====== Category of partitions with small blocks ======

The **category of partitions with small blocks** is a [[category_of_partitions|Banica-Speicher category of partitions]] inducing the corepresentation category of the [[bistochastic group|bistochastic groups]]. 

===== Definition =====
By the **category of partitions with small blocks** one denotes the subcategory of the [[category of all partitions]] $\Pscr$ whose underlying set is the //set of all partitions with small blocks//. 

A partition $p\in \Pscr$ is said to have **small blocks** if every block in $p$ is of size $1$ or $2$.

The set of all pair partitions with small blocks is denoted by $P_b$ in [(:ref:BanSp09)].

===== Canonical generator =====

The category of partitions with small blocks is the subcategory of $\Pscr$ generated by the set $\{\crosspart, \singleton\}$ of partitions. 

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


Via [[tannaka_krein_duality|Tannaka-Krein duality]] for compact quantum groups, the category of all partitions with small blocks corresponds to the family $(B_N)_{N\in \N}$ of [[bistochastic group|bistochastic 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
)]