The freely modified bistochastic quantum groups are the elements of a sequence of compact matrix quantum groups introduced by Banica and Speicher in [BanSp09], although originally under the names . Each can be seen as a free counterpart of the bistochastic group of the corresponding dimension . However, differently from the other matrix groups, actually has two free counterparts, the second being the free modified bistochastic quantum group , defined by Weber in [Web12].
Given , the freely modified 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#.
Note that is designated by in [BanSp09]. It was later renamed by Weber in [Web12] after the discovery of the discovery of the easy quantum group which is now called free modified bistochastic quantum group and commonly given the symbol instead.
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 modified bistochastic group , the subgroup of given by signed bistochastic matrices. Hence, is a compact quantum supergroup of .
Moreover, if is the closed two-sided ideal of generated by the relations for any , where , then is isomorphic to , the algebra of the free modified bistochastic quantum group . Thus, in particular, is a compact quantum supergroup of .
The freely modified 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 of even size with small blocks and even distances between legs that induces the corepresentation categories of . The partition is its canonical generator.