====== Group-theoretical hyperoctahedral easy orthogonal quantum groups ======
The **group-theoretical hyperoctahedral easy orthogonal quantum groups** are a two-parameter family of [[compact matrix quantum group|compact matrix quantum groups]] introduced by Raum and Weber and classified combinatorially in [(:ref:RaWe14)] and algebraically in [(:ref:RaWe15)]. The first parameter is the matrix dimension $N\in \N$ of the fundamental corepresentation. The second one runs through all strongly symmetric reflection groups on $N$ generators excepting $\Z_2$ and the trivial group. They are particular examples of the more general class of [[group-theoretical quantum groups]].
===== Definition =====
For every $N\in \N$ let $\Z_2^{\ast N}$ be the [[wp>free product]] group of $N$ many copies of thy cyclic group $\Z_2\equiv \Z/2\Z$ of order $2$ and for every $k\in\{1,\ldots,N\}$ let $a_k$ be the image of $1\equiv 1+2\Z$ under the embedding of $\Z_2$ as the $k$-th free factor in $\Z_2^{\ast N}$. Then, $\{a_1,\ldots,a_N\}$ generates $\Z_2^{\ast N}$ and any group endomorphism of $\Z_2^{\ast N}$ is uniquely determined by its restriction to $\{a_1,\ldots,a_N\}$. The **strong symmetric semigroup** $\mathrm{sS}_N$ is the subsemigroup of the semigroup $\mathrm{End}(\Z_2^{\ast N})$ of group endomorphisms of $\Z_2^{\ast N}$ generated by the endomorphisms defined by $a_k\mapsto a_{i(k)}$ for all $k\in \{1,\ldots,N\}$ for all mappings $i:\{1,\ldots,N\}\to\{1,\ldots,N\}$. In other words, $\mathrm{sS}_N$ is given by all identifications of letters in words in an alphabet of $N$ letters from $\Z_2$.
A **strongly symmetric reflection group (on** $N$ **generators)** is now a quotient group of $\Z_2^{\ast N}$ by a normal subgroup $A$ which is invariant under the action of $\mathrm{sS}_N$, the latter condition meaning $\varphi(w)\in A$ for all $w\in A$ and $\varphi\in \mathrm{sS}_N$.
Given $N\in \N$ and a strongly symmetric reflection group $\Z^{\ast N}/A$ on $N$ generators __other than__ $\Z_2$ __and the trivial group__, the **group-theoretical hyperoctahedral easy orthogonal quantum group with parameter** $\Z^{\ast N}/A$ **(for dimension** $N$**)** is the [[compact matrix quantum group]] $(C(H_N^{}),u)$ where $u=(u_{i,j})_{i,j=1}^N$ organizes the generators $\{u_{i,j}\}_{i,j=1}^N$ of the (unital) [[wp>Universal_C*-algebra|universal C*-algebra]]
$$C(H_N^{})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,u=\overline u, \, uu^t=u^tu=I_N\otimes 1\,$$
$${\color{white}C(H_N^{})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall_{i,j,l=1}^N: i\neq j\Rightarrow u_{i,l}u_{j,l}=u_{l,i}u_{l,j}=0,$$
$${\color{white}C(H_N^{})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall_{i,j,k,l=1}^N: u_{i,j}^2u_{k,l}=u_{k,l}u_{i,j}^2,$$
$${\color{white}C(H_N^{})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall_{m\in \N}:\, \forall_{t_1,\ldots,t_m=1}^N: \, a_{t_1}\ldots a_{t_m}\in A \Rightarrow$$
$${\color{white}C(H_N^{})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,} \forall_{i_1,j_1,\ldots,i_m,j_m=1}^N: (\forall_{p,q=1}^m: i_p=i_q\Leftrightarrow j_p=j_q\Leftrightarrow t_p=t_q)\Rightarrow$$
$${\color{white}C(H_N^{})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,} u_{i_1,j_1}u_{i_2,j_2}\ldots u_{i_m,j_m}=u_{i_1,j_1}^2u_{i_2,j_2}^2\ldots u_{i_m,j_m}^2\big\rangle,$$
where $\overline u=(u^\ast_{i,j})_{i,j=1}^N$ is the complex conjugate of $u$ and $u^t=(u_{j,i})_{i,j=1}^N$ the transpose, where $I_N$ is the identity $N\!\times \!N$-matrix and where $1$ is the unit of the universal $C^\ast$-algebra.
The defining relations of $H_N^{}$ imply in particular for all $i,j=1,\ldots,N$ that $u_{i,j}^2$ is a projection and that $\sum_{l=1}^N u_{i,l}^2=\sum_{l=1}^N u_{l,j}^2=1$.
Another characterization of $H_N^{}$ is given by writing it as a [[semi-direct product]] with its [[diagonal subgroup of a compact matrix quantum group|diagonal subgroup]] [(:ref:RaWe15)], which is precisely the strongly symmetric reflection group $\Z^{\ast N}/A$: $$C(H_N^{})\cong C^\ast(\Z^{\ast N}/A)\bowtie C(S_N),$$
where $C^\ast(\Z^{\ast N}/A)$ is the [[wp>Group_algebra#The_group_C*-algebra_C*(G)|group C*-algebra]] of $\Z^{\ast N}/A$ and where $C(S_N)$ denotes the continuous functions over the symmetric group of dimension $N$ (considered as the subgroup of $\mathrm{GL}(\C,N)$ given by all [[wp>permutation matrices]]).
Allowing also $\Z_2$ and the trivial group in the above definition yields the two group-theoretical __non-hyperoctahedral__ easy orthogonal quantum groups $S_N'$ and $S_N$, the [[modified symmetric group]] and the symmetric group $S_N$.
===== Basic Properties =====
The fundamental corepresentation matrix $u$ of $H_N^{}$ is in particular //orthogonal//. Hence, $H_N^{}$ is a compact quantum subgroup of the [[free orthogonal quantum group]] $O_N^+$.
Moreover, $u$ is also //cubic// especially, implying that $H_N^{}$ is a compact quantum subgroup of the [[free hyperoctahedral quantum group]] $H_N^{+}$, the free counterpart of the hyperoctahedral group $H_N$.
===== Representation theory =====
===== Cohomology =====
===== Related 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
)]
[( :ref:RaWe16 >>
author: Raum, Sven and Weber, Moritz
title: The full classification of orthogonal easy quantum groups
year: 2016
journal: Communications in Mathematical Physics
volume: 341
issue: 3
pages: 751--779
url: https://doi.org/10.1007/s00220-015-2537-z
archivePrefix: arXiv
eprint :1312.3857
)]
[( :ref:RaWe15 >>
author: Raum, Sven and Weber, Moritz
title: Easy quantum groups and quantum subgroups of a semi-direct product quantum group
year: 2015
journal: Journal of Noncommutative Geometry
volume: 9
issue: 4
pages: 1261--1293
url: https://doi.org/10.4171/JNCG/223
archivePrefix: arXiv
eprint :1311.7630v2
)]
[( :ref:RaWe14 >>
author: Raum, Sven and Weber, Moritz
title: The combinatorics of an algebraic class of easy quantum groups
year: 2014
journal: Infinite Dimensional Analysis, Quantum Probability and related topics
volume: 17
issue: 3
url: https://doi.org/10.1142/S0219025714500167
archivePrefix: arXiv
eprint :1312.1497v1
)]