The square commuting hyperoctahedral quantum groups are a family of compact matrix quantum groups introduced by Raum and Weber in [RaWe16]. Each interpolates the free hyperoctahedral quantum group and any group-theoretical hyperoctahedral easy orthogonal quantum groups of the corresponding dimension . They are a special instance of non-group-theoretical hyperoctahedral easy orthogonal quantum groups.
Given , the square commuting (hyperoctahedral) quantum group for dimension is the compact matrix quantum group where organizes the generators of the (unital) universal C*-algebra
where is the complex conjugate of and the transpose, where is the identity -matrix and where is the unit of the universal -algebra.
The defining relations of imply in particular for all that is a partial isometry and that .
The square commuting quantum group is the maximal quantum group of the descending chain of non-group-theoretical hyperoctahedral easy orthogonal quantum groups of dimension .
The fundamental corepresentation matrix of is in particular orthogonal. Hence, is a compact quantum subgroup of the free orthogonal quantum group .
Moreover, is also cubic especially, implying that is a compact quantum subgroup of the free hyperoctahedral quantum group , the free counterpart of the hyperoctahedral group .
The fundamental corepresentation matrix of any group-theoretical hyperoctahedral easy orthogonal quantum group satisfies the defining relations of . Hence, is a compact quantum supergroup of any such quantum group.
The square commuting quantum groups are an 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 category of partitions that induces the corepresentation categories of . Canonically, it is generated by the partition [RaWe16], where is the partition whose word representation is given by .