Planar algebras: a category theoretic point of view

TL;DR

This paper presents a category-theoretic framework for planar algebras, constructs them from 2-category 1-cells, and analyzes their affine representations, including finiteness and convergence properties.

Contribution

It introduces a new categorical perspective on planar algebras, connecting them to pivotal 2-categories and providing new finiteness and convergence results.

Findings

Finiteness results for affine representations of finite depth planar algebras.

Lower bounds on the radius of convergence related to the modulus.

Construction of planar algebras from 2-category 1-cells.

Abstract

We define Jones's planar algebra as a map of multicategories and constuct a planar algebra starting from a 1-cell in a pivotal strict 2-category. We prove finiteness results for the affine representations of finite depth planar algebras. We also show that the radius of convergence of the dimension of an affine representation of the planar algebra associated to a finite depth subfactor is at least as big as the inverse-square of the modulus.