The dual quantum group for the quantum group analogue of the normalizer of SU(1,1) in SL(2,C)

TL;DR

This paper explicitly describes the dual quantum group of the quantum analogue of the normalizer of SU(1,1) in SL(2,C), revealing its graded structure, generators, and decomposing the regular corepresentation into irreducibles, involving advanced special functions.

Contribution

It provides the first explicit description of the dual quantum group for this non-compact quantum group, including its graded structure and generators, and analyzes its corepresentations.

Findings

Explicit dual quantum group description obtained

Decomposition of corepresentations into irreducibles

New results on special functions of basic hypergeometric type

Abstract

The quantum group analogue of the normalizer of SU(1,1) in SL(2,C) is an important and non-trivial example of a non-compact quantum group. The general theory of locally compact quantum groups in the operator algebra setting implies the existence of the dual quantum group. The first main goal of the paper is to give an explicit description of the dual quantum group for this example involving the quantized enveloping algebra U_q(su(1,1)). It turns out that U_q(su(1,1)) does not suffice to generate the dual quantum group. The dual quantum group is graded with respect to commutation and anticommutation with a suitable analogue of the Casimir operator characterized by an affiliation relation to a von Neumann algebra. This is used to obtain an explicit set of generators. Having the dual quantum group the left regular corepresentation of the quantum group analogue of the normalizer of SU(1,1) in SL(2,C) is decomposed into irreducible corepresentations. Upon restricting the irreducible corepresentations to U_q(su(1,1))-representation one finds combinations of the positive and negative discrete series representations with the strange series representations as well as combinations of the principal unitary series representations. The detailed analysis of this example involves analysis of special functions of basic hypergeometric type and, in particular, some results on these special functions are obtained, which are stated separately. The paper is split into two parts; the first part gives almost all of the statements and the results, and the statements in the first part are independent of the second part. The second part contains the proofs of all the statements.