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Category of non-crossing partitions with small blocks
The category of non-crossing partitions with small blocks is a a Banica-Speicher category of partitions inducing the corepresentation category of the free bistochastic quantum groups.
Definition
By the category of non-crossing partitions with small blocks one denotes the subcategory of the category of all partitions 
whose morphism set is the set of all non-crossing partitions with small blocks. It was introduced by Banica and Speicher in [BanSp09].
- A partition
is said to have small blocks if every block of 
has size 
or 
. - 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 with small blocks is to be taken literally.
Canonical generator
The category of all non-crossing partitions 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 with small blocks corresponds to the family 
of free bistochastic quantum groups.
References
[BanSp09]
Banica, Teodor and Speicher, Roland, 2009. Liberation of orthogonal Lie groups. Advances in Mathematics, 222, pp.1461–150.