[Frohlich and Kerler](http://link.springer.com/book/10.1007%2FBFb0084244) classify categories with $SU(2)\_k$ fusion rules and [Kazdhan-Wenzl](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=Wenzl&s5=Reconstructing&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq) 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?



Fusion Classification of $U_q(sl_N)$ Categories
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).


