By a free hyperoctahedral quantum group one means any element of the one-parameter sequence of compact matrix quantum groups defined by Banica, Bichon and Collins in [BanBichColl07]. Each is a free counterpart of the hyperoctahedral group of the corresponding dimension .
Given , the free hyperoctahedral quantum group is the compact matrix quantum group where organizes the generators of the (unital) universal C*-algebra
where is the complex conjugate and the transpose of , where is the identity -matrix and where is the unit of the universal -algebra.
The definition of is often equivalently expressed by saying that the fundamental corepresentation matrix is cubic.
If denotes the closed two-sided ideal of generated by the relations for any , then is isomorphic to the -algebra of continuous functions on the hyperoctahedral group, the latter realized as the group of orthogonal -matrices with integer entries. Hence, is a compact quantum supergroup of .
The free hyperoctahedral 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 the category of non-crossing partitions with blocks of even size that induces the corepresentation categories of . Its canonical generating partition is .