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An introduction to quantum symmetries

Christian Voigt

Abstract

These notes are an introduction to the theory of quantum symmetries of finite and infinite sets, graphs, and locally compact spaces.

An introduction to quantum symmetries

Abstract

These notes are an introduction to the theory of quantum symmetries of finite and infinite sets, graphs, and locally compact spaces.

Paper Structure

This paper contains 29 sections, 25 theorems, 75 equations, 2 figures.

Table of Contents

  1. Preface
  2. Introduction
  3. What is quantum symmetry?
  4. Quantum automorphisms of graphs
  5. The graph isomorphism game
  6. Exercises
  7. Tensor categories and quantum groups
  8. Constructions with quantum permutations
  9. $C^*$-tensor categories
  10. $C^*$-categories
  11. $C^*$-tensor categories
  12. Discrete quantum groups
  13. Tannaka-Krein duality
  14. Discretisation
  15. Exercises
  16. ...and 14 more sections

Key Result

Lemma 2.1

A matrix $u = (u_{x,y}) \in M_n(\mathbb{C})$ is a permutation matrix if and only if the following conditions are satisfied.

Figures (2)

  • Figure 1: A graph with a cherry
  • Figure 2: The homeomorphisms $\sigma$ and $\tau$

Theorems & Definitions (74)

  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Theorem 2.7: Banica-Collins 2008, Banica-Bichon 2009
  • ...and 64 more