On Flatness of Ocneanu′s Connections on the Dynkin Diagrams and Classification of Subfactors

Kawahigashi, Yasuyuki

doi:10.1006/jfan.1995.1003

Cited by 83 publications

(86 citation statements)

References 16 publications

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“…This deep theorem solves the original problem of Loi of classification of strongly amenable subfactors of type III λ completely. Subfactors, however, often have interesting automorphisms with trivial Loi invariant, as shown in our previous work [11], [12], [21], [22]. Thus we expect an interesting classification theory for automorphisms of subfactors which are not classified by Loi's invariant.…”

Section: Introductionmentioning

confidence: 57%

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Classification of approximately inner automorphisms of subfactors

Kawahigashi¹

1997

Mathematische Annalen

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For classification of approximately inner automorphisms of subfactors, we introduce a new invariant, a higher obstruction. From an algebraic viewpoint, this can be regarded as a generalization of the Connes obstruction, and from an analytic viewpoint, this can be regarded as a generalization of the Jones invariant κ. We have two classification theorems for approximately inner automorphisms of strongly amenable subfactors with known invariants and this new one. In particular, our theorems give a complete classification of automorphisms, up to outer conjugacy, of AFD subfactors of type II 1 with index less than four except for one special case for A 4n−1 and E 6 .

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Section: Introductionmentioning

confidence: 57%

“…We, however, have something different for subfactors from the case of injective type III factors, that is, the obstruction to flatness in the orbifold construction may not vanish -as in [21], [39]. This causes combinatorial difficulty and gives a reason our classification here is not complete in all the cases.…”

Section: Example 63mentioning

confidence: 97%

Classification of approximately inner automorphisms of subfactors

Kawahigashi¹

1997

Mathematische Annalen

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“…This means that in order to get an automorphism ϕ 11 of the commuting square (2.1) the following must be satisfied; see for instance [6], [10]. Definition 2.10.…”

Section: Construction Of Subfactor Automorphismsmentioning

confidence: 99%

Automorphisms of subfactors from commuting squares

Svendsen

2004

Trans. Amer. Math. Soc.

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Abstract. We study an infinite series of irreducible, hyperfinite subfactors, which are obtained from an initial commuting square by iterating Jones' basic construction. They were constructed by Haagerup and Schou and have A∞ as principal graphs, which means that their standard invariant is "trivial". We use certain symmetries of the initial commuting squares to construct explicitly non-trivial outer automorphisms of these subfactors. These automorphisms capture information about the subfactors which is not contained in the standard invariant.

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“…The orbifold construction has originally arisen from the technique in conformal field theory and in solvable lattice model theory. This technique was first used in subfactor theory by Y. Kawahigashi [Ka2]. At first he used it to show the existence of subfactors of the AFD factor of type II 1 with principal graph D 2n and non-existence of those with principal graph D 2n+1 .…”

Section: Proposition 25 Let G Be a Finite Group And α An Outer Actmentioning

confidence: 99%

Commutativity of automorphisms of subfactors modulo inner automorphisms

Goto

1996

Proc. Amer. Math. Soc.

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Abstract. We introduce a new algebraic invariant χa(M, N) of a subfactor N ⊂ M . We show that this is an abelian group and that if the subfactor is strongly amenable, then the group coincides with the relative Connes invariant χ(M, N) introduced by Y. Kawahigashi. We also show that this group is contained in the center of Out(M, N) in many interesting examples such as quantum SU(n) k subfactors with level k (k ≥ n + 1), but not always contained in the center. We also discuss its relation to the most general setting of the orbifold construction for subfactors. §0. Introduction . These facts show that we can generalize this relative Connes invariant to properly infinite subfactors. In this paper we introduce a new algebraic invariant of a subfactor N ⊂ M . It is the intersection of the sets of non-strongly outer automorphisms and automorphisms with trivial Loi invariant modulo inner automorphisms arising from the normalizers. We define Recently Y. Kawahigashi introduced the relative Connes invariantwhere we denote the set of non-strongly outer automorphisms on the subfactor by Ψ (M, N ), the map assigning Loi's invariant by Φ and the set of normalizers of N in M , i.e., {u ∈ U(M)| uN u * = N } by N (M, N ). Note that we define

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On Flatness of Ocneanu′s Connections on the Dynkin Diagrams and Classification of Subfactors

Cited by 83 publications

References 16 publications

Classification of approximately inner automorphisms of subfactors

Classification of approximately inner automorphisms of subfactors

Automorphisms of subfactors from commuting squares

Commutativity of automorphisms of subfactors modulo inner automorphisms

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