Free oriented extensions of subfactor planar algebras

TL;DR

This paper introduces a left adjoint functor called the free oriented extension for subfactor planar algebras and demonstrates its realization in the context of hyperfinite II_1 subfactors.

Contribution

It establishes the existence of the free oriented extension functor and its realization as bimodules for hyperfinite II_1 subfactors, expanding the understanding of planar algebra extensions.

Findings

Existence of a left adjoint functor for oriented extensions

Realization of the projection category as bimodules

Application to hyperfinite II_1 subfactors

Abstract

We show that the restriction functor from oriented factor planar algebras to subfactor planar algebras admits a left adjoint, which we call the free oriented extension functor. We show that for any subfactor planar algebra realized as the standard invariant of a hyperfinite $\rm{II}_1$ subfactor, the projection category of the free oriented extension admits a realization as bimodules of the hyperfinite $\rm{II}_{1}$ factor.