A characterization of depth 2 subfactors of II_1 factors

TL;DR

This paper characterizes depth 2 subfactors of type II_1 factors using weak Kac and C*-Hopf algebra actions, extending known results to non-irreducible cases and describing the structure of the Jones tower.

Contribution

It introduces a new framework for understanding depth 2 subfactors via weak Hopf algebra actions, generalizing previous irreducible case results.

Findings

B=M' ap M_2 has a weak C*-Hopf algebra structure

M is the fixed point algebra under a minimal action of B on M_1

M_2 is isomorphic to the crossed product of M_1 by B

Abstract

We characterize finite index depth 2 inclusions of type II_1 factors in terms of actions of weak Kac algebras and weak C*-Hopf algebras. If N\subset M \subset M_1 \subset M_2 \subset ... is the Jones tower constructed from such an inclusion N\subset M, then B=M^\prime \cap M_2 has a natural structure of a weak C*-Hopf algebra and there is a minimal action of B on M_1 such that M is the fixed point subalgebra of M_1, and M_2 is isomorphic to the crossed product of M_1 and B. This extends the well-known results for irreducible depth 2 inclusions.