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Easy unitary quantum groups are a particular class of compact matrix quantum groups introduced by Freslon and Weber in [FreWeb16], generalizing the definition of easy orthogonal quantum groups given by Banica and Speicher in [BanSp09]. Every easy unitary quantum group is by definition a compact quantum subgroup of a free unitary quantum group.
Informally, a compact matrix quantum group is called easy unitary if it is a compact quantum subgroup of the corresponding free unitary quantum group and if its corepresentation category is generated by a category of partitions. Formally, for every , any compact -matrix quantum group is called an easy unitary quantum group if the corepresentation category of has as objects the set of tuples of arbitrary lengths with two distinct possible entries and and if there exists some category of two-colored partitions such that for all and all the morphism set of is given by
where for all the linear map satisfies for all ,
where is the standard basis of and where for all the symbol is if the kernel, i.e., the induced partition with upper and lower points, of refines and is otherwise.