# Fusion Classification of $U_q(sl_N)$ Categories

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?

The classification says that a fusion category $\mathcal{C}$ with the same fusion rules of $SU(N)_k$ is a twist of $SU(N)_k$ (the twists are determined by an $N$-th root of unity). The point is that $SU(N)_k$ is pivotal (even more is spherical), and a twist of a pivotal fusion category is again pivotal (same for spherical).