Subfactor Planar Algebra Research Articles

Standard $\lambda$-lattices, rigid $\mathrm{C}^{*}$ tensor categories, and (bi)modules

In this article, we construct a 2 -shaded rigid \mathrm{C}^{*} multitensor category with canonical unitary dual functor directly from a standard \lambda -lattice. We use the notions of traceless Markov towers and lattices to define the notion of module and bimodule over standard \lambda -lattice(s), and we explicitly construct the associated module category and bimodule category over the corresponding 2 -shaded rigid \mathrm{C}^{*} multitensor category. As an example, we compute the modules and bimodules for Temperley–Lieb–Jones standard \lambda -lattices in terms of traceless Markov towers and lattices. Translating into the unitary 2-category of bigraded Hilbert spaces, we recover De Commer–Yamashita’s classification of \mathcal{T LJ} module categories in terms of edge weighted graphs, and a classification of \mathcal{T LJ} bimodule categories in terms of biunitary connections on square-partite weighted graphs. As an application, we show that every (infinite depth) subfactor planar algebra embeds into the bipartite graph planar algebra of its principal graph.

Integrable models from singly generated planar algebras

Not all planar algebras can encode the algebraic structure of a Yang–Baxter integrable model described in terms of a so-called homogeneous transfer operator. In the family of subfactor planar algebras, we focus on the ones known as singly generated and find that the only such planar algebras underlying homogeneous Yang–Baxter integrable models are the so-called Yang–Baxter relation planar algebras. According to a result of Liu, there are three such planar algebras: the well-known Fuss–Catalan and Birman–Wenzl–Murakami planar algebras, in addition to one more which we refer to as the Liu planar algebra. The Fuss–Catalan and Birman–Wenzl–Murakami algebras are known to underlie Yang–Baxter integrable models, and we show that the Liu algebra likewise admits a Baxterisation. We also show that the homogeneous transfer operator describing a model underlied by a singly generated Yang–Baxter relation planar algebra is polynomialisable, meaning that it is polynomial in a spectral-parameter-independent element of the algebra.

The Extended Haagerup fusion categories

In this paper we construct two new fusion categories and many new subfactors related to the exceptional Extended Haagerup subfactor. The Extended Haagerup subfactor has two even parts EH1 and EH2. These fusion categories are mysterious and are the only known fusion categories which appear to be unrelated to finite groups, quantum groups, or Izumi quadratic categories. One key technique which has previously revealed hidden structure in fusion categories is to study all other fusion categories in the Morita equivalence class, and hope that one of the others is easier to understand. In this paper we show that there are exactly four categories (EH1, EH2, EH3, EH4) in the Morita equivalence class of Extended Haagerup, and that there is a unique Morita equivalence between each pair. The existence of EH3 and EH4 gives a number of interesting new subfactors. Neither EH3 nor EH4 appears to be easier to understand than the Extended Haaerup subfactor, providing further evidence that Extended Haagerup does not come from known constructions. We also find several interesting intermediate subfactor lattices related to Extended Haagerup. The method we use to construct EH3 and EH4 is interesting in its own right and gives a general computational recipe for constructing fusion categories in the Morita equivalence class of a subfactor. We show that pivotal module $\rm C^*$ categories over a given subfactor correspond exactly to realizations of that subfactor planar algebra as a planar subalgebra of a graph planar algebra. This allows us to construct EH3 and EH4 by realizing the Extended Haagerup subfactor planar algebra inside the graph planar algebras of two new graphs. This technique also answers a long-standing question of Jones: which graph planar algebras contain a given subfactor planar algebra?

Intermediate planar algebra revisited, II

We describe the subfactor planar algebra of an intermediate subfactor [Formula: see text] of an extremal subfactor [Formula: see text] of finite Jones index which is not necessarily irreducible.

Commuting squares and planar subalgebras

We show a close relationship between non-degenerate smooth commuting squares of II1-factors with horizontal inclusions of finite index and inclusions of subfactor planar algebras by showing that each leads to a construction of the other. One direction of this uses the Guionnet--Jones--Shlyakhtenko construction.

