Group-type subfactors and Hadamard matrices

TL;DR

This paper explores the construction of hyperfinite II_1 subfactors from Hadamard matrices, identifying their group-type structure and classifying them via automorphisms and cohomological methods.

Contribution

It introduces a new approach to realize subfactors as group-type inclusions derived from Hadamard matrices and classifies them using automorphism groups and cohomology.

Findings

Subfactors can be expressed as $R^H \\subset R \\rtimes K$ with finite groups H and K.

The automorphism groups generated by H and K are characterized.

Complete classifications are achieved in some cases through cohomological analysis.

Abstract

A hyperfinite $II_1$ subfactor may be obtained from a symmetric commuting square via iteration of the basic construction. For certain commuting squares constructed from Hadamard matrices, we describe this subfactor as a group-type inclusion $R^H \subset R \rtimes K$, where $H$ and $K$ are finite groups with outer actions on the hyperfinite $II_1$ factor $R$. We find the group of outer automorphisms generated by $H$ and $K$, and use the method of Bisch and Haagerup to determine the principal and dual principal graphs. In some cases a complete classification is obtained by examining the element of $H^3(H \ast K / Int R)$ associated with the action.