Introduction to quantum groups

TL;DR

This paper introduces the concept of quantum groups, focusing on the simplest examples like free unitary groups, and discusses their structure, examples, and theoretical foundations relevant to quantum mechanics and statistical mechanics.

Contribution

It provides an accessible introduction to quantum groups, including their basic properties, examples, and foundational theories such as Peter-Weyl and Tannakian duality.

Findings

Development of the basic theory of quantum groups

Examples of quantum groups related to quantum mechanics

Application of classical representation theory to quantum groups

Abstract

This is an introduction to the quantum groups, or rather to the simplest quantum groups. The idea is that the unitary group $U_N$ has a free analogue $U_N^+$, whose standard coordinates $u_{ij}\in C(U_N^+)$ are allowed to be free, and the closed subgroups $G\subset U_N^+$ can be thought of as being the compact quantum Lie groups. There are many interesting examples of such quantum groups, for the most designed in order to help with questions in quantum mechanics and statistical mechanics, and some general theory available as well, including Peter-Weyl theory, Tannakian duality, Brauer theorems and Weingarten integration. We discuss here the basic aspects of all this.