Mathematics – Quantum Algebra
Scientific paper
2009-05-18
IMRN 2010 (7), 1167-1314
Mathematics
Quantum Algebra
124 pages, incl. index, typos corrected
Scientific paper
10.1093/imrn/rnp173
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.
Groenevelt Wolter
Koelink Erik
Kustermans Johan
No associations
LandOfFree
The dual quantum group for the quantum group analogue of the normalizer of SU(1,1) in SL(2,C) does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The dual quantum group for the quantum group analogue of the normalizer of SU(1,1) in SL(2,C), we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The dual quantum group for the quantum group analogue of the normalizer of SU(1,1) in SL(2,C) will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-609345