyl, H.: The Classical Groups, Princeton University Press, 1946. | Zbl 1024.20502

[079] [80] Weyl, H.: Theory of Groups and Quantum Mechanics, Princeton University Press, 1930.

[080] [81] Woronowicz, S. L.: Twisted SU(2) group. An example of noncommutative differential calculus, Publ. RIMS, Kyoto Univ. 23, (1987), 117-181. | Zbl 0676.46050

[081] [82] Woronowicz, S. L.: Compact matrix pseudogroups, Comm. Math. Phys. 111, (1987), 613-665. | Zbl 0627.58034

[082] [83] Woronowicz, S. L.: Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math. 93 (1988), 35-76. | Zbl 0664.58044

[083] [84] Woronowicz, S. L.: Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1989), 125-170. | Zbl 0751.58042

[084] [85] Woronowicz, S. L.: Unbounded elements affiliated with C*-algebras and non-compact quantum groups, Comm. Math. Phys. 136 (1991), 399-432. | Zbl 0743.46080

[085] [86] Woronowicz, S. L.: Compact quantum groups, Preprint, Warsaw University, 1992 | Zbl 0759.17009

[086] [87] Woronowicz, S. L.: C*-algebras generated by unbounded elements, Reviews in Mathematical Physics 7:3 (1995), 481-521. | Zbl 0853.46057

[087] [88] Woronowicz, S. L.: From multiplicative unitaries to quantum groups, Preprint, Warsaw University, March, 1995. | Zbl 0876.46044