From quantum groups to groups (Q2871242)

In this significant and interesting paper, the authors use the recent developments in the representation theory of locally compact quantum groups to assign to each locally compact quantum group \(G\) a locally compact group \(\widetilde{G}\) which is the quantum version of point masses and is an invariant for the latter. The authors further show that ``quantum point masses '' can be identified with several other locally compact groups that can be naturally assigned to the quantum group \(G.\) This assignment \(G \rightarrow \widetilde{G}\) preserves compactness, discreteness and hence finiteness and for large classes of quantum groups amenability. The authors compute this invariant for some of the most well-known examples of non-classical quantum groups. For example, when \(G = SU_{\mu}(2)\), then \(\widetilde{G}= T\).NEWLINENEWLINEIn the final section of the paper the authors investigate the relation between the structure of \(G\) and that of \(\widetilde{G}\) and show that the latter, despite being a simpler object, can carry very important information about \(G\). NEWLINENEWLINENEWLINEThis work is a continuation of the first author's PhD thesis written under the supervision of the second author.