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# From Quantum Groups to Groups

Published:2013-02-06
Printed: Oct 2013
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6
• Matthias Neufang,
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6
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## Abstract

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 $\tilde {\mathbb{G}}$ which 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 $\mathbb{G}$.
 Keywords: locally compact quantum group, locally compact group, von Neumann algebra
 MSC Classifications: 46L89 - Other noncommutative'' mathematics based on $C^*$-algebra theory [See also 58B32, 58B34, 58J22]