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Category of non-crossing partitions of even size
The category of non-crossing partitions of even size is a Banica-Speicher category of partitions inducing the corepresentation category of the free modified symmetric quantum groups.
Definition
By the category of non-crossing partitions of even size one denotes the subcategory of the category of all partitions 
whose morphism class is the set of all non-crossing partitions of even size. It was introduced by Banica and Speicher in [BanSp09].
- For all
, a partition 
is said to be of even size if 
is an even number, i.e., if 
has evenly many points. - 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. - The name set of all non-crossing partitions of even size is to be taken literally.
It is sometimes said that the category of non-crossing partitions of even size is the even part of the category 
of all non-crossing partitions.
Canonical generator
The category of all non-crossing partitions of even size is the subcategory of generated by the set of partitions 
.
Associated easy quantum group
Via Tannaka-Krein duality for compact quantum groups, the category of all non-crossing partitions of even size corresponds to the family 
of free modified symmetric quantum groups.