Table of Contents
Half-liberated hyperoctahedral quantum group
The half-liberated hyperoctahedral quantum groups are the elements of a sequence 
of compact matrix quantum groups introduced by Banica, Curran and Speicher in [BanCuSp10]. Each 
interpolates the hyperoctahedral group 
and the free hyperoctahedral quantum group 
of the corresponding dimension 
.
Definition
Given 
, the half-liberated hyperoctahedral quantum group 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 definition can also be expressed by saying that the fundamental corpresentation matrix of is cubic and satisfies the half-commutation relations.
Basic Properties
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 .
The half-liberated 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 partitions with blocks of even size and parity-balanced legs that induces the corepresentation categories of . Canonically, it is generated by the set 
of partitions.
Representation theory
Cohomology
Related quantum groups
References
