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Classical actions of quantum permutation groups

Amaury Freslon, Frank Taipe, Simeng Wang

Abstract

We describe explicitly all actions of the quantum permutation groups on classical compact spaces. In particular, we show that the defining action is the only non-trivial ergodic one. We then extend these results to all easy quantum groups associated to non-crossing partitions.

Classical actions of quantum permutation groups

Abstract

We describe explicitly all actions of the quantum permutation groups on classical compact spaces. In particular, we show that the defining action is the only non-trivial ergodic one. We then extend these results to all easy quantum groups associated to non-crossing partitions.
Paper Structure (17 sections, 28 theorems, 80 equations)

This paper contains 17 sections, 28 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminiaries
  3. Compact quantum groups
  4. Actions
  5. Quantum permutation groups
  6. Quantum permutations: the ergodic case
  7. Actions with small spectrum
  8. Arbitrary ergodic actions
  9. Quantum permutations: the general case
  10. Orbit structure of the action
  11. The fundamental domain
  12. Classification of arbitrary classical actions
  13. Further examples
  14. Free orthogonal quantum group
  15. Direct products
  16. ...and 2 more sections

Key Result

Theorem 2.2

Any finite-dimensional representation of a compact quantum group splits as a direct sum of irreducible ones, and any irreducible representation is equivalent to a unitary one, i.e. the matrix of any irreducible representation of dimension $n$ is the conjugate of a unitary element of $M_{n}(C(\mathbb

Theorems & Definitions (65)

  • Definition 2.1
  • Theorem 2.2: Woronowicz
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Banica
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • ...and 55 more