From Quantum Groups to Groups

Kalantar, Mehrdad ; Neufang, Matthias

Canadian journal of mathematics, Tome 65 (2013) no. 5, pp. 1073-1094

Résumé

In this paper we use the recent developments in the representation theory of locally compact quantum groups, to assign to each locally compact quantum group $\mathbb{G}$ a locally compact group $\tilde{\mathbb{G}}$ that is the quantum version of point-masses and is an invariant for the latter. We show that “quantum point-masses” can be identified with several other locally compact groups that can be naturally assigned to the quantum group $\mathbb{G}$ . This assignment preserves compactness as well as discreteness (hence also finiteness), and for large classes of quantum groups, amenability. We calculate this invariant for some of the most well-known examples of non-classical quantum groups. Also, we show that several structural properties of $\mathbb{G}$ are encoded by $\tilde{\mathbb{G}}$ the latter, despite being a simpler object, can carry very important information about $\mathbb{G}$ .

Détail

DOI : 10.4153/CJM-2012-047-x

Mots-clés : 46L89, locally compact quantum group, locally compact group, von Neumann algebra