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D.E.: Modular Invariants, Graphs and αInduction for Nets of Subfactors II
 In preparation
"... We analyze the induction and restriction of sectors for nets of subfactors defined by Longo and Rehren. Picking a local subfactor we derive a formula which specifies the structure of the induced sectors in terms of the original DHR sectors of the smaller net and canonical endomorphisms. We also obta ..."
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Cited by 80 (8 self)
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We analyze the induction and restriction of sectors for nets of subfactors defined by Longo and Rehren. Picking a local subfactor we derive a formula which specifies the structure of the induced sectors in terms of the original DHR sectors of the smaller net and canonical endomorphisms. We also obtain a reciprocity formula for induction and restriction of sectors, and we prove a certain homomorphism property of the induction mapping. Developing further some ideas of F. Xu we will apply this theory in a forthcoming paper to nets of subfactors arising from conformal field theory, in particular those coming from conformal embeddings or orbifold inclusions of SU(n) WZW models. This will provide a better understanding of the labeling of modular invariants by certain graphs, in particular of the ADE classification of SU(2) modular invariants.
Multiinterval subfactors and modularity of representations in conformal field theory
 Commun. Math. Phys
"... Dedicated to John E. Roberts on the occasion of his sixtieth birthday We describe the structure of the inclusions of factors A(E) ⊂A(E ′ ) ′ associated with multiintervals E ⊂ R for a local irreducible net A of von Neumann algebras on the real line satisfying the split property and Haag duality. I ..."
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Cited by 63 (26 self)
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Dedicated to John E. Roberts on the occasion of his sixtieth birthday We describe the structure of the inclusions of factors A(E) ⊂A(E ′ ) ′ associated with multiintervals E ⊂ R for a local irreducible net A of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the LongoRehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of A. As a consequence, the index of A(E) ⊂A(E ′ ) ′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is nondegenerate, namely the representations of A form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry.
On αinduction, chiral generators and modular invariants for subfactors
 Commun. Math. Phys
, 1999
"... We consider a type III subfactor N ⊂ M of finite index with a finite system of braided NN morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply αinduction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle ..."
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Cited by 38 (10 self)
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We consider a type III subfactor N ⊂ M of finite index with a finite system of braided NN morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply αinduction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle algebra. Using a new concept of intertwining braiding fusion relations, we show that the chiral generators can be naturally identified with the αinduced sectors. A matrix Z is defined and shown to commute with the S and Tmatrices arising from the braiding. If the braiding is nondegenerate, then Z is a “modular invariant mass matrix ” in the usual sense of conformal field theory. We show that in that case the fusion rule algebra of the dual system of MM morphisms is generated by the images of both kinds of αinduction, and that the structural information about its irreducible representations is encoded in the mass matrix Z. Our analysis sheds further light on the connection between (the classifications of) modular invariants and subfactors, and we will
The Structure of Sectors Associated with the LongoRehren Inclusions I. General Theory
 Commun. Math. Phys
, 1999
"... We investigate the structure of the LongoRehren inclusion for a finite closed system of endomorphisms of factors, whose categorical structure is known to be the same as the asymptotic inclusion of A. Ocneanu. In particular, we obtain a precise description of the sectors associated with the LongoRe ..."
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Cited by 37 (0 self)
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We investigate the structure of the LongoRehren inclusion for a finite closed system of endomorphisms of factors, whose categorical structure is known to be the same as the asymptotic inclusion of A. Ocneanu. In particular, we obtain a precise description of the sectors associated with the LongoRehren inclusions in terms of half braidings, which do not necessarily satisfy the usual condition of braidings. In doing so, we give new proofs to most of the known statements concerning asymptotic inclusions. We construct a complete system of matrix units of the tube algebra using the half braidings, which will be used in the second part to describe concrete examples of the LongoRehren inclusions arising from the Cuntz algebra endomorphisms. We also discuss the case where the original system has a braiding, and generalize Ocneanu and EvansKawahigashi's method for the analysis of the asymptotic inclusions of the Hecke algebra subfactors. 1 Introduction The notion of the asymptotic inclusio...
