Flatness of $\alpha$-induced bi-unitary connections and commutativity of Frobenius algebras

The tensor functor called $\alpha$-induction produces a new unitary fusion category from a Frobenius algebra, or a $Q$-system, in a braided unitary fusion category. A bi-unitary connection, which is a finite family of complex number subject to some axioms, realizes an object in any unitary fusion category. It also gives a characterization of a finite-dimensional nondegenerate commuting square in subfactor theory of Jones and realizes a certain $4$-tensor appearing in recent studies of $2$-dimensional topological order. We study $\alpha$-induction for bi-unitary connections, and show that flatness of the resulting $\alpha$-induced bi-unitary connections implies commutativity of the original Frobenius algebra. This gives a converse of our previous result and answers a question raised by R. Longo. We furthermore give finer correspondence between the flat parts of the $\alpha$-induced bi-unitary connections and the commutative Frobenius subalgebras studied by B\"ockenhauer-Evans.