Projective Versions of Spatial Partition Quantum Groups

Abstract: We generalize categories of spatial partitions in the sense of C\'ebron-Weber by introducing new base partitions. This allows us to construct additional examples of free orthogonal quantum groups but yields the same class of spatial partition quantum groups as before. Further, we use these new base partitions to show that the class of spatial partition quantum groups is closed under taking projective versions and in particular contains the projective version of all easy quantum groups. As an application, we determine the quantum groups corresponding to the categories of all spatial pair partitions and give explicit descriptions of the projective versions of easy quantum groups in terms of spatial partitions.