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](http://arxiv.org/abs/math/0006057v2)). Then the initial subfactor is isomorphic to an intermediate of $(R^H \subset R)$ and by Galois correspondence (see [here](http://arxiv.org/abs/math/0001020)), 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?)



What are the applications of the depth 2 reduction to the subfactors theory?
*Two answers of Dmitri Nikshych:* 



> 
> **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.
> 
> 
> 


