The half-liberated orthogonal quantum groups are the elements of a sequence of compact matrix quantum groups introduced by Banica and Speicher in [BanSp09]. Each interpolates the orthogonal group and the free orthogonal quantum group . It is characterized by the half-commutation relations.
Given , the half-liberated orthogonal quantum group is the compact matrix quantum group where organizes the generators of the (unital) universal C*-algebra
where is the complex conjugate and is the transpose of , where is the identity -matrix and where is the unit of the universal -algebra.
In words, is the compact matrix quantum group whose fundamental corepresentation matrix is orthogonal and whose entries satisfy the half-commutation relations (which one calls the relations say for all entries ). Of course, these latter relations need to be viewed as a generalization of the commutation relations satisfied by all entries of the fundamental corepresentation matrix of .
By definition, the fundamental corepresentation matrix of is orthogonal. That makes a compact quantum subgroup of the free orthogonal quantum group .
If denotes the closed two-sided ideal of generated by the commutation relations for all , then is isomorphic to the -algebra of continuous functions on the orthogonal group . Hence, is a compact quantum supergroup of .
In conclusion, for every . That explains the name “half-liberated”: The half-liberated orthogonal quantum group is a “halfway point” between the orthogonal group and its “liberation”, the free orthogonal quantum group .
The half-liberated orthogonal 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 pair partitions with evenly many crossings, sometimes denoted by , that induces the corepresentation categories of . Its canonical generator partition is .