# What are the applications of the depth 2 reduction to the subfactors theory?

Let $$(N \subset M)$$ be an irreducible finite depth ($$>2$$) finite index unital inclusion of hyperfinite $${\rm II}_1$$ factors, then for $$n$$ sufficiently large the subfactor $$(N \subset M_n)$$ is depth $$2$$ (reducible) and is isomorphic to $$(R^H \subset R)$$ with $$H$$ a weak Kac algebra (also called quantum groupoid, see prop. 9.1.1 p37 here). Then the initial subfactor is isomorphic to an intermediate of $$(R^H \subset R)$$ and by Galois correspondence (see here), it is completely given by the inclusion $$(J \subset H)$$ with $$J$$ a left coideal subalgebra of $$H$$.

Question: what are the applications of this result to the subfactors theory?
(In what this help to better understand the subfactors?)

Dmitri
This reduces the study of subfactors to that of weak Hopf algebras. Equivalently (and more importantly) it relates the theory of finite index finite depth subfactors to the theory of fusion categories and module categories over them.

There are many papers by Snyder, Morrison and others in this direction.

A concrete example: it follows from the theory of fusion categories that the index of a subfactor is a cyclotomic integer.

Sebastien
Yes the fusion category framework helps a lot!

My question is more about the weak Kac algebra framework. An irreducible finite index finite depth subfactor is completely given by an inclusion of a left coideal subalgebra in a weak Kac algebra, i.e. an inclusion of finite dimensional ${\rm C}^{\star}$-algebras with an additional "quantum group"-like structure.
In what this help to better understand the subfactors?

Dmitri
To me weak Hopf algebras are a bridge between subfactors and fusion categories.

The importance of weak Hopf algebra description of finite depth subfactors is that it reduces a seemingly analytic problem (inclusion of von Neumann algebras) to purely algebraic theory. It also folows that weak Hopf algebras are the correct symmetries of subfactors just like groups are symmetries of field extensions.