On the planar algebra of Ocneanu's asymptotic inclusion

TL;DR

This paper provides a diagrammatic construction of the Jones tower for finite-depth planar algebras and describes the planar algebra of Ocneanu's asymptotic inclusion, leading to new fusion algebra computations.

Contribution

It introduces a diagrammatic method to construct the Jones tower and describe the planar algebra of the asymptotic inclusion, connecting to Ocneanu's work.

Findings

Diagrammatic construction of the Jones tower for finite-depth planar algebras

Description of the planar algebra of Ocneanu's asymptotic inclusion

Computation of the reduced fusion algebra of the asymptotic inclusion

Abstract

In recent joint work with V. Jones and D. Shlyakhtenko, we have given a diagrammatic description of Popa's symmetric enveloping inclusion for planar algebra subfactors. In this paper we give a diagrammatic construction of the associated Jones tower, in the case that the planar algebra is finite-depth. We then use this construction to describe the planar algebra of the symmetric enveloping inclusion, which is known to be isomorphic to the planar algebra of Ocneanu's asymptotic inclusion by a result of Popa. As an application we give a planar algebraic computation of the (reduced) fusion algebra of the asymptotic inclusion, recovering some well-known results of Ocneanu and Evans-Kawahigashi.