====== CMQG Wiki ======

This website was created to summarize all known properties of [[compact_quantum_group|compact quantum groups]], in particular [[Compact matrix quantum group|compact matrix quantum groups]] and associated [[Linear category of partitions|partition categories]].

All researchers working on compact quantum groups and related fields are invited to create and edit pages to push this project closer to its goal being the summary of all the current knowledge about compact matrix quantum groups. Before creating or editing pages, you are advised to read the [[Instructions for authors|instructions for authors]].

===== Content =====

In this section, all the central content of this wiki is listed.

==== Compact matrix quantum groups ====

  * [[Easy quantum group|Orthogonal easy quantum groups]]
    * [[free_orthogonal_quantum_group|Free orthogonal quantum group]] ($O^+$)
    * [[Free symmetric quantum group]] ($S^+$)
    * //[[easy summary|...summary]]//
  * [[Unitary easy quantum group|Unitary easy quantum groups]]
    * [[Free unitary quantum group]] ($U^+$)
    * //[[unitary easy summary|...summary]]//
  * $q$-deformations
    * $q$-special unitary quantum group ($SU_2^q$)

==== Partition categories ====

  * [[category_of_partitions|Banica–Speicher easy non-colored categories]]
    * //[[easy summary|summary]]//
  * [[category_of_partitions|Colored categories]]
    * [[Categories of two-colored partitions]]
      * //[[unitary easy summary|summary]]//
    * Partitions with two-self dual colors
    * Partitions with extra singletons
  * [[Linear category of partitions]]
  * [[Graph category]]
  * Examples
    * [[temperley_lieb_category|Temperley–Lieb category – Category of all non-crossing pairings]]
    * [[brauer category|Brauer catagory – category of all pairings]]
    * [[Category of all non-crossing partitions]]
    * [[Category of all partitions]]
    * [[Category of all graphs]]


==== Properties of CMQGs and associated algebras ====

  * [[unimodularity|Unimodularity]]
  * (Co)homology of quantum groups
  * Approximation properties ([[approximation properties summary|summary]])
    * [[Connes embedding property]]
    * [[Kirchberg property|Kirchberg factorization property (F)]]
    * [[Residual finiteness|Residual finiteness]]
    * [[Kazhdan property|Kazhdan property (T)]]
    * [[Amenability]]
    * [[Weak amenability]]
    * [[Haagerup property]]

==== Quantum group constructions ====

  * [[Quantum subgroup|Quantum subgroups and quotients]]
  * [[Intersection of quantum groups]]
  * [[Topological generation]]
  * [[Tensor product]]
  * [[Free product]]
  * [[Glued products]]
  * [[Free wreath product]]
  * Semidirect products
  * Wreath products


===== Literature =====

Here, we recommend some monographies and surveys related to the topic.


  * Teodor Banica, //Free quantum groups and related topics// [[https://banica.u-cergy.fr/a3.pdf|Online notes]]
  * Amaury Freslon, //Applications of non-crossing partitions to quantum groups//, lecture notes, 2019. [[https://www.imo.universite-paris-saclay.fr/~freslon/Documents/Talks/Copenhaguen_2019.pdf|Available online]]
  * Sergey Neshveyev, Lars Tuset, //Compact Quantum Groups and Their Representation Categories//, Société Mathématique de France, 2013. [[https://www.sciencesmaths-paris.fr/upload/Contenu/Fichiers_chaire/livre%20Sergey%20Neyshveyev.pdf|Available online]]
  * Thomas Timmermann. //An Invitation to Quantum Groups and Duality//, Zürich: European Mathematical Society (2008)
  * Moritz Weber, //Introduction to compact (matrix) quantum groups and Banica–Speicher (easy) quantum groups//, [[https://doi.org/10.1007/s12044-017-0362-3|Proceedings – Mathematical Sciences, Vol. 127, No. 5, pp. 881–933, 2017]]. [[https://www.ias.ac.in/article/fulltext/pmsc/127/05/0881-0933|Full text online.]]