A Galois correspondence for II_1 factors and quantum groupoids

TL;DR

This paper develops a Galois correspondence linking finite quantum groupoid actions to subfactors of II_1 factors, providing a new framework to classify and analyze subfactors via quantum groupoids.

Contribution

It introduces a novel Galois correspondence for quantum groupoid actions on II_1 factors and characterizes subfactors as crossed products by quantum groupoids.

Findings

Every finite index, finite depth subfactor is an intermediate subalgebra of a quantum groupoid crossed product.

Subfactors are uniquely determined by a quantum groupoid and a coideal *-subalgebra.

The bimodule category and principal graph are described in terms of quantum groupoid representations.

Abstract

We establish a Galois correspondence for finite quantum groupoid actions on II_1 factors and show that every finite index and finite depth subfactor is an intermediate subalgebra of a quantum groupoid crossed product. Moreover, any such a subfactor is completely and canonically determined by a quantum groupoid and its coideal *-subalgebra. This allows to express the bimodule category of a subfactor in terms of the representation category of a corresponding quantum groupoid and the principal graph as the Bratteli diagram of an inclusion of certain C^*-algebras related to it.