The free modified bistochastic quantum groups are the elements of a sequence of compact matrix quantum groups introduced by Weber in [Web12]. 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 freely modified bistochastic quantum group , defined by Banica and Speicher in [BanSp09].
Given , the free 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 [BanSp09] also studies free easy quantum groups denoted by , which are actually not the free modified bistochastic quantum groups treated in this article. Instead, in [BanSp09] corresponds to , the free freely modified bistochastic quantum group.
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 free freely modified bistochastic quantum group , the other counterpart of the bistochastic 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 .
The free 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 that induces the corepresentation categories of . The partition is its canonical generator.