Intermediate Subfactors Research Articles

Inclusions of simple C⁎-algebras arising from compact group actions

Inclusions of operator algebras have long been studied. In particular, inclusions arising from actions of compact groups on factors were studied by Izumi-Longo-Popa and others. The correspondence between intermediate subfactors and subgroups is called the Galois correspondence. Analogues for actions on C⁎-algebras have been studied by Izumi, Cameron-Smith, Peligrad, and others. In this article, we give examples of compact group actions on simple C⁎-algebras for which the Galois correspondence holds.

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?

Quantum operations on conformal nets

On a conformal net [Formula: see text], one can consider collections of unital completely positive maps on each local algebra [Formula: see text], subject to natural compatibility, vacuum preserving and conformal covariance conditions. We call quantum operations on [Formula: see text] the subset of extreme such maps. The usual automorphisms of [Formula: see text] (the vacuum preserving invertible unital *-algebra morphisms) are examples of quantum operations, and we show that the fixed point subnet of [Formula: see text] under all quantum operations is the Virasoro net generated by the stress-energy tensor of [Formula: see text]. Furthermore, we show that every irreducible conformal subnet [Formula: see text] is the fixed points under a subset of quantum operations. When [Formula: see text] is discrete (or with finite Jones index), we show that the set of quantum operations on [Formula: see text] that leave [Formula: see text] elementwise fixed has naturally the structure of a compact (or finite) hypergroup, thus extending some results of [M. Bischoff, Generalized orbifold construction for conformal nets, Rev. Math. Phys. 29 (2017) 1750002]. Under the same assumptions, we provide a Galois correspondence between intermediate conformal nets and closed subhypergroups. In particular, we show that intermediate conformal nets are in one-to-one correspondence with intermediate subfactors, extending a result of Longo in the finite index/completely rational conformal net setting [R. Longo, Conformal subnets and intermediate subfactors, Comm. Math. Phys. 237 (2003) 7–30].

A short note on relative entropy for a pair of intermediate subfactors

Given a quadruple of finite index subfactors we explicitly compute the Pimsner-Popa probabilistic constant for the pair of intermediate subfactors and relate it with the corresponding Connes-Størmer relative entropy between them. This generalizes an old result of Pimsner and Popa [Ann. Sci. École Norm. Sup. (4) 19 (1986), pp. 57–106].

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.

Lattice of intermediate subalgebras

Analogous to subfactor theory, employing Watatani's notions of index and C ∗ -basic construction of certain inclusions of C ∗ -algebras, (a) we develop a Fourier theory (consisting of Fourier transforms, rotation maps and shift operators) on the relative commutants of any inclusion of simple unital C ∗ -algebras with finite Watatani index, and (b) we introduce the notions of interior and exterior angles between intermediate C ∗ -subalgebras of any inclusion of unital C ∗ -algebras admitting a finite index conditional expectation. Then, on the lines of Bakshi et al. (Trans. Amer. Math. Soc. 371 (2019) 5973–5991), we apply these concepts to obtain a bound for the cardinality of the lattice of intermediate C ∗ -subalgebras of any irreducible inclusion as in (a), and improve Longo's bound for the cardinality of intermediate subfactors of an irreducible inclusion of type I I I factors with finite index. Moreover, we also show that for a fairly large class of inclusions of finite von Neumann algebras, the lattice of intermediate von Neumann subalgebras is always finite.

On a composition of subfactors with group subfactors

Let M be a type II1 factor, G be a finite group, and N ⊆ M be an irreducible subfactor of finite index. We prove that the composed lattice of the intermediate subfactor lattice for the inclusion N ⊆ M and the subgroup lattice of G can be realized as an intermediate subfactor lattice of a certain composed subfactor of finite index, and this subfactor also has finite depth when N ⊆ M has finite depth.

A note on irreducible quadrilaterals of II1 factors

Given any quadruple [Formula: see text] of [Formula: see text]-factors with finite index, the notions of interior and exterior angles between [Formula: see text] and [Formula: see text] were introduced in [An angle between intermediate subfactors and its rigidity, Trans. Amer. Math. Soc. 371(8) (2019) 5973–5991]. We determine the possible values of these angles in terms of the cardinalities of the Weyl groups of the intermediate subfactors when [Formula: see text] is an irreducible quadrilateral and the subfactors [Formula: see text] and [Formula: see text] are both regular. For an arbitrary irreducible quadruple, an attempt is made to determine the values of angles by deriving expressions for the angles in terms of the common norm of two naturally arising auxiliary operators and the indices of the intermediate subfactors of the quadruple. Finally, certain bounds on angles between [Formula: see text] and [Formula: see text] are obtained when [Formula: see text] is regular, which enforce some restrictions on the index of [Formula: see text] in terms of that of [Formula: see text].

