A Locally Compact Quantum Group Analogue of the Normalizer of SU(1,1) in SL(2, ?) |
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Authors: | Erik Koelink Johan Kustermans |
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Institution: | (1) Technische Universiteit Delft, Faculteit ITS, Afdeling Toegepaste Wiskundige Analyse, Mekelweg 4, 2628CD Delft, The Netherlands. E-mail: h.t.koelink@its.tudelft.nl, NL;(2) Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200B, 3001 Leuven, Belgium. E-mail: johan.kustermans@wis.kuleuven.ac.be, BE |
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Abstract: | 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 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 , a new example of a unimodular locally compact quantum group (depending on a parameter 0<q<1) that is a deformation of . After defining the underlying von Neumann algebra of 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 . The proofs of all these results depend on various properties of q-hypergeometric 1ϕ1 functions.
Received: 28 June 2001 / Accepted: 25 July 2002 Published online: 10 December 2002
RID="*"
ID="*" Post-doctoral researcher of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.)
Communicated by L. Takhtajan |
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Keywords: | |
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