====== Half-liberated hyperoctahedral quantum group ======

The **half-liberated hyperoctahedral quantum groups** are the elements of a sequence $(H_N^{\ast})_{N\in \N}$ of [[compact matrix quantum group|compact matrix quantum groups]] introduced by Banica, Curran and Speicher in [(:ref:BanCuSp10)]. Each $H_N^{\ast}$ interpolates the [[wp>hyperoctahedral group]] $H_N$ and the [[free hyperoctahedral quantum group]] $H_N^{+}$ of the corresponding dimension $N$.

===== Definition =====
Given $N\in \N$, the **half-liberated hyperoctahedral quantum group** $H_N^{\ast}$ is the [[compact matrix quantum group]] $(C(H_N^{\ast}),u)$ 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]] 
$$C(H_N^{\ast})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,u=\overline u,\, uu^t=u^tu=I_N\otimes 1, \, \forall_{i,j,k=1}^N: i\neq j\Rightarrow u_{i,k}u_{j,k}=u_{k,i}u_{k,j}=0, \, \forall a,b,c\in \{u_{i,j}\}_{i,j=1}^n: acb=bca\big\rangle,$$
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 definition can also be expressed by saying that the fundamental corpresentation matrix $u$ of $H_N^{\ast}$ is **cubic** and satisfies the **half-commutation relations**.

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

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

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

If $I$ denotes the closed two-sided ideal of $C(H_N^{\ast})$ generated by the relations $u_{i,j}u_{k,l}=u_{k,l}u_{i,j}$ for any $i,j,k,l=1,\ldots, N$, then $C(H_N^{\ast})/I$ is isomorphic to the $C^\ast$-algebra $C(H_N)$ of continuous functions on the [[wp>hyperoctahedral group]] $H_N$, the subgroup  of $\mathrm{GL}(N,\C)$ given by orthogonal matrices with integer entries. Hence, $H_N^{\ast}$ is a compact quantum supergroup of $H_N$.

The half-liberated hyperoctahedral quantum groups $(H_N^{\ast})_{N\in \N}$ 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 the [[category of partitions with blocks of even size and even distances between legs|category of partitions with blocks of even size and parity-balanced legs]] that induces the corepresentation categories of $(H_N^{\ast})_{N\in \N}$. Canonically, it is generated by the set $\{\Pabcabc,\fourpart\}$ of partitions.


===== 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
)]