ICM-90 Satellite Conference Proceedings pp 260-288 | Cite as

# Quantum Groups and q-Orthogonal Polynomials

_{q}(2)

## Abstract

Many links are already known between quantum groups and q-orthogonal polynomials. In this article, we will give a survey on recent works concerning the realization of q-analogues of the Jacobi polynomials as spherical functions on the quantum group SU (2). We would like to emphasize that quantum groups and q-orthogonal polynomials have some characteristics in common and that the interaction between the two fields of mathematics will be important for both of them. The latter half of this article is a review on the works [K3–5] by T.H.Koornwinder and [NM1–5] by K.Mimachi and the author. This article, written from the viewpoint of [NM1–5], may be compared with Koornwinder’s survey [K4].

## Keywords

Orthogonal Polynomial Quantum Group Spherical Function Jacobi Polynomial Hermitian Form## Preview

Unable to display preview. Download preview PDF.

## References

- [A]E. Abe: Hopf algebras, Cambridge tracts in mathematics 74, Cambridge University Press, 1980.Google Scholar
- [AA]G.E. Andrews and R. Askey: Classical orthogonal polynomials, Lecture Notes in Math., 1171, Springer, 1985, 36–62.Google Scholar
- [AW]R. Askey and J. Wilson: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs Amer. Math. Soc. 54(1985), No. 319.Google Scholar
- [D]V.G. Drinfeld: Quantum groups, Proceedings of the International Congress of Mathematicians, Berkely, California, U.S.A., 1986, 798–820.Google Scholar
- [GR]G. Gasper and M. Rahman: Basic hypergeometric series, Encylclopedia of Mathematics and its Applications Vol.35, Cambridge University Press, 1990.Google Scholar
- [J1]M. Jimbo: A q-difference analogue of U(
**of**) and the Yang-Baxter equation, Lett. Math. Phys. 10(1985), 63–69.MathSciNetMATHCrossRefGoogle Scholar - [J2]M. Jimbo: A q-analogue of U(
**of**)ℓ (N+1)), Heck algebra and the Yang-Baxter equation, Lett. Math. Phys. 11(1986), 247–252.MathSciNetMATHCrossRefGoogle Scholar - [KR]A.N. Kirillov and N.Yu. Reshetikhin: Representations of the algebra U
_{q}(sℓ(2)), q-orthogonal polynomials and invariants of links, in Infinite-dimensional Lie algebras and groups, edited by V.G. Kac, World Scientific, 1989, 285–339.Google Scholar - [Koe]H.T. Koelink: The addition formula for continuous q-Legendre polynomials and associated spherical elements on the SU(2) quantum group related to Askey-Wilson polynomials, preprint 1990.Google Scholar
- [KK]H.T. Koelink and T.H. Koornwinder: The Clebsh-Gordan coefficients for the quantum group S
_{μ}U(2) and q-Hahn polynomials, to appear in Nederl. Acad. Wetensch. Proc..Google Scholar - [K1]T.H. Koornwinder: Representations of the twisted SU(2) quantum group and some q-hypergeometric orthogonal polynomials, Nederl. Akad. Wetensch. Proc. Ser.A
**92**(1989), 97–117.MathSciNetGoogle Scholar - [K2]T.H.Koornwinder: The addition formula for little q-Legendre polynomials and the SU(2) quantum group, CWI Rep. AM-R8906, preprint.Google Scholar
- [K3]T.H. Koornwinder: Continuous q-Legendre polynomials as spherical matrix elements of irreducible representations of the quantum SU(2) group, CWI Quaterly, 2(1989), 171–173.MATHGoogle Scholar
- [K4]T.H. Koornwinder: Orthogonal polynomials in connections with quantum groups, in Orthogonal Polynomials, Theory and Practice, edited by P. Nevai, NATO ASI Series, Kluwer Academic Publishers, 257–292, 1990.Google Scholar
- [K5]T.H. Koornwinder: Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group, preprint 1990.Google Scholar
- [Kor]L.I. Korogodsky: Quantum projective spaces, spheres and hyperboloids, preprint 1990.Google Scholar
- [MM1]T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi and K. Ueno: Representations of quantum groups and a q-analogue of orthogonal polynomials, C. R. Acad. Sci. Paris 307(1988), 559–564.MathSciNetMATHGoogle Scholar
- [MM2]T.Masuda, K.Mimachi, Y.Nakagami, M.Noumi and K.Ueno: Representations of the quantum group SU (2) and the little q-Jacobi polynomials, to appear in J. Functional Analysis.Google Scholar
- [NM1]M. Noumi and K. Mimachi: Quantum 2-spheres and big q-Jacobi polynomials, Commun. Math. Phys. 128(1990), 521–531.MathSciNetMATHCrossRefGoogle Scholar
- [NM2]M. Noumi and K. Mimachi: Big q-Jacobi polynomials, q-Hahn polynomials and a family of quantum 3-spheres, Lett. Math. Phys. 19(1990), 299–305.MathSciNetMATHCrossRefGoogle Scholar
- [NM3]M.Noumi and K.Mimachi: Spherical functions on a family of quantum 3-spheres, to appear in Compositio Mathematica.Google Scholar
- [NM4]M.Noumi and K.Mimachi: Rogers’ q-ultraspherical polynomials on a quantum 2-sphere, to appear in Duke Mathematical Journal.Google Scholar
- [NM5]M. Noumi and K. Mimachi: Askey-Wilson polynomials and the quantum group SU (2), Proc. Japan Acad. Ser.A 66(1990), 146–149.MathSciNetMATHCrossRefGoogle Scholar
- [NYM1]M. Noumi, H. Yamada and K. Mimachi: Zonal spherical functions on the quantum homogenous space SU (n+1)/SU (n), Proc. Japan Acad. Ser.A 65(1989), 169–171.MathSciNetMATHCrossRefGoogle Scholar
- [NYM2]M. Noumi, H. Yamada and K. Mamachi: Finite dimensional representations of the quantum group GL
_{q}(n;c) and the zonal spherical functions on U (n-1)\U (n), preprint 1990.Google Scholar - [P]P. Podles: Quantum spheres, Lett. Math. Phys. 14(1987), 193–202.MathSciNetMATHCrossRefGoogle Scholar
- [VS1]L.L. Vaksman and Ya.S. Soibelman: Algebra of functions on the quantum SU(2) group, Funct. Anal, i-ego Pril. 22(1988), 1036 – 1040 (in Russian).MathSciNetGoogle Scholar
- [VS2]L.L.Vaksman and Ya.S.Soibelman: Algebra of functions on quatum group SU(n+1) and odd dimensional quantum spheres, to appear in Algebra and Analysis (in Russian).Google Scholar
- [W1]S. L. Woronowicz: Twisted SU(*2) group. An example of non-commutative differential calculus, Publ. RIMS, Kyoto Univ., 23(1987), 117–181.MathSciNetMATHCrossRefGoogle Scholar
- [W2]S.L. Woronowicz: Compact matrix pseudogroups, Comm. Math. Phys., 111(1987), 613–665.MathSciNetMATHCrossRefGoogle Scholar