====== Compact matrix quantum group ======

**Compact matrix quantum groups** were defined by Woronowicz in [(:ref:Wor87)], originally under the name //compact matrix pseudogroups//. They generalize compact matrix groups in the field of non-commutative geometry. Compact matrix quantum groups are particular instances of [[Compact quantum group|compact quantum groups]], where the comultiplication is given by matrix multiplication. 

===== Definition =====
The term **compact matrix quantum group** (or **CMQG** for short) only makes sense with reference to a certain dimension  $N\in \N$. Two definitions appear in the literature, the orginal one by Woronowicz from [(:ref:Wor87)] and an equivalent alternative formulation. 

Both define a compact matrix quantum group $G$ as a pair $(A,u)$ of a $C^\ast$-algebra $A$ and a matrix $u$ with entries in $A$. In keeping with the general paradigm of non-commutative topology, $A$ is usually referred to as the //algebra of continuous functions// $C(G)$ //on// $G$ even if $A$ is non-commutative.

==== Original version by Woronowicz ====
A **compact** $N\!\times\! N$**-matrix quantum group** is a pair $(A,u)$ such that 
  * $A$ is a unital $C^\ast$-algebra, 
  * $u=(u_{i,j})_{i,j=1}^N$ is an $N\!\times\! N$-matrix of elements $\{u_{i,j}\}_{i,j=1}^N$ of $A$,
  * the $\ast$-subalgebra $\mathscr{A}$ of $A$ generated by $\{u_{i,j}\}_{i,j=1}^N$ is dense in $A$, 
  * there exists a homomorphism $\Delta:A\to A\otimes A$ of unital $C^\ast$-algebras from $A$ to the minimal tensor product  $A\otimes A$  of $C^\ast$-algebras of $A$ with itself with  $\Delta(u_{i,j})=\sum_{k=1}^N u_{i,k}\otimes u_{k,j}$ for all $i,j=1,\ldots,N$ and 
  * there exists a linear antimultiplicative mapping $S:\mathscr{A}\to \mathscr{A}$ with $S(S(a^\ast)^\ast)=a$ for all $a\in \mathscr{A}$ and with $\sum_{k=1}^N S(u_{i,k})u_{k,j}=\delta_{i,j}I$ and $\sum_{k=1}^N u_{i,k}S(u_{k,j})=\delta_{i,j}I$ for all $i,j=1,\ldots,N$, where $I$ is the unit of $A$.

If so, then $\Delta$ and $S$ are uniquely determined by [(:ref:Wor87)]. They are called the //comultiplication// and the //antipode// (or //coinverse//), respectively. And $u$ is called the //fundamental corepresentation (matrix)//.
==== Equivalent alternative version  ====
A **compact** $N\!\times\! N$**-matrix quantum group** is a pair $(A,u)$ such that 
  * $A$ is a unital $C^\ast$-algebra, 
  * $u=(u_{i,j})_{i,j=1}^N$ is an $N\!\times\! N$-matrix of elements $\{u_{i,j}\}_{i,j=1}^N$ of $A$,
  * $A$ is generated as a $C^\ast$-algebra by $\{u_{i,j}\}_{i,j=1}^N$,
  * the unique linear map $\Delta:A\to A\otimes A$ from $A$ to the minimal tensor product $A\otimes A$ of $C^\ast$-algebras with the property that $\Delta(u_{i,j})=\sum_{k=1}^N u_{i,k}\otimes u_{k,j}$ for all $i,j=1,\ldots,N$ is a homomorphism of unital $C^\ast$-algebras 
  * $u$ and $u^t=(u_{j,i})_{i,j=1}^N$ are invertible in the $C^\ast$-algebra $\C^{N\times N}\otimes A$.

Here also, of course, $\Delta$ and //u// are referred to as the //comultiplication// and //fundamental corepresentation (matrix)//, respectively.
===== Compact matrix quantum groups as compact quantum groups =====
Given a compact matrix quantum group $(A,u)$ with comultiplication $\Delta$ the pair $(A,\Delta)$ is a [[compact quantum group]] by [(:ref:Wor98)].

===== Comparing compact matrix quantum groups =====

Two ways of comparing compact matrix quantum groups of the __same matrix dimension__ $N\in\N$ were introduced by Woronowicz in [(:ref:Wor87)].

Any two compact $N\!\times\!N$-matrix quantum groups $(A,u)$ and $(A',u')$ with $u=(u_{i,j})_{i,j=1}^N$ and $u'=(u'_{i,j})_{i,j}^N$ are called **identical** if there exists an isomorphism of unital $C^\ast$-algebras $s:A\to A'$ with $s(u_{i,j})=u'_{i,j}$ for all $i,j=1,\ldots,N$.

And we say that $(A,u)$ and $(A',u')$ are **similar** if there exists an isomorphism of unital $C^\ast$-algebras $s:A\to A'$ as well as an invertible matrix $T\in \C^{N\times N}$ (a //similarity transformation//) such that $u'(T\otimes I')=(T\otimes I')u_s$, where $u_s=(s(u_{i,j}))_{i,j=1}^N$ and where $I'$ is the unit of $A'$.


===== References =====

[( :ref:Wor87 >>
author   :Stanisław L. Woronowicz
title    :Compact matrix pseudogroups
journal  :Communications in Mathematical Physics
year     :1987
volume   :111
number   :4
pages    :613--665
url      :http://dx.doi.org/10.1007/BF01219077
)]

[( :ref:Wor98 >>
author   :Stanisław L. Woronowicz
title	 :Compact quantum groups
booktitle: Quantum symmetries/Symétries quantiques. Proceedings of the Les Houches summer school, Session LXIV, Les Houches, France, August 1st -- September 8th, 1995
editor   :Connes, Alain
pages    :845--884
year     :1998
)]