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.
The term compact matrix quantum group (or CMQG for short) only makes sense with reference to a certain dimension . 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 as a pair of a -algebra and a matrix with entries in . In keeping with the general paradigm of non-commutative topology, is usually referred to as the algebra of continuous functions on even if is non-commutative.
A compact -matrix quantum group is a pair such that
If so, then and are uniquely determined by [Wor87]. They are called the comultiplication and the antipode (or coinverse), respectively. And is called the fundamental corepresentation (matrix).
A compact -matrix quantum group is a pair such that
Here also, of course, and u are referred to as the comultiplication and fundamental corepresentation (matrix), respectively.
Given a compact matrix quantum group with comultiplication the pair is a compact quantum group by [Wor98].
Two ways of comparing compact matrix quantum groups of the same matrix dimension were introduced by Woronowicz in [Wor87].
Any two compact -matrix quantum groups and with and are called identical if there exists an isomorphism of unital -algebras with for all .
And we say that and are similar if there exists an isomorphism of unital -algebras as well as an invertible matrix (a similarity transformation) such that , where and where is the unit of .