Hypergroup structures associated with Gel'fand pairs of compact quantum groups

Vainerman, Leonid

Recent advances in operator algebras - Orléans, 1992, Astérisque, no. 232 (1995), pp. 231-242.

MR Zbl

@incollection{AST_1995__232__231_0,
     author = {Vainerman, Leonid},
     title = {Hypergroup structures associated with {Gel'fand} pairs of compact quantum groups},
     booktitle = {Recent advances in operator algebras - Orl\'eans, 1992},
     series = {Ast\'erisque},
     pages = {231--242},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {232},
     year = {1995},
     mrnumber = {1372536},
     zbl = {0848.43007},
     language = {en},
     url = {http://www.numdam.org/item/AST_1995__232__231_0/}
}

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Vainerman, Leonid. Hypergroup structures associated with Gel'fand pairs of compact quantum groups, dans Recent advances in operator algebras - Orléans, 1992, Astérisque, no. 232 (1995), pp. 231-242. http://www.numdam.org/item/AST_1995__232__231_0/

Bibliographie
Cité par

[1] E. Abe. Hopf algebras, volume 74 of Cambridge Tracts in Math. Cambridge University Press, 1980. | MR | Zbl

[2] G. E. Andrews and R. Askey. Enumeration of partitions: the role of Eulerian series and $q$ -orthogonal polynomials. In Higher Combinatorics, M. Aigner, ed., pages 3-26, 1977. | DOI | MR | Zbl

[3] R. Askey and J. Wilson. Some basic hypergeometrical orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc., 319, 1985. | MR | Zbl

[4] S. Baaj and G. Skandalis. Unitaires multiplicatifs et dualité pour les produits croisés de $C^{*}$ -algèbres. Ann. Sci. Ecole Norm. Sup., 4 série, 26 : 425-488, 1993. | DOI | EuDML | Numdam | MR | Zbl

[5] Y. Berezanskii and A. Kalyuzhnyi. Hypercomplex systems with locally compact basis. Selecta Math. Sov., 4 (2) : 151-200, 1985. | Zbl

[6] Y. Berezanskii and A. Kalyuzhnyi. Harmonic analysis in hypercomplex systems. Naukova Dumka, Kiev, 1992. Russian. | MR | Zbl

[7] K. Bragiel. On the Spherical and Zonal Spherical Functions on a Compact Quantum Group. Letters in Math. Phys., 22 : 195-201, 1991. | DOI | MR | Zbl

[8] Y. Chapovsky and L. Vainerman. Hypergroup structures, associated with a pair of quantum groups $(S U_{q} (n), U_{q} (n - 1))$ . In Methods of functional analysis in problems of mathematical physics, pages 47-69. Inst. Math. Acad. Sci. Ukraine, Kiev, 1992. | MR

[9] M. S. Dijkhuizen and T. H. Koornwinder. Quantum homogeneous spaces, duality, and quantum $2$ -spheres. Report AM-R9309, Centrum voor Wiskunde en Informatica, Amsterdam, September 1993 (to appear in Geom. Dedicata). | MR | Zbl

[10] M. Enock and J.-M. Vallin. $C^{*}$ -algèbres de Kac et algèbres de Kac. Proc. London Math. Soc., 66 : 619-650, 1993. | DOI | MR | Zbl

[11] J. Faraut. Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques. In Analyse Harmonique, pages 315-446. CIMPA, Nice, 1982. | Zbl

[12] P. J. A. Floris. Gelfand Pair Criteria for Compact Matrix Quantum Groups. Mathematical Institute University of Leiden, Report No. W93-20, 1993 (to appear in Indag. Math. (N.S.)). | MR | Zbl

[13] R. Jewett. Spaces with an abstract convolution of measures. Adv. in Math., 18(1) : 1-101 1975. | DOI | MR | Zbl

[14] H. Koelink. On quantum groups and $q$ -special functions, Ph.D. Thesis, Rijksuniversiteit Leiden, 1991.

[15] H. Koelink. Askey-Wilson polynomials and the quantum $S U (2)$ group: survey and applications. Katholieke Universiteit Leuven, Preprint, 1994. | MR | Zbl

[16] H. Koelink and T. Koornwinder. The Clebsch-Gordan coefficients for the quantum group $S U_{μ} (2)$ and $q$ -Hahn polynomials. Nederl. Acad. Wetensch. Proc. Ser : A, 92, 1989. | MR | Zbl

[17] T. Koornwinder. Orthogonal polynomials in connection with quantum groups. In P. Nevai, editor, Orthogonal Polynomials: Theory and Practice, volume 294, pages 257-292, Norwell, MA, 1990. NATO-ASI Series C, Kluwer. | DOI | MR | Zbl

