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Ocneanu rigidity implies that there are finitely many (braided) fusioncategories with a given set of fusion rules. While there is no method fordetermining all such categories up to equivalence, there are a few cases forwhich can. For example, Kazhdan and Wenzl described all fusion categories withfusion rules isomorphic to those of $SU(N)_k$. In this paper we extend theirresults to a statement about braided fusion categories, and obtain similarresults for certain subcategories of $SU(N)_k$.

Reconstructing Braided Subcategories of $SU(N)_k$