Table of Contents
Half-liberated bistochastic quantum group
The half-liberated bistochastic quantum groups are the elements of a sequence 
of compact matrix quantum groups introduced by Weber in [Web12]. Each 
interpolates the modified bistochastic group 
(and thus the bistochastic group 
) and the freely modified bistochastic quantum group 
of the corresponding dimension 
.
Definition
Given 
, the half-liberated bistochastic 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 bistochastic# 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 bistochastic# especially, implying that is a compact quantum subgroup of the freely modified bistochastic quantum group , one of the free counterparts of the bistochastic group (and the modified bistochastic 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 modified bistochastic group , the subgroup 
of 
given by signed bistochastic matrices. Hence, is a compact quantum supergroup of .
The half-liberated bistochastic 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 of even size with small blocks and even distances between legs that induces the corepresentation categories of 
. Canonically, it is generated by the set 
of partitions.
Representation theory
Cohomology
Related quantum groups
References
