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109
Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index
"... this paper a noncommutative probability approach (in the sense considered by D. Voiculescu in [28]) to the algebras that are associated to certain amalgamated free products. In this way we find that the type II 1 ..."
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Cited by 55 (5 self)
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this paper a noncommutative probability approach (in the sense considered by D. Voiculescu in [28]) to the algebras that are associated to certain amalgamated free products. In this way we find that the type II 1
From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories
 J. Pure Appl. Alg
, 2003
"... We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground f ..."
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Cited by 52 (6 self)
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We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = FVect, where F is a field. An object X ∈ A with twosided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1
On qanalog of McKay correspondence and ADE classification of sl (2) conformal field theories
 Adv. Math
"... Abstract. The goal of this paper is to classify “finite subgroups in Uq(sl2)” where q = e πi/l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of Uq(sl2); we show that this definition is a natural generalization of the notion of a subgroup i ..."
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Cited by 46 (4 self)
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Abstract. The goal of this paper is to classify “finite subgroups in Uq(sl2)” where q = e πi/l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of Uq(sl2); we show that this definition is a natural generalization of the notion of a subgroup in a reductive group, and that it is also related with extensions of the chiral (vertex operator) algebra corresponding to ̂ sl2 at level k = l − 2. We show that “finite subgroups in Uq(sl2) ” are classified by Dynkin diagrams of types An, D2n, E6, E8 with Coxeter number equal to l, give a description of this correspondence similar to the classical McKay correspondence, and discuss relation with modular invariants in ( ̂ sl2)k conformal field theory.
The many faces of ocneanu cells
 Nuclear Phys. B
"... We define generalised chiral vertex operators covariant under the Ocneanu “double triangle algebra” A, a novel quantum symmetry intrinsic to a given rational 2d conformal field theory. This provides a chiral approach, which, unlike the conventional one, makes explicit various algebraic structures e ..."
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Cited by 44 (4 self)
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We define generalised chiral vertex operators covariant under the Ocneanu “double triangle algebra” A, a novel quantum symmetry intrinsic to a given rational 2d conformal field theory. This provides a chiral approach, which, unlike the conventional one, makes explicit various algebraic structures encountered previously in the study of these theories and of the associated critical lattice models, and thus allows their unified treatment. The triangular Ocneanu cells, the 3jsymbols of the weak Hopf algebra A, reappear in several guises. With A and its dual algebra Â one associates a pair of graphs, G and ˜G. While G are known to encode complete sets of conformal boundary states, the Ocneanu graphs ˜G classify twisted torus partition functions. The fusion algebra of the twist operators provides the data determining Â. The study of bulk field correlators in the presence of twists reveals that the Ocneanu graph quantum symmetry gives also an information on the field operator algebra.
Orbifold subfactors from Hecke algebras
 Comm. Math. Phys
, 1994
"... A. Ocneanu has observed that a mysterious orbifold phenomenon occurs in the system of the M∞M ∞ bimodules of the asymptotic inclusion, a subfactor analogue of the quantum double, of the Jones subfactor of type A2n+1. We show that this is a general phenomenon and identify some of his orbifolds with ..."
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Cited by 39 (23 self)
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A. Ocneanu has observed that a mysterious orbifold phenomenon occurs in the system of the M∞M ∞ bimodules of the asymptotic inclusion, a subfactor analogue of the quantum double, of the Jones subfactor of type A2n+1. We show that this is a general phenomenon and identify some of his orbifolds with the ones in our sense as subfactors given as simultaneous fixed point algebras by working on the Hecke algebra subfactors of type A of Wenzl. That is, we work on their asymptotic inclusions and show that the M∞M ∞ bimodules are described by certain orbifolds (with ghosts) for SU(3)3k. We actually compute several examples of the (dual) principal graphs of the asymptotic inclusions. As a corollary of the identification of Ocneanu’s orbifolds with ours, we show that a nondegenerate braiding exists on the even vertices of D2n, n>2. 1
Quantum geometry of algebra factorisations and coalgebra bundles
 Commun. Math. Phys
, 2000
"... We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices M2(C) = CZ2 · CZ2. We also further extend the coalgebra version of theory introduced previously, to include frame bundles and el ..."
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Cited by 34 (15 self)
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We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices M2(C) = CZ2 · CZ2. We also further extend the coalgebra version of theory introduced previously, to include frame bundles and elements of Riemannian geometry. As an example, we construct qmonopoles on all the Podle´s quantum spheres S 2 q,s. 1.
On flatness of Ocneanu’s connections on the Dynkin diagrams and classification of subfactors
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Classification of local conformal nets. Case c < 1
"... We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of AD2nE6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We f ..."
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Cited by 27 (13 self)
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We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of AD2nE6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We first identify the nets generated by irreducible representations of the Virasoro algebra for c<1 with certain coset nets. Then, by using the classification of modular invariants for the minimal models by CappelliItzyksonZuber and the method of αinduction in subfactor theory, we classify all local irreducible extensions of the Virasoro nets for c<1 and infer our main classification result. As an application, we identify in our classification list certain concrete coset nets studied in the literature.
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.
SINGLY GENERATED PLANAR ALGEBRAS OF SMALL DIMENSION
 VOL. 101, NO. 1 DUKE MATHEMATICAL JOURNAL
, 2000
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