====== Category of non-crossing two-colored pair partitions with neutral blocks ======

The **category of non-crossing two-colored pair partitions with neutral blocks** is a [[categories of two-colored partitions|category of two-colored partitions]] inducing the co-representation categories of the [[free unitary quantum group]].

===== Definition =====

By the **category of non-crossing two-colored pair partitions with neutral blocks** one denotes the subcategory of the [[category of all two-colored partitions]] $\Pscr^{\circ\bullet}$ whose morphism class is the //set of all non-crossing pair partitions with neutral blocks//. It was introduced by Tarrago and Weber in [(:ref:TaWe18)], Proposition 3.3 (a)  under the name $\mathcal{O}_{\mathrm{loc}}$. 

  * A two-colored partition $p\in\Pscr^{\circ\bullet}$ is called a **pair partition** (see [[category of all pair partitions]] in the un-colored case), if every block $B$ of $p$ satisfies $|B|=2$.
  * It 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]] in the un-colored case.)
  * $p$ is said to have **neutral blocks** if every block $B$ of $p$ has vanishing [[color sum]] $\sigma_p(B)=0$. In other words, the numbers of, on the one hand, upper $\bullet$-colored plus lower $\circ$-colored legs of $B$ and, on the other hand, upper $\circ$-colored plus lower $\bullet$-colored legs of $B$ coincide. 
  * The name **set of all non-crossing pair partitions partitions with neutral blocks** is to be taken literally.

The set of non-crossing two-colored pair partitions with neutral blocks is the morphism set of the subcategory of $\Pscr^{\circ\bullet}$ generated by the empty set $\emptyset$ of two-colored partitions. It is the minimal [[categories of two-colored partitions|category of two-colored partitions]]. 

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

The category of non-crossing two-colored pair partitions with neutral blocks induces the co-representation categories of the [[free unitary quantum group|free unitary quantum groups]] $(U_N^+)_{N\in \N}$, defined by Wang in [(:ref:Wang95)], Example 4.2.

===== References =====


[( :ref:TaWe18 >>
author:  Tarrago, Pierre and Weber, Moritz
title:   The classification of tensor categories of two-colored non-crossing partitions
year:    2018
journal: Journal of Combinatorial Theory, Series A
volume:  154
month:   February
pages:   464--506
url:     https://doi.org/10.1016/j.jcta.2017.09.003
archivePrefix: arXiv
eprint   :1509.00988 
)]

[( :ref:Wang95 >>
author   : Shuzhou Wang
title    : Free products of compact quantum groups
journal  : Communications in Mathematical Physics
year     : 1995
volume   : 167
number   : 3
pages    : 671--692
url      : http://dx.doi.org/10.1007/BF02101540
)]