Abstract view
From Quantum Groups to Groups


Published:20130206
Printed: Oct 2013
Mehrdad Kalantar,
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6
Matthias Neufang,
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6
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 pointmasses, and is an
invariant for the latter. We show that ``quantum pointmasses"
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
wellknown examples of
nonclassical 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 $\mathbb{G}$.