Classification of Paragroup Actions on Subfactors

Kawahigashi, Yasuyuki

doi:10.2977/prims/1195164051

Cited by 36 publications

(30 citation statements)

References 43 publications

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“…It is straightforward to check that the M -M sectors, labelling the even vertices in Figure 4, obey in fact the fusion rules determined by Kawahigashi [25] as the correct fusion table of the five possibilities given in [2]. Another result of [25], namely that this fusion al-…”

Section: Conformal Embeddings Of Su (2) Revisitedmentioning

confidence: 99%

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

Böckenhauer¹,

Evans²

1999

Commun. Math. Phys.

113

321

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In this paper we further develop the theory of α-induction for nets of subfactors, in particular in view of the system of sectors obtained by mixing the two kinds of induction arising from the two choices of braiding. We construct a relative braiding between the irreducible subsectors of the two "chiral" induced systems, providing a proper braiding on their intersection. We also express the principal and dual principal graphs of the local subfactors in terms of the induced sector systems. This extended theory is again applied to conformal or orbifold embeddings of SU (n) WZW models. A simple formula for the corresponding modular invariant matrix is established in terms of the two inductions, and we show that it holds if and only if the sets of irreducible subsectors of the two chiral induced systems intersect minimally on the set of marked vertices, i.e. on the "physical spectrum" of the embedding theory, or if and only if the canonical endomorphism sector of the conformal or orbifold inclusion subfactor is in the full induced system. We can prove either condition for all simple current extensions of SU (n) and many conformal inclusions, covering in particular all type I modular invariants of SU (2) and SU (3), and we conjecture that it holds also for any other conformal inclusion of SU (n) as well. As a by-product of our calculations, the dual principal graph for the conformal inclusion SU (3) 5 ⊂ SU (6) 1 is computed for the first time.

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Section: Conformal Embeddings Of Su (2) Revisitedmentioning

confidence: 99%

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

Böckenhauer¹,

Evans²

1999

Commun. Math. Phys.

113

321

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“…§3 0 Main Results For the definition and the basic properties of strongly outer and strongly free automorphisms, we refer the reader to [1,20,10,23]. Our first lemma is a direct corollary of the results in [1,8,19]. Let /J be another strongly outer action of Z m on B CIA such that J3 st has period m and that a st and @ st are conjugate.…”

Section: §1 Introductionmentioning

confidence: 82%

“…Let A/"CM be a finite depth inclusion of AFD type Ilk factors AfCM, X ^ (0, 1), with a determines the conjugacy class of a. Thi idea of using the commuting square of the simultaneous crossed product algebras to classify finite group actions on inclusions of factors was suggested in [8] .…”

Section: §1 Introductionmentioning

confidence: 99%

Commuting Squares and the Classification of Finite Depth Inclusions of AFD Type $\mathrm{III}_λ$ Factors, $λ \in (0, 1)$

Loi¹

1998

Publ. Res. Inst. Math. Sci.

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We give a new proof of the classification result due to Sorm Popa that a finite depth inclusion of AFD type III; factois N^Al, A^ (0, 1), with a common discrete decomposition (jV^CM 00 , 0} is classified, up to isomorphism, by the type II core A^ciM 00 and the standard invariant of 6. §1. IntroductionLet X& (0, 1) and N^M be an inclusion of AFD type Ilk factors with finite index and with a common discrete decomposition (A 700 c M°°, 6} . It is an interesting problem to classify up to isomorphism inclusions of this kind with the same index.In [20] Popa has shown that if N C M is strongly amenable then such an inclusion is classified by the isomorphism class of N^^-M 00 and the standard invariant 6 st of 6. This result was obtained as a consequence of a powerful classification theorem that properly/strongly outer actions of countable discrete amenable groups on strongly amenable inclusions of type II factors are classified by their standard invariants together with their modules if the factors are of type IIoo. By [18,19], finite depth inclusions of AFD type II factors are strongly amenable.In this note, we will give a different proof of the finite depth case of this classification result of Popa by proving directly that the isomorphism class of N M is classified by that of N°°^M°° and the conjugacy class of 8 st .

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“…Kawahigashi [Kaw95] proved that the dual even part of the GHJ subfactor (i.e. (A even 11 ) * E even 6 ) has fusion ring R .…”

Section: Uniqueness Of the Ghj Subfactormentioning

confidence: 99%

Subfactors of Index Less Than 5, Part 3: Quadruple Points

Izumi

Jones

Morrison

et al. 2012

Commun. Math. Phys.

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Abstract. One major obstacle in extending the classification of small index subfactors beyond 3 + √ 3 is the appearance of infinite families of candidate principal graphs with 4-valent vertices (in particular, the "weeds" Q and Q from Part 1 [MS10]). Thus instead of using triple point obstructions to eliminate candidate graphs, we need to develop new quadruple point obstructions. In this paper we prove two quadruple point obstructions. The first uses quadratic tangles techniques and eliminates the weed Q immediately. The second uses connections, and when combined with an additional number theoretic argument it eliminates both weeds Q and Q . Finally, we prove the uniqueness (up to taking duals) of the 3311 Goodman-de la Harpe-Jones subfactor using a combination of planar algebra techniques and connections.

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Classification of Paragroup Actions on Subfactors

Cited by 36 publications

References 43 publications

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

Commuting Squares and the Classification of Finite Depth Inclusions of AFD Type $\mathrm{III}_λ$ Factors, $λ \in (0, 1)$

Subfactors of Index Less Than 5, Part 3: Quadruple Points

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