Orthogonal-polynomial structures and fusion algebras of rational conformal field theories

A large class of fusion algebras, isomorphic to rings of orthogonal polynomials in one real variable, is studied. It includes all SU(2) WZW and all minimal model fusion algebras. All the algebras in this class having structure constants limited to 0 or 1 are classified. Two series consistent with both modular and duality constraints are found. Numerical searches for structure constants also larger than 1 seem to indicate that the whole classification is exhausted by the aforementioned series and an additional one. Relations of these structures with the SU(2) group are discussed.