When is a quantum space not a group?

TL;DR

This paper surveys techniques from quantum group theory to identify quantum spaces that cannot be endowed with a quantum group structure, illustrating the methods with well-known examples and analyzing algebraic properties.

Contribution

It provides a comprehensive overview of criteria and methods to determine when quantum spaces lack quantum group structures, including new examples and theoretical tools.

Findings

Certain quantum spaces do not admit quantum group structures

Character and trace properties can obstruct quantum group structures

Nuclearity and other algebraic properties influence quantum group existence

Abstract

We give a survey of techniques from quantum group theory which can be used to show that some quantum spaces (objects of the category dual to the category of $\mathrm{C}^*$-algebras) do not admit any quantum group structure. We also provide a number of examples which include some very well known quantum spaces. Our tools include several purely quantum group theoretical results as well as study of existence of characters and traces on $\mathrm{C}^*$-algebras describing the considered quantum spaces as well as properties such as nuclearity.