Quantum groups, from a functional analysis perspective

TL;DR

This paper reviews the foundational tools and advanced topics related to the study of closed quantum subgroups of free unitary quantum groups, bridging functional analysis and quantum group theory.

Contribution

It provides a comprehensive overview of the basic and advanced methods for analyzing quantum subgroups within free unitary quantum groups.

Findings

Introduction of key tools for quantum group analysis

Discussion of properties of closed quantum subgroups

Inclusion of advanced topics in quantum group theory

Abstract

It is well-known that any compact Lie group appears as closed subgroup of a unitary group, $G\subset U_N$. The unitary group $U_N$ has a free analogue $U_N^+$, and the study of the closed quantum subgroups $G\subset U_N^+$ is a problem of general interest. We review here the basic tools for dealing with such quantum groups, with all the needed preliminaries included, and we discuss as well a number of more advanced topics.