====== Diagonal subgroup of a compact matrix quantum group ======

The **diagonal subgroup** is an algebraic invariant of [[compact matrix quantum group|compact matrix quantum groups]] introduced by Raum and Weber in [(:ref:RaWe15)].

===== Definition =====

Given a [[compact matrix quantum group]] $G\cong(C(G),u)$ of dimension $N\in \N$ the **diagonal subgroup** of $G$ is defined as the unique (classical) group $\Gamma$ such that a [[wp>Group_algebra#The_group_C*-algebra_C*(G)|group C*-algebra]] $C^\ast(\Gamma)$ of $\Gamma$ is isomorphic as a $C^\ast$-algebra to the quotient $C(G)/I$, where $I$ is the closed two-sided ideal of $C(G)$ generated by the relations $\{u_{i,j}=0\,\vert\, i,j\in\N,\, i\neq j\}$. It is denoted by $\mathrm{diag}(G)\colon\hspace{-0.66em}=\Gamma$.

If $G$ is in its [[compact_quantum_group#the_universal_C*-algebra|maximal compact quantum group]] version, then $\mathrm{diag} (G)$ is in its maximal group $C^\ast$-algebra version.  [(:ref:RaWe15)].

===== Basic Properties =====



===== Importance to group-theoretical easy quantum groups =====



===== References =====
[( :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
)]