The Annular Structure of Subfactors

TL;DR

This paper explores the structure of subfactors through planar algebras, establishing equivalences of module notions, classifying modules with invariant inner products, and constructing important subfactors like E6 and E8.

Contribution

It introduces a new classification of modules over planar algebras and provides a planar construction for key subfactors, advancing the understanding of subfactor theory.

Findings

Equivalence between operadic modules and modules over a universal annular algebra

Classification of modules with invariant inner products in the generic region

Planar construction of E6 and E8 subfactors

Abstract

Given a planar algebra we show the equivalence of the notions of a module over this algebra (in the operadic sense), and module over a universal annular algebra. We classify such modules, with invariant inner products, in the generic region and give applications to subfactorss, including a planar construction of the $E_6$ and $E_8$ subfactors.