Frohlich and Kerler classify categories with $SU(2)_k$ fusion rules and Kazdhan-Wenzl expand this to $SU(N)_k$ categories. In both cases, unless I am missing something, the classification are determined by the dimension of the generating object which seems to imply that there's a hidden pivotal structure lurking about.
Is this true? If so, how does the classification refine down to the fusion level?