The module embedding theorem via towers of algebras

Jones and Penneys showed that a finite depth subfactor planar algebra embeds in the bipartite graph planar algebra of its principal graph, via a Markov towers of algebras approach. We relate several equivalent perspectives on the notion of module over a subfactor planar algebra, and show that a Markov tower is equivalent to a module over the Temperley-Lieb-Jones planar algebra. As a corollary, we obtain a classification of semisimple pivotal C⁎ modules over Temperley-Lieb-Jones in terms of pointed graphs with a Frobenius-Perron vertex weighting. We then generalize the Markov towers of algebras approach to show that a finite depth subfactor planar algebra embeds in the bipartite graph planar algebra of the fusion graph of any of its cyclic modules.

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.

Ore's theorem on subfactor planar algebras

This article proves that an irreducible subfactor planar algebra with a distributive biprojection lattice admits a minimal 2-box projection generating the identity biprojection. It is a generalization (conjectured in 2013) of a theorem of Oystein Ore on distributive intervals of finite groups (1938), and a corollary of a natural subfactor extension of a conjecture of Kenneth S. Brown in algebraic combinatorics (2000). We deduce a link between combinatorics and representations in finite group theory.

Continuum Limits of Homogeneous Binary Trees and the Thompson Group.

Tree tensor network descriptions of critical quantum spin chains are empirically known to reproduce correlation functions matching conformal field theory (CFT) predictions in the continuum limit. It is natural to seek a more complete correspondence, additionally incorporating dynamics. On the CFT side, this is determined by a representation of the diffeomorphism group of the circle. In a remarkable series of papers, Jones outlined a research program where the Thompson group T takes the role of the latter in the discrete setting, and representations of T are constructed from certain elements of a subfactor planar algebra. He also showed that, for a particular example of such a construction, this approach only yields-in the continuum limit-a representation which is highly discontinuous and hence unphysical. Here we show that the same issue arises generically when considering tree tensor networks: the set of coarse-graining maps yielding discontinuous representations has full measure in the set of all isometries. This extends Jones's no-go example to typical elements of the so-called tensor planar algebra. We also identify an easily verified necessary condition for a continuous limit to exist. This singles out a particular class of tree tensor networks. Our considerations apply to recent approaches for introducing dynamics in holographic codes.

Quantum double inclusions associated to a family of Kac algebra subfactors

In \cite{Sde2018} we defined the notion of \textit{quantum double inclusion} associated to a finite-index and finite-depth subfactor and studied the quantum double inclusion associated to the Kac algebra subfactor $R^H \subset R$ where $H$ is a finite-dimensional Kac algebra acting outerly on the hyperfinite $II_1$ factor $R$ and $R^H$ denotes the fixed-point subalgebra. In this article we analyse quantum double inclusions associated to the family of Kac algebra subfactors given by $\{ R^H \subset R \rtimes \underbrace{H \rtimes H^* \rtimes \cdots}_{{\text{$m$ times}}} : m \geq 1 \}$. For each $m > 2$, we construct a model $\mathcal{N}^m \subset \mathcal{M}$ for the quantum double inclusion of $\{ R^H \subset R \rtimes \underbrace{H \rtimes H^* \rtimes \cdots}_{{\text{$m-2$ times}}} \}$ with $\mathcal{N}^m = ((\cdots \rtimes H^{-2} \rtimes H^{-1}) \otimes (H^m \rtimes H^{m+1} \cdots))^{\prime \prime}, \mathcal{M} = (\cdots \rtimes H^{-1} \rtimes H^0 \rtimes H^1 \rtimes \cdots)^{\prime \prime}$ and where for any integer $i$, $H^i$ denotes $H$ or $H^*$ according as $i$ is odd or even. In this article, we give an explicit description of $P^{\mathcal{N}^m \subset \mathcal{M}}$ ($m > 2$), the subfactor planar algebra associated to $\mathcal{N}^m \subset \mathcal{M}$, which turns out to be a planar subalgebra of $^{*(m)}\!P(H^m)$ (the adjoint of the $m$-cabling of the planar algebra of $H^m$). We then show that for $m > 2$, depth of $\mathcal{N}^m \subset \mathcal{M}$ is always two. Observing that $\mathcal{N}^m \subset \mathcal{M}$ is reducible for all $m > 2$, we explicitly describe the weak Hopf $C^*$-algebra structure on $(\mathcal{N}^m)^{\prime} \cap \mathcal{M}_2$, thus obtaining a family of weak Hopf $C^*$-algebras starting with a single Kac algebra $H$.

Universal skein theory for group actions

Given a group action on a finite set, we define the group-action model which consists of tensor network diagrams which are invariant under the group symmetry. In particular, group-action models can be realized as the even part of group-subgroup subfactor planar algebras. Moreover, all group-subgroup subfactor planar algebras arise in this way from transitive actions. In this paper, we provide a universal skein theory for those planar algebras. With the help of this skein theory, we give a positive answer to a question asked by Vaughan Jones in the late nineties.

