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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
.