====== 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 [[category_of_partitions|Banica-Speicher category of partitions]] inducing the corepresentation category of the [[free modified bistochastic quantum group|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]] $\Pscr$ whose morphism set is the //set of all non-crossing partitions of even size with small blocks//. It was introduced by Weber in [(:ref:Web12)]. 

  * 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. 
  * We say that $p$ has **small blocks** if every block of $p$ has size $1$ or $2$. 
  * The partition $p$ is said to be **non-crossing** if there exist no blocks $B$ and $B'$ of $p$ with $B\neq B'$ and no legs $i,j\in B$ and $i',j'\in B'$ such that $i\prec i'\prec j$ and $i'\prec j\prec j'$ with respect to the cyclic order of $p$. 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 $\Pscr$ generated by the partition $\Labac$.

===== Associated easy quantum group =====

Via [[tannaka_krein_duality|Tannaka-Krein duality]] for compact quantum groups, the category of all non-crossing partitions of even size with small blocks corresponds to the family $(B^{\prime +}_N)_{N\in \N}$ of [[free modified bistochastic quantum group|free modified bistochastic quantum 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
)]

[( :ref:Web12 >>
author:  Weber, Moritz
title:   On the classification of easy quantum groups
year:    2013
journal: Advances in Mathematics
volume:  245
pages:   500--533
url:     https://doi.org/10.1016/j.aim.2013.06.019
archivePrefix: arXiv
eprint   :1201.4723v2
)]