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Quantum symmetries of Cayley graphs of abelian groups

Daniel Gromada

Abstract

We study Cayley graphs of abelian groups from the perspective of quantum symmetries. We develop a general strategy for determining the quantum automorphism groups of such graphs. Applying this procedure, we find the quantum symmetries of the halved cube graph, the folded cube graph and the Hamming graphs.

Quantum symmetries of Cayley graphs of abelian groups

Abstract

We study Cayley graphs of abelian groups from the perspective of quantum symmetries. We develop a general strategy for determining the quantum automorphism groups of such graphs. Applying this procedure, we find the quantum symmetries of the halved cube graph, the folded cube graph and the Hamming graphs.

Paper Structure

This paper contains 11 sections, 7 theorems, 24 equations.

Table of Contents

  1. Preliminaries
  2. Notation
  3. Compact matrix quantum groups
  4. Representation categories and Tannaka--Krein reconstruction
  5. Partitions
  6. Anticommutative deformations
  7. Exterior products -- classical case
  8. Exterior products -- anticommutative case
  9. Projective versions
  10. Warm-up: Quantum symmetries of the classical hypercube
  11. Quantum automorphism group of a graph

Key Result

Theorem 1

We determine the quantum automorphism groups of the following graphs.

Theorems & Definitions (13)

  • Theorem
  • Remark 1.1
  • Theorem 1.2: Woronowicz--Tannaka--Krein
  • Theorem 1.3: Jon94
  • Proposition 1.4
  • proof
  • Remark 1.5
  • Proposition 1.6
  • proof
  • Proposition 1.7
  • ...and 3 more