====== Group-theoretical quantum groups =====

**Group-theoretical quantum groups** are a particular class of [[compact matrix quantum group|compact matrix quantum groups]] introduced by Raum and Weber in [(:ref:RaWe15)].
===== Definition =====

For every $N\in \N$ any compact $N\times N$-matrix quantum group $G\cong(C(G),u)$ is called **group-theoretical** if $G$ is [[homogeneous compact matrix quantum group|homogeneous]] and the squares of the entries $\{u_{i,j}\}_{i,j=1}^N$ of the fundamental corepresentation $u$ of $G$ are central projections in $C(G)$, i.e., if $u_{i,j}^2=(u_{i,j}^2)^\ast=(u_{i,j}^2)^2$ and $u_{i,j}^2u_{k,l}=u_{k,l}u_{i,j}^2$ for all $i,j,k,l\in\{1,\ldots,N\}$.

===== References =====

[( :ref:RaWe16 >>
author:  Raum, Sven and Weber, Moritz 
title:   The full classification of orthogonal easy quantum groups
year:    2016
journal: Communications in Mathematical Physics
volume:  341
issue:   3
pages:   751--779
url:     https://doi.org/10.1007/s00220-015-2537-z
archivePrefix: arXiv
eprint   :1312.3857
)]


[( :ref:RaWe15 >>
author:  Raum, Sven and Weber, Moritz 
title:   Easy quantum groups and quantum subgroups of a semi-direct product quantum group
year:    2015
journal: Journal of Noncommutative Geometry
volume:  9
issue:   4
pages:   1261--1293
url:     https://doi.org/10.4171/JNCG/223 
archivePrefix: arXiv
eprint   :1311.7630v2
)]

[( :ref:RaWe14 >>
author:  Raum, Sven and Weber, Moritz 
title:   The combinatorics of an algebraic class of easy quantum groups
year:    2014
journal: Infinite Dimensional Analysis, Quantum Probability and related topics
volume:  17
issue:   3
url:     https://doi.org/10.1142/S0219025714500167
archivePrefix: arXiv
eprint   :1312.1497v1
)]