  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.
