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We consider a type III subfactor $N\subset M$ of finite index with a finitesystem of braided $N$-$N$ morphisms which includes the irreducible constituentsof the dual canonical endomorphism. We apply $\alpha$-induction and, developingfurther some ideas of Ocneanu, we define chiral generators for the doubletriangle algebra. Using a new concept of intertwining braiding fusionrelations, we show that the chiral generators can be naturally identified withthe $\alpha$-induced sectors. A matrix $Z$ is defined and shown to commute withthe S- and T-matrices arising from the braiding. If the braiding isnon-degenerate, then $Z$ is a ``modular invariant mass matrix'' in the usualsense of conformal field theory. We show that in that case the fusion rulealgebra of the dual system of $M$-$M$ morphisms is generated by the images ofboth kinds of $\alpha$-induction, and that the structural information about itsirreducible representations is encoded in the mass matrix $Z$. Our analysissheds further light on the connection between (the classifications of) modularinvariants and subfactors, and we will construct and analyze modular invariantsfrom $SU(n)_k$ loop group subfactors in a forthcoming publication, includingthe treatment of all $SU(2)_k$ modular invariants.Comment: 66 pages, latex, epic, eepic; minor changes, typos fixed, references  adde

On α-Induction, Chiral Generators and¶Modular Invariants for Subfactors