On flatness of Ocneanu’s connections on the Dynkin diagrams and classification of subfactors
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Galois theory for braided tensor categories and the modular closure
 Adv. Math
, 2000
"... Given a braided tensor ∗category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a tensor ∗category with conjugates and an irreducible unit. (A ∗category is a category enriched over VectC ..."
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Cited by 29 (6 self)
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Given a braided tensor ∗category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a tensor ∗category with conjugates and an irreducible unit. (A ∗category is a category enriched over VectC with positive ∗operation.) A Galois correspondence is established between intermediate categories sitting between C and C ⋊S and closed subgroups of the Galois group Gal(C⋊S/C) = AutC(C⋊S) of C, the latter being isomorphic to the compact group associated to S by the duality theorem of Doplicher and Roberts. Denoting by D ⊂ C the full subcategory of degenerate objects, i.e. objects which have trivial monodromy with all objects of C, the braiding of C extends to a braiding of C⋊S iff S ⊂ D. Under this condition C⋊S has no nontrivial degenerate objects iff S = D. If the original category C is rational (i.e. has only finitely many isomorphism classes of irreducible objects) then the same holds for the new one. The category C ≡ C ⋊ D is called the modular closure of C since in the rational case it is modular, i.e. gives rise to a unitary representation of the modular group SL(2, Z). (In passing we prove that every braided tensor ∗category with conjugates automatically is a ribbon category, i.e. has a twist.) If all simple objects of S have dimension one the structure of the category C ⋊ S can be clarified quite explicitly in terms of group cohomology. 1
Exotic subfactors of finite depth with Jones indices (5 + √13)/2 and (5 + √17)/2
, 1998
"... We prove existence of subfactors of finite depth of the hyperfinite II1 factor with indices (5 + √ 13)/2 = 4.302 · · · and (5 + √ 17)/2 = 4.561 · · ·. The existence of the former was announced by the second named author in 1993 and that of the latter has been conjectured since then. These are the ..."
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Cited by 20 (2 self)
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We prove existence of subfactors of finite depth of the hyperfinite II1 factor with indices (5 + √ 13)/2 = 4.302 · · · and (5 + √ 17)/2 = 4.561 · · ·. The existence of the former was announced by the second named author in 1993 and that of the latter has been conjectured since then. These are the only known subfactors with finite depth which do not arise from classical groups, quantum groups or rational conformal field theory.
Centrally trivial automorphisms and an analogue of Connes’ χ(M) for subfactors
 Duke Math. J
, 1993
"... Abstract. We study a class of centrally trivial automorphisms for subfactors, and get an upper bound for the order of the group they make (modulo normalizers) in terms of the “dual ” principal graph for AFD type II1 subfactors with trivial relative commutant, finite index and finite depth. We prove ..."
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Cited by 17 (8 self)
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Abstract. We study a class of centrally trivial automorphisms for subfactors, and get an upper bound for the order of the group they make (modulo normalizers) in terms of the “dual ” principal graph for AFD type II1 subfactors with trivial relative commutant, finite index and finite depth. We prove that this upper bound is attained for many known subfators. We also introduce χ(M,N) for subfactors N ⊂ M as the relative version of Connes ’ invariant χ(M), and compute this group for many AFD type II1 subfactors with finite index and finite depth including all the cases with index less than 4 and many Hecke algebra subfactors of Wenzl. In these finite depth cases, the group χ(M,N) is always finite and abelian, and we realize all the finite abelian groups as χ(M,N). Analogy between this topic and modular structure of type III factors is also discussed. As an application, we give some classification results for Aut(M,N). For example, for the subfactors of type A2n+1, there are two and only two outer actions of Z2. One is of the “standard” form and the other is given by the “orbifold ” action arising from the paragroup symmetry. As preliminaries, we also prove several statements on central sequence subfactors announced by A. Ocneanu.