====== Category of pair partitions with even distances between legs ======

The **category of pair partitions with even distances between legs** is a [[category_of_partitions|Banica-Speicher category of partitions]] inducing the corepresentation category of the [[half-liberated orthogonal quantum group|half-liberated orthogonal quantum groups]]. It is a proper subcategory of the [[Brauer category]]. Its unique proper subcategory is the [[temperley_lieb_category|Temperley-Lieb category]].

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

What it means for a partition $p\in \Pscr$ to belong to this set has been said 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 set of all pair partitions with even distances between legs is denoted by $P_o^\ast$ in [(:ref:BanSp09)].

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

The category of pair partitions with even distances between legs is the subcategory of $\Pscr$ generated by the partition $\Pabcabc$. This canonical generator embodies the **half-commutation relations** $acb=bca$.

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

Via [[tannaka_krein_duality|Tannaka-Krein duality]] for compact quantum groups, the category of all pair partitions with even distances between legs corresponds to the family $(O^\ast_N)_{N\in \N}$ of [[half-liberated orthogonal quantum group|half-liberated orthogonal 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
)]