Realizations of algebra objects and discrete subfactors

We give a characterization of extremal irreducible discrete subfactors (N⊆M,E) where N is type II1 in terms of connected W*-algebra objects in rigid C*-tensor categories. We prove an equivalence of categories where the morphisms for discrete inclusions are normal N−N bilinear ucp maps which preserve the state τ∘E, and the morphisms for W*-algebra objects are categorical ucp morphisms.As an application, we get a well-behaved notion of the standard invariant of an extremal irreducible discrete subfactor, together with a subfactor reconstruction theorem. Thus our equivalence provides many new examples of discrete inclusions (N⊆M,E), in particular, examples where M is type III coming from non Kac-type discrete quantum groups and associated module W*-categories. Finally, we obtain a Galois correspondence between intermediate subfactors of an extremal irreducible discrete inclusion and intermediate W*-algebra objects.

An angle between intermediate subfactors and its rigidity

We introduce a new notion of an angle between intermediate subfactors and prove various interesting properties of the angle and relate it to the Jones index. We prove a uniform 60 60 to 90 90 degree bound for the angle between minimal intermediate subfactors of a finite index irreducible subfactor. From this rigidity we can bound the number of minimal (or maximal) intermediate subfactors by the kissing number in geometry. As a consequence, the number of intermediate subfactors of an irreducible subfactor has at most exponential growth with respect to the Jones index. This answers a question of Longo’s published in 2003.

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.

The classification of $3^n$ subfactors and related fusion categories

We investigate a (potentially infinite) series of subfactors, called 3^n subfactors , including A_4 , A_7 , and the Haagerup subfactor as the first three members corresponding to n=1,2,3 . Generalizing our previous work for odd n , we further develop a Cuntz algebra method to construct 3^n subfactors and show that the classification of the 3^n subfactors and related fusion categories is reduced to explicit polynomial equations under a mild assumption, which automatically holds for odd n . In particular, our method with n=4 gives a uniform construction of 4 finite depth subfactors, up to dual, without intermediate subfactors of index 3+\sqrt{5} . It also provides a key step for a new construction of the Asaeda–Haagerup subfactor due to Grossman, Snyder, and the author.

Examples of Subfactors from Conformal Field Theory

Conformal field theory (CFT) in two dimensions provide a rich source of subfactors. The fact that there are so many subfactors coming from CFT have led people to conjecture that perhaps all finite depth subfactors are related to CFT. In this paper we examine classes of subfactors from known CFT. In particular we identify the so called $3^{\Z_2\times \Z_2}$ subfactor with an intermediate subfactor from conformal inclusion, and construct new subfactors from recent work on holomorphic CFT with central charge $24$.

Generalized orbifold construction for conformal nets

Let [Formula: see text] be a conformal net. We give the notion of a proper action of a finite hypergroup [Formula: see text] acting by vacuum preserving unital completely positive (so-called stochastic) maps on [Formula: see text] which generalizes the proper action of a finite group [Formula: see text]. Taking the fixed point under such an action gives a finite index subnet [Formula: see text] of [Formula: see text], which generalizes the [Formula: see text]-orbifold net. Conversely, we show that if [Formula: see text] is a finite inclusion of conformal nets, then [Formula: see text] is a generalized orbifold [Formula: see text] of the conformal net [Formula: see text] by a unique finite hypergroup [Formula: see text]. There is a Galois correspondence between intermediate nets [Formula: see text] and subhypergroups [Formula: see text] given by [Formula: see text]. In this case, the fixed point of [Formula: see text] is the generalized orbifold by the hypergroup of double cosets [Formula: see text].If [Formula: see text] is a finite index inclusion of completely rational nets, we show that the inclusion [Formula: see text] is conjugate to an intermediate subfactor of a Longo–Rehren inclusion. This implies that if [Formula: see text] is a holomorphic net, and [Formula: see text] acts properly on [Formula: see text], then there is a unitary fusion category [Formula: see text] which is a categorification of [Formula: see text] and [Formula: see text] is braided equivalent to the Drinfel’d center [Formula: see text]. More generally, if [Formula: see text] is a completely rational conformal net and [Formula: see text] acts properly on [Formula: see text], then there is a unitary fusion category [Formula: see text] extending [Formula: see text], such that [Formula: see text] is given by the double cosets of the fusion ring of [Formula: see text] by the Verlinde fusion ring of [Formula: see text] and [Formula: see text] is braided equivalent to the Müger centralizer of [Formula: see text] in the Drinfel’d center [Formula: see text].

