Examples of non-compact quantum group actions

TL;DR

This paper provides specific examples of non-regular locally compact quantum group actions on homogeneous spaces, emphasizing a purely C*-algebraic approach and discussing the continuity and construction of quantum homogeneous spaces.

Contribution

It introduces new examples of quantum group actions on homogeneous spaces using a C*-algebraic framework and details a general construction method for quantum homogeneous spaces as quotients.

Findings

Examples of non-regular quantum group actions presented

Continuity of actions analyzed

Construction of quantum homogeneous spaces detailed

Abstract

We present two examples of actions of non-regular locally compact quantum groups on their homogeneous spaces. The homogeneous spaces are defined in a way specific to these examples, but the definitions we use have the advantage of being expressed in purely $\mathrm{C}^*$-algebraic language. We also discuss continuity of the obtained actions. Finally we describe in detail a general construction of quantum homogeneous spaces obtained as quotients by compact quantum subgroups.