A Characterization of a Finite-Dimensional Commuting Square Producing a Subfactor of Finite Depth

Abstract

Abstract We give a characterization of a finite-dimensional commuting square of $C^*$-algebras with a normalized trace that produces a hyperfinite type II$_1$ subfactor of finite index and finite depth in terms of Morita equivalent unitary fusion categories. This type of commuting squares was studied by N. Sato and we show that a slight generalization of his construction covers the fully general case of such commuting squares. We also give a characterization of such a commuting square that produces a given hyperfinite type II$_1$ subfactor of finite index and finite depth. These results also give a characterization of certain 4-tensors that appear in recent studies of matrix product operators in 2D topological order.