Hilbert space representations of the quantum *-algebra $\mathcal{U}_q(su_{1,1})$

Abstract

The Hilbert space representation theory of the $q$-deformed quantum $*$-algebra $\mathcal{U}_q(su_{1,1})$ is studied using the inducing procedure. As a result we obtain four series of irreducible induced $*$-representations on Hilbert spaces. Furthermore, we show that there is a one-to-one correspondence between the induced series and the series of irreducible well-behaved $*$-representations of $\mathcal{U}_q(su_{1,1})$ computed by Burban and Klymik.