Free oriented extensions of subfactor planar algebras

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 ${\mathrm{II}}_1$ subfactor, the projection category of the free oriented extension admits a realization as bimodules of the hyperfinite ${\mathrm{II}}_1$ factor.

29 pages, several figures, this version includes the complete statement and proof of universal property of free product of categories. We also re-defined projection category in terms of unitary Karaubi envelope and several other errors and typos are corrected. To appear in Int. J. Math