Chiral structure of modular invariants for subfactors

In this paper we further analyze modular invariants for subfactors, in part icular the structure of the chiral induced systems of M-hl morphisms. The r elative braiding between the chiral systems restricts to a proper braiding on their "ambichiral" intersection, and we show that the ambichiral braidin g is non-degenerate if the original braiding of the N-N morphisms is. Moreo ver, in this case the dimensions of the irreducible representations of the chiral fusion rule algebras are given by the chiral branching coefficients which describe the ambichiral contribution in the irreducible decomposition of ct-induced sectors. We show that modular invariants come along naturall y with several non-negative integer valued matrix representations of the or iginal N-N Verlinde fusion rule algebra, and we completely determine their decomposition into its characters. Finally the theory is illustrated by var ious examples, including the treatment of all SU(2)(k) modular invariants.