From Quantum Groups to Groups

TL;DR

This paper introduces a method to associate a classical locally compact group to a quantum group, capturing key structural properties and invariants, thus bridging quantum and classical group theories.

Contribution

It defines a new invariant for locally compact quantum groups, linking quantum point-masses to classical groups and preserving key properties like compactness, discreteness, and amenability.

Findings

The invariant can be explicitly calculated for well-known quantum groups.

Structural properties of quantum groups are encoded in the associated classical group.

The invariant preserves properties such as compactness, discreteness, and amenability.

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 $\mathbb{G}$.