On a correspondence between quantum SU(2), quantum E(2) and extended quantum SU(1,1)

TL;DR

This paper explores the relationships between quantum SU(2), quantum E(2), and extended quantum SU(1,1) by applying a construction method to their actions on specific quantum spaces, revealing new quantum group connections.

Contribution

It demonstrates a novel application of a construction method to relate quantum SU(2), quantum E(2), and extended quantum SU(1,1) through their actions on quantum spaces.

Findings

Derived the extended SU(1,1) quantum group from quantum SU(2) action on quantum projective plane

Identified a quantum groupoid connecting the three quantum groups

Extended the understanding of quantum group actions on type I-factors

Abstract

In a previous paper, we showed how one can obtain from the action of a locally compact quantum group on a type I-factor a possibly new locally compact quantum group. In another paper, we applied this construction method to the action of quantum SU(2) on the standard Podles sphere to obtain Woronowicz' quantum E(2). In this paper, we will apply this technique to the action of quantum SU(2) on the quantum projective plane (whose associated von Neumann algebra is indeed a type I-factor). The locally compact quantum group which then comes out at the other side turns out to be the extended SU(1,1) quantum group, as constructed by Koelink and Kustermans. We also show that there exists a (non-trivial) quantum groupoid which has at its corners (the duals of) the three quantum groups mentioned above.