The classification of subfactors with index at most $5 \frac{1}{4}$

Narjess, Afzaly,; Morrison, Scott; Penneys, David

doi:10.48550/arxiv.1509.00038

Cited by 11 publications

(22 citation statements)

References 49 publications

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“…The standard invariant has been axiomatized in many equivalent ways; in finite depth, we have Ocneanu's paragroups [Ocn88] and Popa's canonical commuting squares [Pop90]. Popa's λ-lattices [Pop95a] give an axiomatization of the standard invariant in the general case, and Jones' equivalent planar algebras [Jon99] provide a powerful graphical calculus for the construction [BMPS12] and classification [JMS14,AMP15] of standard invariants. Using Longo's Q-systems [Lon89], the article [Müg03] re-interprets the standard invariant as a unitary Frobenius algebra object in a rigid C*-tensor category [LR97,Yam04].…”

Section: Introductionmentioning

confidence: 99%

Realizations of algebra objects and discrete subfactors

Jones¹,

Penneys²

2019

Advances in Mathematics

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We give a characterization of extremal irreducible discrete subfactors (N ⊆ M, E) where N is type II 1 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. Corollary B. The realization |A| H of the connected W*-algebra object A coming from a discrete quantum group G = (C, F) together with a fully faithful representation H : C → Bim sp bf (N ) is type III if and only if G is not Kac-type. (See Section 6.3 for more details.) This corollary is related to [Vae05, Thm. 3.5] in the case of compact quantum groups. De Commer and Yamashita's classification of T LJ (δ)-module W*-categories [DCY15] gives many more such examples (see Section 6.4 for more details). BackgroundWe refer the reader to [HP17b, §2.1-2.2] and [JP17a, §2.1-2.3] for background on rigid C*tensor categories. Of particular importance is their bi-involutive structure, consisting of two commuting involutions (the adjoint * and conjugate · ) together with coherence data. As usual, we will suppress all associators, unitors, and the involutive structure morphisms.We fix a rigid C*-tensor category C. As in [JP17a, §2.4], Vec(C) denotes the involutive tensor category of linear functors C op → Vec with linear transformations. Note that Vec(C) = ind(C ), where C denotes the involutive tensor category obtained from C by forgetting the adjoint. As in [JP17a, §2.6], Hilb(C) denotes the bi-involutive tensor W*-category of linear dagger functors C op → Hilb with bounded natural transformations. Note that Hilb(C) is the Neshveyev-Yamashita unitary ind-category of C from [NY16]. 6Proposition 5.42. The map κ gives a natural isomorphism id ⇒ | · | H .

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Section: Introductionmentioning

confidence: 99%

Realizations of algebra objects and discrete subfactors

Jones¹,

Penneys²

2019

Advances in Mathematics

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“…The Haagerup subfactor is the first subfactor that is not directly related to either an ordinary group or a quantum group, and whether its Drinfeld center is related to a quantum group (conformal field theory) or not is an interesting open problem. At the time of writing, the classification of finite depth subfactors is completed up to index 5+1/4 (see [32], [29], [1], and references therein), and it turns out that three subfactors actually exist among Haagerup's list, namely the Haagerup subfactor, Asaeda-Haagerup subfactors constructed in [2], and the extended Haagerup subfactor constructed in [3]. The original construction of the Haagerup subfactor in [2] used computation of connections, a special type of 6j-symbols.…”

Section: Introductionmentioning

confidence: 99%

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

Izumi

2018

Quantum Topol.

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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 + √ 5. It also provides a key step for a new construction of the Asaeda-Haagerup subfactor due to Grossman, Snyder, and the author.

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“…He established a notion of index for subfactors, which is an invariant (hence opened the way to classification questions) and surprisingly quantized for values between 1 and 4 (Jones' rigidity theorem). Since then, the major efforts have been devoted to the study of finite index (finite depth) subfactors and a complete classification has been achieved for subfactors with index at most 5 + 1 4 [JMS14], [AMP15], using techniques of [Pop95a] and [Jon99]. At the same time, the analyses of QFT extensions [LR95] and of theories with defects and boundaries [BKLR16] cover the finite index case only, both being based on the notion of Q-system (which is tightly connected to the existence of conjugate morphisms ῑ of the inclusion morphism ι : N ֒→ M for a subfactor N ⊂ M, hence to the finiteness of the dimension of ι in the sense of [LR97]).…”

Section: Introductionmentioning

confidence: 99%

Infinite index extensions of local nets and defects

Vecchio

Giorgetti

2018

Rev. Math. Phys.

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Subfactor theory provides a tool to analyze and construct extensions of Quantum Field Theories, once the latter are formulated as local nets of von Neumann algebras. We generalize some of the results of [LR95] to the case of extensions with infinite Jones index. This case naturally arises in physics, the canonical examples are given by global gauge theories with respect to a compact (non-finite) group of internal symmetries. Building on the works of Izumi, Longo, Popa [ILP98] and Fidaleo, Isola [FI99], we consider generalized Q-systems (of intertwiners) for a semidiscrete inclusion of properly infinite von Neumann algebras, which generalize ordinary Q-systems introduced by Longo [Lon94] to the infinite index case. We characterize inclusions which admit generalized Q-systems of intertwiners and define a braided product among the latter, hence we construct examples of QFTs with defects (phase boundaries) of infinite index, extending the family of boundaries in the grasp of [BKLR16].Comment: 50 page

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The classification of subfactors with index at most $5 \frac{1}{4}$

Cited by 11 publications

References 49 publications

Realizations of algebra objects and discrete subfactors

Realizations of algebra objects and discrete subfactors

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

Infinite index extensions of local nets and defects

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