[18] T. Koornwinder. The addition formula for little $q$ -Legendre polynomials and the $S U (2)$ quantum group. SIAM J. Math. Anal., 22(1) : 295-301, 1991. | DOI | MR | Zbl

[19] T. Koornwinder. Positive convolution structures associated with quantum groups. In Probability Measures on Groups, X, pages 249-268. Plenum, 1991. | DOI | MR | Zbl

[20] T. Koornwinder. Discrete hypergroups associated with compact quantum Gelfand pairs. Applications of hypergroups and related measure algebras, Contemporary Math. (W. Connett, O. Geburer, A. Schwartz, eds.) (to appear). | MR | Zbl

[21] T. Koornwinder. Askey-Wilson polynomials as zonal spherical functions on the $S U (2)$ quantum group. SIAM J. Math. Anal., 24(3) : 795-813, 1993. | DOI | MR | Zbl

[22] B. Levitan. Generalized translation operators and some of their applications. Jerusalem, New York, 1964. | MR | Zbl

[23] M. Noumi. Macdonalds symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces. Preprint, 1993 (to appear in Adv. in Math.). | MR | Zbl

[24] M. Noumi, H. Yamada, and K. Mimachi. Finite dimensional representations of the quantum group $G L_{q} (n; C)$ and the zonal spherical functions on $U_{q} {(n - 1)}_{q} (n)$ . Japanese J. Math. (N.S.) 19 : 31-80, 1993. | DOI | MR | Zbl

[25] N. Reshetikhin, L. Takhtadjan, and L. Faddeev. Quantization of Lie groups and Lie algebras. Algebra and Analysis, 1 (1) : 178-206, 1989. English transl. in Leningrad Math. J., v.1, 1990, pp.193-225. | MR | Zbl

[26] L. Vainerman. Duality for algebras with involution and generalized shift operators. Journal of Soviet Math., 42 (6) : 52-59, 1988. | DOI

[27] L. Vainerman. Harmonic Analysis on Hypercomplex Systems with Compact and Discrete Basis. In Spectral Theory of Operators and Infinite-Dimensional Analysis, pages 19-31. Inst. Math. Acad. Sci. Ukraine, Kiev, 1984. English transl in Selecta Math. Sov., v.10, No.2, 1991, pp. 181-193. | MR | Zbl

L. Vainerman. Harmonic Analysis on Hypercomplex Systems with Compact and Discrete Basis. Inst. Math. Acad. Sci. Ukraine, Kiev, 1984. English transl in Selecta Math. Sov., v.10, No.2, 1991, pp. 181-193. | MR | Zbl

[28] L. Vainerman. Gel'fand pairs of quantum groups, hypergroups and $q$ -special functions. Applications of hypergroups and related measure algebras, Contemporary Math. (W. Connett, O. Geburer, A. Schwartz, eds.) (to appear). | MR | Zbl

[29] L. Vainerman. Gel'fand pair associated with the quantum group of motions of the plane and $q$ -Bessel functions. Reports of Math. Phys. (to appear). | MR | Zbl

[30] L. Vainerman and Y. Weiss. Hypergroup associated with double cosets of the quantum group $S U_{q} (2)$ . In Applications of the functional analysis methods in math. physics, pages 52-59. | MR | Zbl

L. Vainerman and Y. Weiss. Hypergroup associated with double cosets of the quantum group $S U_{q} (2)$ . Inst. Math. Acad. Sci. Ukraine, Kiev, 1991. Russian ; to be transl. in Selecta Math. Sov. | MR

[31] L. Vaksman and Y. Soibelman. An algebra of functions on the quantum group $S U (2)$ . Funktsional. Anal. i Prilozen., 22(3) : 1-14, 1988. English transl. in Funct. Anal. Appl., v.22, 1988, pp.170-181. | Zbl

[32] L. Vaksman and Y. Soibelman. Algebra of functions on quantum group $S U (n + 1)$ and odd dimensional spheres. Algebra i Analyz, 2(5) : 101-120, 1990. English transl. in Leningrad Math. J., v.2, No.5, 1991, pp.1023-1042. | MR | Zbl

[33] N. Vilenkin. Special functions and the theory of group representations, volume 22 of Translations of Math. Monografs. Amer. Math. Soc., 1968. | MR | Zbl

[34] S. Woronowicz. Compact matrix pseudogroups. Commun. Math. Phys., 111 : 613-665, 1987. | DOI | MR | Zbl

[35] S. Woronowicz. Twisted $S U (2)$ group. an example of a non-commutative differential calculus. Publ. RIMS Kyoto Univ., 23 : 117-181, 1987. | DOI | MR | Zbl