UC San Diego

We classify ribbon categories with the tensor product rules of the finite-dimensional com-plex representations ofSO(N), forN≥5 andN= 3. The strategy is to study repre-sentations of the braid group which appear in End(X⊗k), whereXcorresponds to thedefining representation. The fusion rules serve to define path bases for these algebras, andwe prove that the matrix representations of the braid elements are uniquely determinedby the eigenvalues of a braid operator onX⊗X. We use this to show that the equivalenceclass of a category withSO(N) fusion rules depends only on one of the eigenvalues of thebraid operator. The classification applies both to genericSO(N) tensor product rules,and to certain fusion rings having only finitely many simple objects.