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Discrete quantum subgroups of free unitary quantum groups

Amaury Freslon, Moritz Weber

Abstract

We classify all compact quantum groups whose C*-algebra sits inside that of the free unitary quantum groups $U_{N}^{+}$. In other words, we classify all discrete quantum subgroups of $\widehat{U}_{N}^{+}$, thereby proving a quantum variant of Kurosh's theorem to some extent. This yields interesting families which can be described using free wreath products and free complexifications. They can also be seen as quantum automorphism groups of specific quantum graphs which generalize finite rooted regular trees, providing explicit examples of quantum trees.

Discrete quantum subgroups of free unitary quantum groups

Abstract

We classify all compact quantum groups whose C*-algebra sits inside that of the free unitary quantum groups . In other words, we classify all discrete quantum subgroups of , thereby proving a quantum variant of Kurosh's theorem to some extent. This yields interesting families which can be described using free wreath products and free complexifications. They can also be seen as quantum automorphism groups of specific quantum graphs which generalize finite rooted regular trees, providing explicit examples of quantum trees.
Paper Structure (23 sections, 31 theorems, 68 equations)

This paper contains 23 sections, 31 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Colored partitions
  4. Easy quantum groups
  5. Partitions and linear maps
  6. Tannaka-Krein duality and compact quantum groups
  7. Modules of projective partitions
  8. Definition and first properties
  9. Link with dual quantum subgroups
  10. Standard modules
  11. Classification in the orthogonal case
  12. Blocks of size at most two
  13. Blocks of arbitrary size
  14. Classification of modules in the unitary case
  15. The three series
  16. ...and 8 more sections

Key Result

Proposition 2.4

A partition $p\in P^{\circ\bullet}(k, k)$ is projective if and only if there exists a partition $r\in P^{\circ\bullet}(k, k)$ such that $r^{*}r = p$.

Theorems & Definitions (77)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Definition 2.8
  • Example 2.9
  • Theorem 2.10: Woronowicz
  • ...and 67 more