Subfactors and planar algebras

TL;DR

This paper explores the use of planar algebras as a pictorial framework to analyze the standard invariants of subfactors, leading to new structural insights in the theory of II$_1$ factors.

Contribution

It introduces a pictorial approach to subfactor invariants via planar algebras, providing novel structural results for subfactors.

Findings

Planar algebra description of subfactor invariants

New structural results for subfactors

Enhanced understanding of II$_1$ factor inclusions

Abstract

An inclusion of II$_1$ factors $N \subset M$ with finite Jones index gives rise to a powerful set of invariants that can be approached successfully in a number of different ways. We describe Jones' pictorial description of the standard invariant of a subfactor as a so-called planar algebra and show how this point of view leads to new structure results for subfactors.