109 research outputs found
Addition formulas for q-special functions
A general addition formula for a two-parameter family of Askey-Wilson
polynomials is derived from the quantum group theoretic interpretation.
This formula contains most of the previously known addition formulas for
-Legendre polynomials as special or limiting cases. A survey of the
literature on addition formulas for -special functions using quantum groups
and quantum algebras is given
8 Lectures on quantum groups and q-special functions
Lecture notes for an eight hour course on quantum groups and -special
functions at the fourth Summer School in Differential Equations and Related
Areas, Universidad Nacional de Colombia and Universidad de los Andes, Bogot\'a,
Colombia, July 22 -- August 2, 1996. The lecture notes contain an introduction
to quantum groups, -special functions and their interplay. After
generalities on Hopf algebras, orthogonal polynomials and basic hypergeometric
series we work out the relation between the quantum SU(2) group and the
Askey-Wilson polynomials out in detail as the main example. As an application
we derive an addition formula for a two-parameter subfamily of Askey-Wilson
polynomials. A relation between the Al-Salam and Chihara polynomials and the
quantised universal enveloping algebra for is given. Finally, more
examples and other approaches as well as some open problems are given.Comment: AMS-TeX, 82 page
One-parameter orthogonality relations for basic hypergeometric series
The second order hypergeometric q-difference operator is studied for the
value c=-q. For certain parameter regimes the corresponding recurrence relation
can be related to a symmetric operator on the Hilbert space l^2(Z). The
operator has deficiency indices (1,1) and we describe as explicitly as possible
the spectral resolutions of the self-adjoint extensions. This gives rise to
one-parameter orthogonality relations for sums of two 2\phi1-series. In
particular, we find that the Ismail-Zhang q-analogue of the exponential
function satisfies certain orthogonality relations analogous to the Fourier
cosine transform.Comment: 18 pages, to appear in Indagationes Mathematica
A locally compact quantum group analogue of the normalizer of SU(1,1) in SL(2,C)
S.L. Woronowicz proved in 1991 that quantum SU(1,1) does not exist as a
locally compact quantum group. Results by L.I. Korogodsky in 1994 and more
recently by Woronowicz gave strong indications that the normalizer N of SU(1,1)
in SL(2,C) is a much better quantization candidate than SU(1,1) itself. In this
paper we show that this is indeed the case by constructing N_q, a new example
of a unimodular locally compact quantum group (depending on a parameter q) that
is a deformation of N. After defining the underlying von Neumann algebra of N_q
we use a certain class of q-hypergeometric functions and their orthogonality
relations to construct the comultiplication. The coassociativity of this
comultiplication is the hardest result to establish. We define the Haar weight
and obtain simple formulas for the antipode and its polar decomposition. As a
final result we produce the underlying C*-algebra of N_q. The proofs of all
these results depend on various properties of q-hypergeometric 1\phi1
functions.Comment: 48 pages, 1 figur
Harmonic analysis on the SU(2) dynamical quantum group
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 satisfied by 6j-symbols. We investigate
one of the simplest examples, generalizing the standard SU(2) quantum group.
The matrix elements for its corepresentations are identified with Askey-Wilson
polynomials, and the Haar measure with the Askey-Wilson measure. The discrete
orthogonality of the matrix elements yield the orthogonality of q-Racah
polynomials (or quantum 6j-symbols). The Clebsch-Gordan coefficients for
representations and corepresentations are also identified with q-Racah
polynomials. This results in new algebraic proofs of the Biedenharn-Elliott
identity satisfied by quantum 6j-symbols.Comment: 51 pages; minor correction
Modular properties of matrix coefficients of corepresentations of a locally compact quantum group
We give a formula for the modular operator and modular conjugation in terms
of matrix coefficients of corepresentations of a quantum group in the sense of
Kustermans and Vaes. As a consequence, the modular autmorphism group of a
unimodular quantum group can be expressed in terms of matrix coefficients. As
an application, we determine the Duflo-Moore operators for the quantum group
analogue of the normaliser of SU(1,1) in ).Comment: 22 pages. To appear in Journal of Lie Theor
- …