The Asaeda–Haagerup fusion categories

Abstract The classification of subfactors of small index revealed several new subfactors. The first subfactor above index 4, the Haagerup subfactor, is increasingly well understood and appears to lie in a (discrete) infinite family of subfactors where the ℤ \mathbb{Z} /3 ℤ \mathbb{Z} symmetry is replaced by other finite abelian groups. The goal of this paper is to give a similarly good description of the Asaeda–Haagerup subfactor which emerged from our study of its Brauer–Picard groupoid. More specifically, we construct a new subfactor 𝒮 {\mathcal{S}} which is a ℤ \mathbb{Z} /4 ℤ \mathbb{Z} × \times ℤ \mathbb{Z} /2 ℤ \mathbb{Z} analogue of the Haagerup subfactor and we show that the even parts of the Asaeda–Haagerup subfactor are higher Morita equivalent to an orbifold quotient of 𝒮 {\mathcal{S}} . This gives a new construction of the Asaeda–Haagerup subfactor which is much more symmetric and easier to work with than the original construction. As a consequence, we can settle many open questions about the Asaeda–Haagerup subfactor: calculating its Drinfeld center, classifying all extensions of the Asaeda–Haagerup fusion categories, finding the full higher Morita equivalence class of the Asaeda–Haagerup fusion categories, and finding intermediate subfactor lattices for subfactors coming from the Asaeda–Haagerup categories. The details of the applications will be given in subsequent papers.

Symmetries of subfactors motivated by Aschbacher–Guralnick conjecture

Aschbacher–Guralnick conjecture states that the number of conjugacy classes of maximal subgroups of a finite group is bounded by the number of conjugacy classes of the group. Attempting to generalize this to the more general framework of subfactors, we are led to investigate the actions of two finite symmetry groups of a subfactor on the first non-trivial relative commutant of the subfactor which are compatible with their actions on the intermediate subfactors. Our first main result is that the actions of these two groups commute, and thus we can formulate a sensible subfactor generalization of Aschbacher–Guralnick conjecture which reduces to the original Aschbacher–Guralnick conjecture in the group subfactor case. In the case of group-subgroup subfactors, our conjecture can be stated in group theory terms as a relative version of Aschbacher–Guralnick conjecture, and we prove that this conjecture is true for solvable groups.

On maximal subfactors from quantum groups

Motivated by a conjecture about maximal intermediate subfactors, in this paper we show that subfactors from representations of universal quantum enveloping algebras with deformation parameter greater than zero are maximal. This gives continuous family of non-amenable subfactors which verify our conjecture. Using the same idea we also give a group theoretical description of our conjecture for subfactors related to representations of finite group.

The little desert? Some subfactors with index in the interval $(5, 3+\sqrt{5})$

Progress on classifying small index subfactors has revealed an almost empty landscape. In this paper we give some evidence that this desert continues up to index [Formula: see text]. There are two known quantum-group subfactors with index in this interval, and we show that these subfactors are the only way to realize the corresponding principal graphs. One of these subfactors is 1-supertransitive, and we demonstrate that it is the only 1-supertransitive subfactor with index between 5 and [Formula: see text]. Computer evidence shows that any other subfactor in this interval would need to have rank at least 38. We prove our uniqueness results by showing that there is a unique flat connection on each graph. The result on 1-supertransitive subfactors is proved by an argument using intermediate subfactors, running the "odometer" from the FusionAtlas' Mathematica package and paying careful attention to dimensions. This is the published version of arXiv:1205.2742.

On a problem about tensor products of subfactors

In this paper we make progress on the tensor product conjecture about minimal intermediate subfactors in tensor products of subfactors which are not of tensor product form. This conjecture is motivated by a subfactor generalization of Wall’s conjecture from the theory of finite groups. We reduce the tensor product conjecture to a conjecture about a class of what we call very simple Kac algebras. Such a reduction gives a proof of tensor product conjecture for group–subgroup subfactors. At present the only known such Kac algebras come from simple groups with possible twist, and we verify our conjecture in such cases.

On Examples of Intermediate Subfactors from Conformal Field Theory

Motivated by our subfactor generalization of Wall's conjecture, in this paper we determine all intermediate subfactors for conformal subnets corresponding to four infinite series of conformal inclusions, and as a consequence we verify that these series of subfactors verify our conjecture. Our results can be stated in the framework of Vertex Operator Algebras. We also verify our conjecture for Jones-Wassermann subfactors from representations of Loop groups extending our earlier results.