====== Easy orthogonal quantum group ======

**Easy orthogonal quantum groups** are a particular class of [[compact matrix quantum group|compact matrix quantum groups]] introduced by Banica and Speicher in [(:ref:BanSp09)], actually under the name //"easy quantum groups"//. The qualifier //"orthogonal"// is used in this wiki to distinguish this class of compact matrix quantum groups from the class of [[unitary easy quantum group|unitary easy quantum groups]]. Every easy orthogonal quantum group is by definition a compact quantum subgroup of a [[free orthogonal quantum group]]. All easy orthogonal quantum groups are known and they provide a great wealth of examples of compact matrix groups.

===== Definition =====

Informally, a [[compact matrix quantum group]] is called **easy** if it is a compact quantum subgroup of the corresponding free orthogonal quantum group and if its corepresentation category is generated by a category of partitions. Formally, for every $N\in \N$, any compact $N\times N$-matrix quantum group $G\cong (C(G),u)$ is called an **easy orthogonal quantum group** if the [[corepresentation category]] $\FundRep(G)$ of $G$ has as objects the set $\N\cup \{0\}$ and if there exists some [[category_of_partitions|category of (uncolored) partitions]] $\Cscr\subseteq \Pscr$ such that for all $k,\ell\in\N\cup \{0\}$ the morphism set $k\to \ell$ of $\FundRep(G)$ is given by 
$$\mathrm{Hom}(k,\ell)=\spanlin_\C(\{ T_p\,\vert\, p\in \Cscr(k,l)\}),$$
where for all $p\in \Cscr(k,l)$ the linear map $T_p:\,(\C^N)^{\otimes k}\to (\C^N)^{\otimes \ell}$ satisfies for all $j_1,\ldots,j_k\in N$,
$$T_p(e_{j_1}\otimes\cdots\otimes e_{j_k})=\sum_{i_1,\dots,i_\ell=1}^N\delta_p(j_1,\ldots,j_k,i_1,\ldots,i_\ell)(e_{i_1}\otimes\cdots\otimes e_{i_\ell}),$$
where $(e_i)_{i=1}^N$ is the standard basis of $\C^N$ and where for all $i_1,\ldots,i_\ell\in N$ the symbol $\delta_p(j_1,\ldots,j_k,i_1,\ldots,i_\ell)$ is $1$ if the kernel, i.e., the induced partition with $k$ upper and $\ell$ lower points, of $(j_1,\ldots,j_k,i_1,\ldots,i_\ell)$ refines $p$ and is $0$ otherwise.

===== Taxonomy =====

There are several systems to divide the class of of easy orthogonal quantum groups into cases. Let $G\cong(G(G),u)$ be an easy orthogonal quantum group and let $\Cscr\subseteq\Pscr$ be the category of partitions generating its corepresentation category. We say that $G$ is

  * Classical case distinction:
    * **orthogonal** or **case** $O$: if $\singleton\notin \Cscr$ and $\fourpart\notin \Cscr$,
    * **bistochastic** or **case** $B$: if $\singleton\in \Cscr$ and $\fourpart\notin \Cscr$,
    * **symmetric** or **case** $S$: if $\singleton\in \Cscr$ and $\fourpart\in \Cscr$,
    * **hyperoctahedral** or **case** $H$: if $\singleton\notin \Cscr$ and $\fourpart\in \Cscr$,
  * Liberty distinction:
    * **free** or **liberated**: if all partitions of $\Cscr$ are [[category_of_all_non-crossing_partitions|non-crossing]],
    * **half-liberated**: if $\Pabcabc\in \Cscr$ and $\Pabab\notin\Cscr$,
    * **classical** or **commutative** or **group case**: if $\Pabab\in\Cscr$,
  * Group theoreticity distinction:
    * **group-theoretical**: if $\Paabaab\in\Cscr$,
    * **non-group-theoretical**: if $\Paabaab\notin\Cscr$.
===== Classification =====

===== 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
)]