A notion of Morita equivalence between subfactors

Abstract

We will review a notion of Morita equivalence between subfactors, which is a variation of Morita equivalence in ring and module theory. The main result is stated as follows: for arbitrary two Morita equivalent subfactors of hyperfinite $\mathrm{II}_1$ factors with finite Jones index and finite depth we can always choose a finite dimensional nondegenerate commuting square which gives rise to the subfactors isomorphic to the original ones. As an application of Morita equivalence between subfactors in connection with recent developments of theory of finite dimensional weak $C^*$-Hopf algebras, we will make a brief comment about the 3-dimensional topological quantum field theories obtained from subfactors with finite index and finite depth.