====== Category of non-crossing partitions of even size with small blocks and even distances between legs ======

The **category of non-crossing partitions of even size with small blocks and even distances between legs** is a [[category_of_partitions|Banica-Speicher category of partitions]] inducing the corepresentation category of the [[freely modified bistochastic quantum group|freely modified bistochastic quantum groups]]. 

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

A partition $p\in \Pscr$ belongs to this set if the following conditions are met:
  * $p$ has **small blocks**, meaning that every block in $p$ is of size $1$ or $2$.
  * $p$ is of **even size**, i.e., if $k,l\in \{0\}\cup \N$ are such that $p\in \Pscr(k,l)$, then $k+l$ is an even number, which is to say that $p$ has evenly many points. Given that $p$ has small blocks, $p$ is of even size if and only if it has an even number of singleton blocks.
  * $p$ is **non-crossing**, which means that 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]].
  * $p$ has **even distances between legs**. This property has been expressed in three different but equivalent ways:
    * For any given block $B$ of $p$ only evenly many blocks $B'$ of $p$ with $B\neq B'$ exist which cross $B$, i.e., such that one can find $i,j\in B$ and $i',j'\in B'$ with $i\prec i'\prec j$ and $i'\prec j\prec j'$ (where $\cdot\!\prec\!\cdot\!\prec\!\cdot$ is the cyclic order of $p$).
    * For any block $B$ of $p$ and any two legs $i,j\in B$ there is an even number of points located between $i$ and $j$, i.e. in the interval $]i,j[_p$ given by the set $\{ k\,\vert\, i\prec k\prec j\}$.
    * If one labels the points of $p$ in alternating fashion with one of two symbols $\oplus$ and $\ominus$ along the cyclic order of $p$, then blocks of $p$ may only join points with unequal labels.
  * The name **set of all non-crossing partitions of even size with small blocks and even distances between legs** is to be taken literally.

===== Canonical Generator =====

The category of non-crossing partitions of even size with small blocks and even distances between legs is the subcategory of $\Pscr$ generated by the partition $\singleton\otimes \singleton$.

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

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