Planar algebras

We introduce a notion of planar algebra, the simplest example of which is a vector space of tensors, closed under planar contractions. A planar algebra with suitable positivity properties produces a finite index subfactor of a II1 factor, and vice versa.

Planar algebras, quantum information theory and subfactors

We define generalized notions of biunitary elements in planar algebras and show that objects arising in quantum information theory such as Hadamard matrices, quantum Latin squares and unitary error bases are all given by biunitary elements in the spin planar algebra. We show that there are natural subfactor planar algebras associated with biunitary elements.

Free oriented extensions of subfactor planar algebras

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 [Formula: see text] subfactor, the projection category of the free oriented extension admits a realization as bimodules of the hyperfinite [Formula: see text] factor.

Non-semisimple planar algebras from the representation theory of Ūq(𝔰𝔩2)

We describe the generators and prove a number of relations for the construction of a planar algebra from the restricted quantum group [Formula: see text]. This is a diagrammatic description of [Formula: see text], where [Formula: see text] is a two-dimensional [Formula: see text] module.