The quantum group of plane motions and basic Bessel functions
type: Item
instance of: scholarly article
title: The quantum group of plane motions and basic Bessel functions
zbMATH Open document ID: 0833.17011
DOI: 10.1016/0019-3577(95)91243-O
published in: Indagationes Mathematicae. New Series
publication date: 1996-03-12T00:00:00Z
review text: The author considers matrix elements of irreducible representations of the quantum group of plane motions determined in some weak sense (generalized matrix elements), that is, as functionals on the quantized universal enveloping algebra. They have relative bi-invariance properties with respect to infinitesimally characterized quantum subgroups. This relative bi-invariance is the consequence of the fact that we can determine a basis of eigenvectors for different eigenvalues of a certain selfadjoint operator in \(l^2 (\mathbb{Z})\). The eigenvectors are described in terms of Jackson's \(q\)-Bessel functions. The author finds Hansen- Lommel orthogonality relations for these \(q\)-Bessel functions up to a scalar as a consequence of relative bi-invariance. The action of the comultiplication on generalized matrix elements is investigated. From this action the author derives a \(q\)-analogue of Graf's addition theorem for \(q\)-Bessel functions which are expressible as a \(q\)-hypergeometric \({}_2 \varphi_1\)-series. A sketch is given showing how one can formally interpret the orthogonality relations for the \(q\)-Jacobi functions as Hankel type orthogonality relations.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 33D80
zbMATH DE Number: 792825
zbMATH Keywords: irreducible representations
zbMATH Keywords: quantum group of plane motions
zbMATH Keywords: relative bi- invariance
zbMATH Keywords: eigenvectors
zbMATH Keywords: selfadjoint operator
zbMATH Keywords: Jackson's \(q\)-Bessel functions
zbMATH Keywords: Hansen-Lommel orthogonality
zbMATH Keywords: comultiplication
zbMATH Keywords: generalized matrix elements
zbMATH Keywords: \(q\)-analogue of Graf's addition theorem
author: Erik Koelink
reviewed by: A. U. Klimyk
MaRDI profile type: Publication
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full work available at URL: 0019-3577(95)91243-o
OpenAlex ID: W2083027520
description: scientific article; zbMATH DE number 792825