====== Non-group-theoretical hyperoctahedral easy orthogonal quantum groups ======

The **non-group-theoretical hyperoctahedral easy orthogonal quantum groups** are a family $(H_N^{[\pi_k]})_{N\in \N,\,k\in \{2,3\ldots,\infty\}}$ of [[compact matrix quantum group|compact matrix quantum groups]] introduced by Raum and Weber in [(:ref:RaWe16)]. Each $H_N^{[\pi_k]}$ interpolates the [[free hyperoctahedral quantum group]] $H_N^+$ and any [[group-theoretical hyperoctahedral easy orthogonal quantum groups]] of the corresponding dimension $N$.

===== Definition =====
Given $N\in \N$ and $k\in\N\cup \{\infty\}$ with $k\geq 2$, the **non-group-theoretical hyperoctahedral easy orthogonal quantum group**  $H_N^{[\pi_k]}$ **with parameter** $k$ **for dimension** $N$ is the [[compact matrix quantum group]] $(C(H_N^{[\pi_k]}),u)$ 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]] 
$$C(H_N^{[\pi_k]})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,u=\overline u, \, uu^t=u^tu=I_N\otimes 1\,$$
$${\color{white}C(H_N^{[\pi_k]})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall_{i,j,l=1}^N: i\neq j\Rightarrow u_{i,l}u_{j,l}=u_{l,i}u_{l,j}=0,$$
$${\color{white}C(H_N^{[\pi_k]})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall_{m\in \N}: m<k\Rightarrow\forall a,b_1,\ldots,b_{m}\in \{u_{i,j}\}_{i,j=1}^n: $$
$${\color{white}C(H_N^{[\pi_k]})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,} a^2 \ (b_1b_2\ldots b_m)\ (b_m b_{m-1}\ldots b_1)= (b_1b_2\ldots b_m) \ (b_mb_{m-1}\ldots b_1) \ a^2\big\rangle,$$

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 $H_N^{[ \pi_k]}$ imply in particular for all $i,j=1,\ldots,N$ that $u_{i,j}^2$ is a partial isometry and that $\sum_{l=1}^N u_{i,l}^2=\sum_{l=1}^N u_{l,j}^2=1$.

The non-group-theoretical hyperoctahedral easy orthogonal quantum groups  $(H_N^{[\pi_2]})_{N\in\N}$ with parameter $2$ are also called the [[square commuting hyperoctahedral quantum group|square commuting hyperoctahedral quantum groups]] $(H_N^\diamond)_{N\in \N}$.

Had one allowed $k=1$ in the definition, one would have obtained the [[free hyperoctahedral quantum group]] $H_N^{[\pi_1]}\colon\hspace{-0.66em}=H_N^+$. 

===== Basic Properties =====


The fundamental corepresentation matrix $u$ of $H_N^{[\pi_k]}$ is in particular //orthogonal//. Hence, $H_N^{[\pi_k]}$ is a compact quantum subgroup of the [[free orthogonal quantum group]] $O_N^+$.

Moreover, $u$ is also //cubic// especially, implying that $H_N^{[\pi_k]}$ is a compact quantum subgroup of the [[free hyperoctahedral quantum group]] $H_N^{+}$, the free counterpart of the hyperoctahedral group $H_N$.

For all $k,k'\in \N\cup\{\infty\}$ with $k<k'$ the fundamental corepresentation matrix of $H_N^{[\pi_{k'}]}$ also satisfies the relations of $H_N^{[\pi_k]}$, meaning that $H_N^{[\pi_k']}$ is a compact quantum subgroup of $H_N^{[\pi_k]}$. Especially, the sequence $(H_N^{[\pi_{k}]})_{k\in \N\cup\{\infty\}$ forms a descending chain.

Conversely, the fundamental corepresentation matrix of any [[group-theoretical hyperoctahedral easy orthogonal quantum group]] satisfies the defining relations of $H_N^{[\pi_k]}$ for any $k\in \N\cup \{\infty\}$. Hence, $H_N^{[\pi_k]}$ is a compact quantum supergroup of any such quantum group for every $k$. 

For every $k\in \N\cup\{\infty\}$ with $k\geq 2$ the non-group-theoretical hyperoctahedral easy orthogonal quantum groups $(H_N^{[\pi_k]})_{N\in \N}$ with parameter $k$ are an [[easy_quantum_group|easy]] family of compact matrix quantum groups, i.e., the intertwiner spaces of their corepresentation categories are induced by a [[category of partitions]]. More precisely, it is a [[non-group-theoretical hyperoctahedral categories of partitions|non-group-theoretical hyperoctahedral category of partitions]] that induces the corepresentation categories of $(H_N^{[\pi_k]})_{N\in \N}$. Canonically, it is generated by the partition $\pi_k$ if $k<\infty$ and by the set $\{\pi_{l}\,\vert\,l\in\N\}$ of partitions if $k=\infty$ [(:ref:RaWe16)], where $\pi_l$ is the partition whose [[partition#word_representation|word representation]] is given by $\mathsf{a}_1\mathsf{a}_2\cdots\mathsf{a}_l\mathsf{a}_l\cdots\mathsf{a}_2\mathsf{a}_1\mathsf{a}_1\mathsf{a}_2\cdots\mathsf{a}_l\mathsf{a}_l\cdots\mathsf{a}_2\mathsf{a}_1$. 


===== 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
)]