====== Category of two-colored partitions ======

**Categories of two-colored partitions** are certain strict monoidal involutive categories, introduced by Freslon and Weber in [(:ref:FreWeb16)], Definition 6.1. Each such category induces the co-representation categories of a sequence of [[unitary_easy_quantum_group|unitary easy quantum groups]]. Categories of two-colored partitions generalize [[category_of_partitions|categories of (uncolored) partitions]] as defined by Banica and Speicher in [(:ref:BanSp09)].

===== Definition =====

The original definition of Freslon and Weber in [(:ref:FreWeb16)], Definition 6.1 was later equivalently reformulated by Tarrago and Weber in [(:ref:TaWe18)], Section 1.3. In this formulation, a **category of two-colored partitions** is a subset $\Cscr\subseteq \Pscr^{\circ\bullet}$ of the set $\Pscr^{\circ\bullet}$ of all [[two-colored partition|two-colored partitions]] satisfying the following conditions with respect to the [[operations for two-colored partitions]]:
  * $\{\Partition{\Pline (1,0.125) (1,0.875) \Ppoint 0.125 \Pw:1 \Ppoint 0.875 \Pw:1},\Partition{\Pline (1,0.125) (1,0.875) \Ppoint 0.125 \Pb:1 \Ppoint 0.875 \Pb:1},\raisebox{0.125em}{\LPartition{\Pw:1;\Pb:2}{0.6:1,2}},\raisebox{0.125em}{\LPartition{\Pb:1;\Pw:2}{0.6:1,2}}\}\subseteq \Cscr$.
  * $pp'\in \Cscr$ for all $p,p'\in\Cscr$ such that $(p,p')$ is composable.
  * $p_1\otimes p_2\in \Cscr$ for all $p_1,p_2\in\Cscr$.
  * $p^\ast\in \Cscr$ for every $p\in\Cscr$.

===== References =====


[( :ref:FreWeb16 >>
author:  Freslon, Amaury and Weber, Moritz
title:   On the representation theory of partition (easy) quantum groups
year:    2016
journal: Journal für die reine und angewandte Mathematik [Crelle's Journal]
volume:  2016
issue:   720
url:     https://doi.org/10.1515/crelle-2014-0049
archivePrefix: arXiv
eprint   :1308.6390v2
)]


[( :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: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
)]