====== Free hyperoctahedral quantum group ======
By a **free hyperoctahedral quantum group** one means any element of the one-parameter sequence $(H_N^+)_{N\in \N}$ of [[compact matrix quantum group|compact matrix quantum groups]] defined by Banica, Bichon and Collins in [(:ref:BanBichColl07)]. Each $H_N^+$ is a [[free_orthogonal_easy_quantum_groups|free]] counterpart of the [[wp>hyperoctahedral group]] $H_N$ of the corresponding dimension $N$. 

===== Definition =====

Given $N\in \N$, the **free hyperoctahedral quantum group** $H_N^+$ 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]] 
$$C(H_N^+)\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert u=\overline u,\, uu^t=u^tu=I_N\otimes 1,\,\forall_{i,j,k=1}^N: i\neq j\Rightarrow u_{i,k}u_{j,k}=u_{k,i}u_{k,j}=0\big\rangle,$$
where $\overline u=(u_{i,j}^\ast)_{i,j=1}^N$ is the complex conjugate and $u^t=(u_{j,i})_{i,j=1}^N$ the transpose of $u$, where $I_N$ is the identity $N\!\times\!N$-matrix and where $1$ is the unit of the universal $C^\ast$-algebra.

The definition of $H_N^+$ is often equivalently expressed by saying that the fundamental corepresentation matrix $u$ is **cubic**.

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

If $I$ denotes the closed two-sided ideal of $C(H_N^+)$ 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 $C(H_N^+)/I$ is isomorphic to the $C^\ast$-algebra $C(H_N)$ of continuous functions on the [[wp>hyperoctahedral group]], the latter realized as the group of orthogonal $N\!\times\!N$-matrices with integer entries. Hence, $H_N^+$ is a compact quantum supergroup of $H_N$.

The free hyperoctahedral quantum groups $(H_N^+)_{N\in \N}$ 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 non-crossing partitions with blocks of even size]] that induces the corepresentation categories of $(H_N^+)_{N\in \N}$. Its canonical generating  partition is $\fourpart$. 
===== Representation theory =====
===== Cohomology =====
===== Related quantum groups =====

===== References =====

[( :ref:BanBichColl07 >>
author   : Banica Teodor and Bichon Julien and Collins Benoit
title    : The hyperoctahedral quantum group
journal  : Journal of the Ramanujan Mathematical Society
year     : 2007
volume   : 22
number   : 4
pages    : 345--384
archivePrefix: arXiv
eprint   :0701859
)]