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

We consider a type III subfactor $N\subset M$ of finite index with a finite system of braided $N$-$N$ morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply $\alpha$-induction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle algebra. Using a new concept of intertwining braiding fusion relations, we show that the chiral generators can be naturally identified with the $\alpha$-induced sectors. A matrix $Z$ is defined and shown to commute with the S- and T-matrices arising from the braiding. If the braiding is non-degenerate, then $Z$ is a ``modular invariant mass matrix'' in the usual sense of 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 kinds of $\alpha$-induction, and that the structural information about its irreducible representations is encoded in the mass matrix $Z$. Our analysis sheds further light on the connection between (the classifications of) modular invariants and subfactors, and we will construct and analyze modular invariants from $SU(n)_k$ loop group subfactors in a forthcoming publication, including the treatment of all $SU(2)_k$ modular invariants.