# Askey-Wilson polynomials and the quantum SU(2) group: Survey and applications

@article{Koelink1994AskeyWilsonPA, title={Askey-Wilson polynomials and the quantum SU(2) group: Survey and applications}, author={H. T. Koelink}, journal={Acta Applicandae Mathematica}, year={1994}, volume={44}, pages={295-352} }

Generalised matrix elements of the irreducible representations of the quantum SU(2) group are defined using certain orthonormal bases of the representation space. The generalised matrix elements are relatively infinitesimal invariant with respect to Lie algebra like elements of the quantised universal enveloping algebra of sl(2). A full proof of the theorem announced by Noumi and Mimachi [Proc. Japan Acad. Sci. Ser. A66 (1990), 146–149] describing the generalised matrix elements in terms of the…

## 82 Citations

QUANTUM GROUPS AND q-SPECIAL FUNCTIONS

- Mathematics
- 1996

The lecture notes contains an introduction to quantum groups, q-special func- tions and their interplay. After generalities on Hopf algebras, orthogonal polynomials and basic hypergeometric series we…

8 Lectures on quantum groups and q-special functions

- Mathematics
- 1996

The lectures contain an introduction to quantum groups, q-special functions and their interplay. After generalities on Hopf algebras, ortogonal polynomials and basic hypergeometric series we work out…

Harmonic Analysis on the SU(2) Dynamical Quantum Group

- Mathematics
- 2000

Dynamical quantum groups were recently introduced by Etingof and Varchenko as an algebraic framework for studying the dynamical Yang–Baxter equation, which is precisely the Yang–Baxter equation…

Fourier transforms on the quantum SU (1, 1) group

- Mathematics
- 1999

The main goal is to interpret the Askey-Wilson function and the corresponding transform pair on the quantum SU(1,1) group. A weight on the C^*-algebra of continuous functions vanishing at infinity on…

Askey-Wilson Functions and Quantum Groups

- Mathematics
- 2003

Eigenfunctions of the Askey-Wilson second order q-difference operator for 0 < q < 1 and |q| = 1 are constructed as formal matrix coefficients of the principal series representation of the quantized…

A Quantum Algebra Approach to Multivariate Askey–Wilson Polynomials

- MathematicsInternational Mathematics Research Notices
- 2019

We study matrix elements of a change of basis between two different bases of representations of the quantum algebra ${\mathcal{U}}_q(\mathfrak{s}\mathfrak{u}(1,1))$. The two bases, which are…

Coupling coefficients for tensor product representations of quantum SU(2)

- Mathematics
- 2014

We study tensor products of infinite dimensional irreducible *-representations (not corepresentations) of the SU(2) quantum group. We obtain (generalized) eigenvectors of certain self-adjoint…

Dual properties of orthogonal polynomials of discrete variables associated with the quantum algebra Uq(su(2))

- Mathematics
- 2007

We show that for every set of discrete polynomials yn(x(s)) on the lattice x(s), defined on a finite interval (a, b), it is possible to construct two sets of dual polynomials zk(ξ(t)) of degrees k =…

DUAL PROPERTIES OF ORTHOGONAL POLYNOMIALS OF DISCRETE VARIABLES ASSOCIATED WITH THE QUANTUM ALGEBRA U q (su(2))

- Mathematics
- 2007

We show that for every set of discrete polynomials yn(x(s)) on the lattice x(s), defined on a finite interval (a,b), it is possible to construct two sets of dual polynomials zk( (t)) of degrees k = s…

The dynamical U(n) quantum group

- MathematicsInt. J. Math. Math. Sci.
- 2006

Using the Laplace expansions it is proved that the dynamical quantum determinant is almost central, and adjoining an inverse the antipode can be defined, which results in the dynamICAL GL(n) quantum group associated to the dynamicals R-matrix.

## References

SHOWING 1-10 OF 51 REFERENCES

Representations of the twisted SU(2) quantum group and some q-hypergeometric orthogonal polynomials

- Mathematics
- 1988

Askey-Wilson polynomials as zonal spherical functions on the SU (2) quantum group

- Mathematics
- 1990

On the $SU(2)$ quantum group the notion of (zonal) spherical element is generalized by considering left and right invariance in the infinitesimal sense with respect to twisted primitive elements of…

Orthogonal polynomials in connection with quantum groups

- Mathematics
- 1990

This is a survey of interpretations of q-hypergeometric orthogonal polynomials on quantum groups. The first half of the paper gives general background on Hopf algebras and quantum groups. The…

Quantum homogeneous spaces, duality and quantum 2-spheres

- Mathematics, Physics
- 1993

For a quantum groupG the notion of quantum homogeneousG-space is defined. Two methods to construct such spaces are discussed. The first one makes use of quantum subgroups, the second more general one…

Quantum Groups and q-Orthogonal Polynomials

- Mathematics
- 1991

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…

The addition formula for continuous q -Legendre polynomials and associated spherical elements on the SU (2) quantum group related to Askey-Wilson polynomials

- Mathematics
- 1994

The known interpretation of a two-parameter family of Askey–Wilson polynomials as spherical elements on the $SU(2)$ quantum group is extended to an interpretation of a three-parameter (one discrete…

Models of Q -algebra representations: the group of plane motions

- Mathematics
- 1994

This paper continues a study of one- and two-variable function space models of irreducible representations of q-analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by…

The addition formula for little q-Legendre polynomials and the SU(2) quantum group

- Mathematics
- 1989

From the interpretation of little q-Jacobi polynomials as matrix elements of the irreducible unitary representations of the ${\operatorname{SU}}(2)$ quantum group an addition formula is derived for…