About 10 years ago, S.L. Woronowicz recognized that quantum $SU(1,1)$ does not exist as a locally compact quantum group. But research done by L.I. Korogodsky in 1994 and more recently, by Woronowicz suggests heavily that an extension $\widetilde{SU}(1,1)$ of $SU(1,1)$ can be deformed into a genuine locally compact quantum group. Through the use of $q$-special functions, Erik Koelink and the speaker showed in the first half of this year that this is indeed the case. In this talk, we will first look at the construction of this new locally quantum group. In a last part, the quantized universal enveloping Lie algebra and its precise action on the canonical Hilbert space of quantum $\widetilde{SU}(1,1)$ will be the focus of our attention.