====== Category of partitions of even size with small blocks ======

The **category of partitions of even size with small blocks** is a [[category_of_partitions|Banica-Speicher category of partitions]] inducing the corepresentation category of the [[modified bistochastic group|modified bistochastic groups]]. 

===== Definition =====
By the **category of partitions of even size with small blocks** one denotes the subcategory of the [[category of all partitions]] $\Pscr$ whose underlying set is the //set of all partitions of even size with small blocks//. It was introduced by Banica and Speicher in [(:ref:BanSp09)]. 

This name is to be taken literally.
  * For all $k,l\in \{0\}\cup \N$, a partition $p\in \Pscr(k,l)$ is said to be **of even size** if $k+l$ is an even number, i.e., if $p$ has evenly many points. 
  * And $p$ is said to have **small blocks** if every block in $p$ is of size $1$ or $2$.

In particular, the set of all partitions of even size with small blocks is the intersection of the morphism sets of two larger categories, the [[category of partitions of even size]] and the [[category of all partitions with small blocks.]]

===== Canonical generator =====

The category of partitions of even size with small blocks is the subcategory of $\Pscr$ generated by the set $\{\crosspart, \singleton\otimes\singleton\}$ of partitions. 

===== Associated easy quantum groups =====


Via [[tannaka_krein_duality|Tannaka-Krein duality]] for compact quantum groups, the category of all partitions of even size with small blocks corresponds to the family $(B_N')_{N\in \N}$ of [[modified bistochastic group|modified bistochastic groups]].

===== References =====

[( :ref:BanSp09 >>
author:  Banica, Teodor and Speicher, Roland
title:   Liberation of orthogonal Lie groups
year:    2009
journal: Advances in Mathematics
volume:  222
issue:   4
pages:   1461--150
url:     https://doi.org/10.1016/j.aim.2009.06.009
archivePrefix: arXiv
eprint   :0808.2628
)]