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.