Reconstructing braided subcategories of SU(N)k

Ocneanu rigidity implies that there are finitely many (braided) fusion categories with a given set of fusion rules. While there is no method for determining all such categories up to equivalence, there are a few cases for which one can. For example, Kazhdan and Wenzl described all fusion categories with fusion rules isomorphic to those of SU(N)k. In this paper we extend their results to a statement about braided fusion categories, and obtain similar results for certain subcategories of SU(N)k.