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The higher hyperoctahedral series is a family of compact matrix quantum groups introduced by Banica, Curran and Speicher in [BanCuSp10]. Each interpolates the quantum group of the hyperoctahedral series with parameter and the free hyperoctahedral quantum group , both of the corresponding dimension .
Given and with , the quantum group of the hyperoctahedral series with parameter 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.
If , the definition can also be expressed by saying that the fundamental corpresentation matrix of is cubic and satisfies the -mixing relations. In particular, satisfies the ultracubic relations, which is to say for all .
Had one allowed in the definition, one would have obtained the hyperoctahedral group .
The quantum groups of the higher hyperoctahedral series are group-theoretical hyperoctahedral orthogonal easy quantum groups and can therefore be written as a semi-direct product with its diagonal subgroup [RaWe15]:
for all and with , where denotes the continuous functions over the symmetric group of dimension (considered as the subgroup of given by all permutation matrices).
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 .
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 subgroup of given by orthogonal matrices with integer entries. Hence, is a compact quantum supergroup of .
Similarly, if is the closed two-sided ideal of generated by the relations for any , then is isomorphic to the -algebra of the half-liberated hyperoctahedral quantum group . Hence, is a compact quantum supergroup of .
For every with the quantum groups of the higher hyperoctahedral series with parameter 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 group-theoretical hyperoctahedral category of partitions that induces the corepresentation categories of . Canonically, if , it is generated by the set of partitions [RaWe14], where is the partition whose word representation is given by . See also categories of the higher hyperoctahedral series. The corepresentation categories of are induced by .