Given two groups , embedded into a larger one, we can compute their intersection , which is again a group. In particular, we can take compact matrix groups , represented by matrices of the same size and ask, what is their intersection – the largest subgroup of both. This concept can be generalized to the case of compact matrix quantum groups.
Let and be compact matrix quantum groups with fundamental representations and of the same size. We denote by the intersection of and defined as the largest quantum subgroup of both and . That is is defined by the fact that
The quantum group is unique and always exists as follows from the following characterization.
Suppose , , and are compact matrix quantum groups with unitary fundamental representations of the same size. The following are equivalent.