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.

Jones Representations of Thompson’s Group F Arising from Temperley–Lieb–Jones Algebras

Following a procedure due to Jones, using suitably normalized elements in a Temperley–Lieb–Jones (planar) algebra, we introduce a 3-parametric family of unitary representations of the Thompson’s group $F$ equipped with canonical (vacuum) vectors and study some of their properties. In particular, we discuss the behavior at infinity of their matrix coefficients, thus showing that these representations do not contain any finite-type component. We then focus on a particular representation known to be quasi-regular and irreducible and show that it is inequivalent to itself once composed with a classical automorphism of $F$. This allows us to distinguish three equivalence classes in our family. Finally, we investigate a family of stabilizer subgroups of $F$ indexed by subfactor Jones indices that are described in terms of the chromatic polynomial. In contrast to the 1st non-trivial index value for which the corresponding subgroup is isomorphic to the Brown–Thompson’s group $F_3$, we show that when the index is large enough, this subgroup is always trivial.

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.

Intermediate planar algebra revisited

In this paper, we explicitly work out the subfactor planar algebra [Formula: see text] for an intermediate subfactor [Formula: see text] of an irreducible subfactor [Formula: see text] of finite index. We do this in terms of the subfactor planar algebra [Formula: see text] by showing that if [Formula: see text] is any planar tangle, the associated operator [Formula: see text] can be read off from [Formula: see text] by a formula involving the so-called biprojection corresponding to the intermediate subfactor [Formula: see text] and a scalar [Formula: see text] carefully chosen so as to ensure that the formula defining [Formula: see text] is multiplicative with respect to composition of tangles. Also, the planar algebra of [Formula: see text] can be obtained by applying these results to [Formula: see text]. We also apply our result to the example of a semi-direct product subgroup-subfactor.

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.

On Jones’ subgroup of R. Thompson’s group T

Jones introduced unitary representations for the Thompson groups [Formula: see text] and [Formula: see text] from a given subfactor planar algebra. Some interesting subgroups arise as the stabilizer of certain vector, in particular the Jones subgroups [Formula: see text] and [Formula: see text]. Golan and Sapir studied [Formula: see text] and identified it as a copy of the Thompson group [Formula: see text]. In this paper, we completely describe [Formula: see text] and show that [Formula: see text] coincides with its commensurator in [Formula: see text], implying that the corresponding unitary representation is irreducible. We also generalize the notion of the Stallings 2-core for diagram groups to [Formula: see text], showing that [Formula: see text] and [Formula: see text] are not isomorphic, but as annular diagram groups they have very similar presentations.

The Brauer–Picard groups of fusion categories coming from the ADE subfactors

We compute the group of Morita auto-equivalences of the even parts of the [Formula: see text] subfactors, and Galois conjugates. To achieve this, we study the braided auto-equivalences of the Drinfeld centers of these categories. We give planar algebra presentations for each of these Drinfeld centers, which we leverage to obtain information about the braided auto-equivalences of the corresponding categories. We also perform the same calculations for the fusion categories constructed from the full [Formula: see text] subfactors. Of particular interest, the even part of the [Formula: see text] subfactor is shown to have Brauer–Picard group [Formula: see text]. We develop combinatorial arguments to compute the underlying algebra objects of these invertible bimodules.