====== Category of partitions with blocks of even size and parity-balanced legs ======

The **category of partitions with blocks of even size and parity-balanced legs** is a [[category_of_partitions|Banica-Speicher category of partitions]] inducing the corepresentation category of the [[half-liberated hyperoctahedral quantum group|half-liberated hyperoctahedral quantum groups]]. 

===== Definition =====
By the **category of partitions with blocks of even size and parity-balanced legs** one denotes the subcategory of the [[category of all partitions]] $\Pscr$ whose morphism class is the //set of partitions with blocks of even size and even distances between legs//. It was introduced by Banica, Curran and Speicher in [(:ref:BanCuSp10)]. 

A partition $p\in \Pscr$ belongs to this set if the following conditions are met:
  * $p$ has **blocks of even size**, i.e., every block of $p$ has an even number of legs.
  * $p$ has **parity-balanced legs**, i.e., for any block of $p$ when counting from an arbitrary point $i$ of the partition, the number of legs of $B$ at even distances from $i$ is equal to the number of legs of $B$ at odd distances from $i$.
  * The name **set of partitions with blocks of even size and parity-balanced legs** is to be taken literally.

A partition with blocks of even size is in particular of even size itself.

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

The category of partitions with blocks of even size and parity-balanced legs is the subcategory of $\Pscr$ generated by the set $\{\Pabcabc,\fourpart\}$ of partitions.

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

Via [[tannaka_krein_duality|Tannaka-Krein duality]] for compact quantum groups, the category of partitions with blocks of even size and parity-balanced legs corresponds to the family $(H^{\ast}_N)_{N\in \N}$ of [[half-liberated hyperoctahedral quantum group|half-liberated hyperoctahedral 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
)]

[( :ref:BanCuSp10 >>
author:  Banica, Teodor and Curran, Stephen and Speicher, Roland
title:   Classification results for easy quantum groups
year:    2010
journal: Pacific Journal of Mathematics
volume:  247
issue:   1
pages:   1-26
url:     https://doi.org/10.2140/pjm.2010.247.1
archivePrefix: arXiv
eprint   :0906.3890v1
)]