Easy orthogonal quantum group

Easy orthogonal quantum groups are a particular class of compact matrix quantum groups introduced by Banica and Speicher in [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 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 (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 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 $\Paabaab\notin\Cscr$ .