The Haagerup property for Drinfeld doubles

TL;DR

This paper proves that the Drinfeld double construction preserves the Haagerup property for certain locally compact quantum groups, expanding understanding of their analytical properties.

Contribution

It demonstrates that the Drinfeld double construction maintains the Haagerup property across a range of important quantum groups.

Findings

Drinfeld doubles of specific quantum groups have the Haagerup property.

The construction preserves the Haagerup property for locally compact quantum groups.

Includes examples like $C_{0}( ext{F}_2)$, $SU_q(2)$, and others.

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.