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Compact matrix quantum group

Compact matrix quantum groups were defined by Woronowicz in [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 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 [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. 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.

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 [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 $A'$ .