The free unitary quantum groups are the members of a sequence of compact matrix quantum groups introduced by Wang in [Wang95], Example 4.2. Each is a free counterpart of the unitary group of the corresponding dimension .
Given , the free unitary quantum group is the compact matrix quantum group where organizes the generators of the (unital) universal C*-algebra
where is the complex conjugate transpose of , where is the identity -matrix and where is the unit of the universal -algebra.
The definition can also be expressed by saying that the fundamental corpresentation matrix of is unitary.
If denotes the closed two-sided ideal of generated by the relations for any , then is isomorphic to the -algebra of continuous functions on the unitary group , the subgroup of given by all unitary matrices. Hence, is a compact quantum supergroup of .
The free unitary quantum groups are a (unitary) easy family of compact matrix quantum groups; i.e., the intertwiner spaces of their corepresentation categories are induced by a category of (two-colored) partitions. More precisely, it is the category of non-crossing two-colored pair partitions with neutral blocks that induces the corepresentation categories of . Its canonical generating set of partitions is .