×

Big \(q\)-Jacobi polynomials, \(q\)-Hahn polynomials, and a family of quantum 3-spheres. (English) Zbl 0749.33010

Let \(G\) be the quantum group \(SU_ q(2)\). A quantum \(G\)-space \(M\) is constructed closely related to Podles’ space of quantum 2-spheres [P. Podles, Lett. Math. Phys. 14, 193-202 (1987; Zbl 0634.46054)]. \(M\) can be viewed as a deformation family of quantum 3-spheres. The algebra of functions \(A(M)\) is provided with a left \(A(G)\)-comodule structure and a right \(A(K)\) comodule structure for a certain subgroup \(K\) of \(G\). The irreducible decomposition of \(A(M)\) as a left \(A(G)\)-comodule is given and used to construct a \(G\)-invariant functional and a pair of Hermitian forms on \(A(M)\). Orthogonal basis vectors with respect to the forms are given and also explicit expressions in terms of big \(q\)-Jacobi polynomials and \(q\)-Hahn polynomials. These orthogonal polynomials are viewed as quantum spherical functions.

MSC:

33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Citations:

Zbl 0634.46054
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andrews, G. E. and Askey, R., Classical orthogonal polynomials, Lecture Notes in Math. 1171, Springer, 1985, 36-62. · doi:10.1007/BFb0076530
[2] Koornwinder, T. H., Representations of the twisted SU(2) quantum group and some q-hypergeometric orthogonal polynomials, Nederl. Akad. Wetensch. Proc. Ser. A. 92, 97-117 (1989). · Zbl 0681.22020
[3] Masuda, T., Mimachi, K., Nakagami, Y., Noumi, M. and Ueno, K., Representations of the quantum group SU q (2) and the little q-Jacobi polynomials, to appear in J. Funct. Anal. · Zbl 0737.33012
[4] Noumi, M. and Mimachi, K., Quantum 2-spheres and big q-Jacobi polynomials, to appear in Commun. Math. Phys. · Zbl 0699.33005
[5] Podles, P., Quantum spheres, Lett. Math. Phys. 14, 193-202 (1987). · Zbl 0634.46054 · doi:10.1007/BF00416848
[6] Vaksman, L. L. and Soibelman, Ja. S., Algebra of functions on quantum group SU(2), Funktsional. Anal. i Prilozhen. 22, 1-14 (1988). · Zbl 0667.58018 · doi:10.1007/BF01077717
[7] Woronowicz, S. L., Twisted SU(2) group. An example of non-commutative differential calculus, Publ. RIMS, Kyoto Univ. 23, 117-181 (1987). · Zbl 0676.46050 · doi:10.2977/prims/1195176848
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.