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Category of non-crossing two-colored pair partitions with neutral blocks
The category of non-crossing two-colored pair partitions with neutral blocks is a category of two-colored partitions inducing the co-representation categories of the free unitary quantum group.
Definition
By the category of non-crossing two-colored pair partitions with neutral blocks one denotes the subcategory of the category of all two-colored partitions 
whose morphism class is the set of all non-crossing pair partitions with neutral blocks. It was introduced by Tarrago and Weber in [TaWe18], Proposition 3.3 (a) under the name 
.
- A two-colored partition
is called a pair partition (see category of all pair partitions in the un-colored case), if every block 
of 
satisfies 
. - It is said to be non-crossing if there exist no blocks and
of with 
and no legs 
and 
such that 
and 
with respect to the cyclic order of . (See also category of all non-crossing partitions in the un-colored case.) - is said to have neutral blocks if every block of has vanishing color sum
. In other words, the numbers of, on the one hand, upper 
-colored plus lower 
-colored legs of and, on the other hand, upper -colored plus lower -colored legs of coincide. - The name set of all non-crossing pair partitions partitions with neutral blocks is to be taken literally.
The set of non-crossing two-colored pair partitions with neutral blocks is the morphism set of the subcategory of generated by the empty set 
of two-colored partitions. It is the minimal category of two-colored partitions.
Associated unitary easy quantum groups
The category of non-crossing two-colored pair partitions with neutral blocks induces the co-representation categories of the free unitary quantum groups 
, defined by Wang in [Wang95], Example 4.2.