The Haagerup property for Drinfeld doubles

Abstract

We show that Drinfeld's double group construction for locally compact quantum groups preserves the Haagerup property. This shows that the Drinfeld doubles of the quantum groups, $C_{0}(\mathbb{F}_{2})$, $SU_{q}(2)$, $SU_{q}(1,1)_{\text{ext}}$, quantum $ax+b$, quantum $az+b$, and $E_{q}(2)$ have the Haagerup property.