$\alpha$-induction for bi-unitary connections

<jats:p> The tensor functor called <jats:inline-formula> <jats:tex-math>\alpha</jats:tex-math> </jats:inline-formula> -induction produces a new unitary fusion category from a Frobenius algebra object, or a <jats:inline-formula> <jats:tex-math>Q</jats:tex-math> </jats:inline-formula> -system, in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of <jats:inline-formula> <jats:tex-math>N</jats:tex-math> </jats:inline-formula> to <jats:inline-formula> <jats:tex-math>M</jats:tex-math> </jats:inline-formula> arising from a subfactor <jats:inline-formula> <jats:tex-math>N\subset M</jats:tex-math> </jats:inline-formula> of finite index and finite depth, which gives a braided fusion category of endomorphisms of <jats:inline-formula> <jats:tex-math>N</jats:tex-math> </jats:inline-formula> . It is also understood in terms of Ocneanu’s graphical calculus. We study this <jats:inline-formula> <jats:tex-math>\alpha</jats:tex-math> </jats:inline-formula> -induction for bi-unitary connections, which provides a characterization of finite-dimensional nondegenerate commuting squares, and present certain <jats:inline-formula> <jats:tex-math>4</jats:tex-math> </jats:inline-formula> -tensors appearing in recent studies of <jats:inline-formula> <jats:tex-math>2</jats:tex-math> </jats:inline-formula> -dimensional topological order. We show that the resulting <jats:inline-formula> <jats:tex-math>\alpha</jats:tex-math> </jats:inline-formula> -induced bi-unitary connections are flat if we start with a commutative Frobenius algebra, or a local <jats:inline-formula> <jats:tex-math>Q</jats:tex-math> </jats:inline-formula> -system. Examples related to chiral conformal field theory and the Dynkin diagrams are presented. </jats:p>