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Category of non-crossing partitions of even size with small blocks
The category of non-crossing partitions of even size with small blocks is a a Banica-Speicher category of partitions inducing the corepresentation category of the free modified bistochastic quantum groups.
Definition
By the category of non-crossing partitions of even size with small blocks one denotes the subcategory of the category of all partitions 
whose morphism set is the set of all non-crossing partitions of even size with small blocks. It was introduced by Weber in [Web12].
- For all
, a partition 
is said to be of even size if 
is an even number, i.e., if 
has evenly many points. - We say that has small blocks if every block of has size
or 
. - The partition 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 with small blocks is to be taken literally.
Canonical generator
The category of all non-crossing partitions of even size with small blocks is the subcategory of generated by the partition 
.
Associated easy quantum group
Via Tannaka-Krein duality for compact quantum groups, the category of all non-crossing partitions of even size with small blocks corresponds to the family 
of free modified bistochastic quantum groups.
References
