Table of Contents
Free bistochastic quantum group
The free bistochastic quantum groups are the elements of a sequence 
of compact matrix quantum groups introduced by Banica and Speicher in [BanSp09]. Each 
is a free counterpart of the bistochastic group 
of the corresponding dimension 
.
Definition
Given 
, the free 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 (or doubly stochastic), which is to say that is orthogonal and each of its rows and columns sums to .
Basic Properties
The fundamental corepresentation matrix of is in particular orthogonal. Hence, is a compact quantum subgroup of the free orthogonal quantum 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 bistochastic group , the subgroup of 
given by all bistochastic matrices. Hence, is a compact quantum supergroup of .
The free 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 all non-crossing partitions with small blocks that induces the corepresentation categories of 
. Its canonical generating partition is 
.
Representation theory
Cohomology
Related quantum groups
References
