A complete set of numerical invariants for a subfactor

We show that certain numerical invariants associated naturally to a subfactor planar algebra constitute a complete family in the sense of determining the isomorphism class of the subfactor planar algebra.In the course of the proof, we show also that planar algebra isomorphisms of subfactor planar algebras can always be chosen to be ∗-preserving. This latter statement generalises the fact that ‘Hopf algebra isomorphisms of finite-dimensional Kac algebras can be chosen to be ∗-preserving’.     

Connections for small vertex models

This paper is a first attempt at classifying connections on small vertex models i.e., commuting squares of the form displayed in (1.2) below. More precisely, if we letB(k,n) denote the collection of matricesW for which (1.2) is a commuting square then, we: (i) obtain a simple model form for a representative from each equivalence class inB(2,n), (ii) obtain necessary conditions for two such ‘model connections’ inB(2,n) to be themselves equivalent, (iii) show thatB(2,n) contains a (3n - 6)-parameter family of pairwise inequivalent connections, and (iv) show that the number (3n - 6) is sharp. Finally, we deduce that every graph that can arise as the principal graph of a finite depth subfactor of index 4 actually arises for one arising from a vertex model corresponding toB(2,n) for somen.     

Guionnet–Jones–Shlyakhtenko subfactors associated to finite-dimensional Kac algebras

We analyse the Guionnet–Jones–Shlyakhtenko construction for the planar algebra associated to a finite-dimensional Kac algebra and identify the factors that arise as finite interpolated free group factors.     

The planar algebra of group-type subfactors

If G is a countable, discrete group generated by two finite subgroups H and K and P is a II₁ factor with an outer G-action, one can construct the group-type subfactor PH⊂P?K introduced by Haagerup and the first author to obtain numerous examples of infinite depth subfactors whose standard invariant has exotic growth properties. We compute the planar algebra of this subfactor and prove that any subfactor with an abstract planar algebra of “group type” arises from such a subfactor. The action of Jones' planar operad is determined explicitly.     

On Ocneanu's theory of double triangle algebras for subfactors and classification of irreducible connections on the Dynkin diagrams

We give an exposition of Ocneanu's theory of double triangle algebras for subfactors and its application to the classification of irreducible bi-unitary connections on the Dynkin diagrams _An$A_n$, _Dn$D_n$, _E6$E_6$, _E7$E_7$ and _E8$E_8$. More precisely, we give a detailed proof of the complete classification of irreducible K–L$K–L$ bi-unitary connections up to gauge choice, where K and L   represent the two horizontal graphs which are among the A–D–E$A–D–E$ Dynkin diagrams. The result also provides a simple proof of the flatness of _D2n$D_{2n}$, _E6$E_6$ and _E8$E_8$ connections as well as an easy computation of the flat part of _E7$E_7$ as an application.     

The planar algebra associated to a Kac algebra

We obtain (two equivalent) presentations — in terms of generators and relations — of the planar algebra associated with the subfactor corresponding to (an outer action on a factor by) a finite-dimensional Kac algebra. One of the relations shows that the antipode of the Kac algebra agrees with the ‘rotation on 2-boxes’.     

Mutation Pairs in Abelian Categories

A notion of mutation pairs of subcategories in an abelian category is defined in this article. For an extension closed subcategory 𝒵 and a rigid subcategory 𝒟 ? 𝒵, the subfactor category 𝒵/[𝒟] is also a triangulated category whenever (𝒵, 𝒵) forms a 𝒟-mutation pair. Moreover, if 𝒟 and 𝒵 satisfy certain conditions in modΛ, the category of finitely generated Λ-modules over an artin algebra Λ, the triangulated category 𝒵/[𝒟] has a Serre functor.     

Subfactor Categories of Triangulated Categories

Let 𝒳 ? 𝒜 be subcategories of a triangulated category 𝒯, and 𝒳 a functorially finite subcategory of 𝒜. If 𝒜 has the properties that any 𝒳-monomorphism of 𝒜 has a cone and any 𝒳-epimorphism has a cocone, then the subfactor category 𝒜/[𝒳] forms a pretriangulated category in the sense of [4 Beligiannis , A. , Reiten , I. ( 2007 ). Homological and Homotopical Aspects of Torsion Theories . Memoirs of the AMS 883 : 426 – 454 . [Google Scholar]]. Moreover, the above pretriangulated category 𝒜/[𝒳] with 𝒯(𝒳, 𝒳[1]) = 0 becomes a triangulated category if and only if (𝒜, 𝒜) forms an 𝒳-mutation pair and 𝒜 is closed under extensions.     

Local conformal nets arising from framed vertex operator algebras

We apply an idea of framed vertex operator algebras to a construction of local conformal nets of (injective type III₁) factors on the circle corresponding to various lattice vertex operator algebras and their twisted orbifolds. In particular, we give a local conformal net corresponding to the moonshine vertex operator algebras of Frenkel-Lepowsky-Meurman. Its central charge is 24, it has a trivial representation theory in the sense that the vacuum sector is the only irreducible DHR sector, its vacuum character is the modular invariant J-function and its automorphism group (the gauge group) is the Monster group. We use our previous tools such as α-induction and complete rationality to study extensions of local conformal nets.     

Representations of group planar algebras

We explicitly find out the irreducible representations of the planar algebra corresponding to the subfactor arising from the action of a finite group. We also answer the question posed by Vaughan Jones on the radius of convergence of the dimension of a representation in the affirmative for the case of group planar algebras.