$\alpha$-induction for bi-unitary connections

The tensor functor called $\alpha$-induction arises from a Frobenius algebra object, or a Q-system, in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of $N$ to $M$ arising from a subfactor $N\subset M$ of finite index and finite depth giving a braided fusion category of endomorpshisms of $N$. It is also understood in terms of Ocneanu's graphical calculus. We study this $\alpha$-induction for bi-unitary connections, which give a characterization of finite-dimensional nondegenerate commuting squares and gives certain 4-tensors appearing in recent studies of 2-dimensional topological order. We show that the resulting $\alpha$-induced bi-unitary connections are flat if we have a commutative Frobenius algebra, or a local Q-system. Examples related to chiral conformal field theory and the Dynkin diagrams are presented.