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Free unitary quantum group
The free unitary quantum groups are the members of a sequence 
of compact matrix quantum groups introduced by Wang in [Wang95]. Each 
is a free counterpart of the unitary group 
of the corresponding dimension 
.
Definition
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.
Basic Properties
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 
.
Representation theory
Cohomology
Related quantum groups
References