Classification of Thurston relation subfactor planar algebras

Bisch and Jones proposed the classification of planar algebras by simple generators and relations. They investigated with the second author the classification of planar algebras generated by 2-boxes. In this paper, we classify singly-generated Thurston-relation planar algebras, defined as subfactor planar algebras generated by a 3-box satisfying a relation proposed by Dylan Thurston. Our main result shows that such subfactor planar algebras are either the E_6 subfactor planar algebras or belong to a two-parameter family of planar algebras arising from the representations of type A quantum groups. We introduce a new method for determining positivity of the Markov trace of planar algebras in this family.

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.

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.

Euler totient of subfactor planar algebras

We extend the Euler's totient function (from arithmetic) to any irreducible subfactor planar algebra, using the Mobius function of its biprojection lattice, as Hall did for the finite groups. We prove that if it is nonzero then there is a minimal 2-box projection generating the identity biprojection. We explain a relation with a problem of K.S. Brown. As an application, we define the dual Euler totient of a finite group and we show that if it is nonzero then the group admits a faithful irreducible complex representation. We also get an analogous result at depth 2, involving the central biprojection lattice.

Ore’s theorem on cyclic subfactor planar algebras and beyond

Ore proved that a finite group is cyclic if and only if its subgroup lattice is distributive. Now, since every subgroup of a cyclic group is normal, we call a subfactor planar algebra cyclic if all its biprojections are normal and form a distributive lattice. The main result generalizes one side of Ore's theorem and shows that a cyclic subfactor is singly generated in the sense that there is a minimal 2-box projection generating the identity biprojection. We conjecture that this result holds without assuming the biprojections to be normal, and we show that it is true for small lattices. We finally exhibit a dual version of another theorem of Ore and a non-trivial upper bound for the minimal number of irreducible components for a faithful complex representation of a finite group.

From skein theory to presentations for Thompson group

Jones introduced some unitary representations of Thompson group F constructed from a given subfactor planar algebra, and all unoriented links arise as matrix coefficients of these representations. Moreover, all oriented links arise as matrix coefficients of a subgroup F→ which is the stabilizer of a certain vector. Later Golan and Sapir determined the subgroup F→ and showed many interesting properties. In this paper, we investigate into a large class of groups which arises as subgroups of Thompson group F and reveal the relation between the skein theory of the subfactor planar algebra and the presentations of subgroups related to the corresponding unitary representation. Specifically, we answer a question by Jones about the 3-colorable subgroup.

On fixed point planar algebras

To a weighted graph can be associated a bipartite graph planar algebra P. We construct and study the symmetric enveloping inclusion of P. We show that this construction is equivariant with respect to the automorphism group of P. We consider subgroups G of the automorphism of P such that the G-fixed point space PG is a subfactor planar algebra. As an application we show that if G is amenable, then PG is amenable as a subfactor planar algebra. We define the notions of a cocycle action of a Hecke pair on a tracial von Neumann algebra and the corresponding crossed product. We show that a large class of symmetric enveloping inclusions of subfactor planar algebras can be described by such a crossed product.

Quotients of A2 * T2

AbstractWe study unitary quotients of the free product unitary pivotal category A2 * T2. We show that such quotients are parametrized by an integer n ≥ 1 and an 2n–th root of unity ω. We show that for n = 1, 2, 3, there is exactly one quotient and ω = 1. For 4 ≤ n ≤ 10, we show that there are no such quotients. Our methods also apply to quotients of T2 * T2, where we have a similar result.The essence of our method is a consistency check on jellyfish relations. While we only treat the specific cases of A2 × T2 and T2 . T2, we anticipate that our technique can be extended to a general method for proving the nonexistence of planar algebras with a specified principal graph.During the preparation of this manuscript, we learnt of Liu's independent result on composites of A3 and A4 subfactor planar algebras (arxiv:1308.5691). In 1994, BischHaagerup showed that the principal graph of a composite of A3 and A4 must fit into a certain family, and Liu has classified all such subfactor planar algebras. We explain the connection between the quotient categories and the corresponding composite subfactor planar algebras. As a corollary of Liu's result, there are no such quotient categories for n ≥ 4.This is an abridged version of arxiv:1308.5723.