On alpha-induction, chiral generators and modular invariants for subfactors

We consider a type III subfactor N subset of M of finite index with a finit e system of braided N-N morphisms which includes the irreducible constituen ts of the dual canonical endomorphism. We apply alpha-induction and, develo ping further some ideas of Ocneanu, we define chiral generators for the dou ble triangle algebra. Using a new concept of intertwining braiding fusion r elations, we show that the chiral generators can be naturally identified wi th the alpha-induced sectors. A matrix Z is defined and shown to commute wi th the S- and T-matrices arising from the braiding. If the braiding is nond egenerate, then Z is a "modular invariant mass matrix" in the usual sense o f conformal field theory. We show that in that case the fusion rule algebra of the dual system of M-M morphisms is generated by the images of both kin ds of alpha-induction, and that the structural information about its irredu cible representations is encoded in the mass matrix Z. Our analysis sheds f urther light on the connection between (the classifications of) modular inv ariants and subfactors, and we will construct and analyze modular invariant s from SU(n)(k) loop group subfactors in a forthcoming publication, includi ng the treatment of all SU(2)(k) modular invariants.