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13
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 54 (7 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
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 40 (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
Classification of paragroup actions on subfactors
 Publ. RIMS, Kyoto Univ
, 1995
"... We define “a crossed product by a paragroup action on a subfactor ” as a certain commuting square of type II1 factors and give their complete classification in a strongly amenable case (in the sense of S. Popa) in terms of a new combinatorial object which generalizes Ocneanu’s paragroup. In addition ..."
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Cited by 10 (7 self)
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We define “a crossed product by a paragroup action on a subfactor ” as a certain commuting square of type II1 factors and give their complete classification in a strongly amenable case (in the sense of S. Popa) in terms of a new combinatorial object which generalizes Ocneanu’s paragroup. In addition to the standard axioms of paragroups, we have the intertwining YangBaxter equation as the new additional axiom. We will show that Rational Conformal Field Theory in the sense of MooreSeiberg and orbifold construction in the sense of D. E. Evans, the author, and F. Xu produce paragroup actions on 1 subfactors in the canonical form. As applications, we show that the subfactor N ⊂ M of Goodmande la HarpeJones with index 3 + √ 3 is not conjugate to its dual M ⊂ M1 by showing the fusion algebras of NN bimodules and MM bimodules are different, although the principal graph and the dual principal graph are the same. This is the first example of such a subfactor. We also determine the topological quantum field theory of this subfactor. Finally, we make an analogue of the coset construction in RCFT for subfactors in our settings. 1
The E7 commuting squares produce D10 as principal graph
 Publ. RIMS. Kyoto Univ
, 1994
"... Abstract. We prove that the (two) connections, or commuting squares, on the CoxeterDynkin diagram E7 produce a subfactor with principal graph D10. This was conjectured by J.B. Zuber in connection with modular invariants in conformal field theory, and solve the last case of computing the flat parts ..."
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Cited by 8 (7 self)
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Abstract. We prove that the (two) connections, or commuting squares, on the CoxeterDynkin diagram E7 produce a subfactor with principal graph D10. This was conjectured by J.B. Zuber in connection with modular invariants in conformal field theory, and solve the last case of computing the flat parts of the connections on CoxeterDynkin diagrams with index less than 4. Since V. F. R. Jones initiated a systematic study of subfactors in [Jo], more and more connections of the subfactor theory with topology and quantum field theory have been pointed out. Our aim in this paper is to provide an evidence of a deeper relation between a notion of flatness in subfactor theory and modular invariants in
Orbifold subfactors, central sequences and the relative Jones invariant κ
 Internat. Math. Res. Notices
, 1995
"... the relative Jones invariant κ ..."
Paragroups as quantized Galois groups of subfactors
"... revolutionized the theory of operator algebras. Furthermore, he found the Jones polynomial, an invariant of links, based on the theory of subfactors in 1984 in [28], and it has created a stream of new mathematics involving quantum groups, conformal field theory, solvable lattice models, and low dim ..."
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Cited by 3 (3 self)
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revolutionized the theory of operator algebras. Furthermore, he found the Jones polynomial, an invariant of links, based on the theory of subfactors in 1984 in [28], and it has created a stream of new mathematics involving quantum groups, conformal field theory, solvable lattice models, and low dimensional topology. This discovery gave him a Fields medal in 1990 as well known. From an operator algebraic viewpoint, the best machinery to understand algebraic and combinatorial structures of subfactors is the paragroup theory which was introduced by A. Ocneanu [43] in 1987. As an analogue of the classical Galois theory in which Galois groups describe relations between fields and their subfields, paragroups describe relations between certain kinds of algebras of (bounded linear) operators (on Hilbert spaces) and their subalgebras. Passing from function algebras to noncommutative operator algebras is often called “quantization”, and in this sense, a paragroup is regarded as a “quantized Galois group”. It is very similar to a fusion algebra with braiding/fusion matrices in rational conformal field theory (RCFT), and it is also related to a representation theory of quantum groups at roots of unity, an
Paragroup and their actions on subfactors
 in “Subfactors — Proceedings of the Taniguchi Symposium, Katata
, 1994
"... Our aim in this work is to make a “double quantization ” of studies of group actions on von Neumann algebras in the setting of Ocneanu’s paragroup theory. Detailed proofs will be given in [33]. Since the pioneering work [27] of V. F. R. Jones, the theory of subfactors has ..."
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Cited by 1 (1 self)
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Our aim in this work is to make a “double quantization ” of studies of group actions on von Neumann algebras in the setting of Ocneanu’s paragroup theory. Detailed proofs will be given in [33]. Since the pioneering work [27] of V. F. R. Jones, the theory of subfactors has
Subfactors and Conformal Field Theory
, 1992
"... Abstract. We discuss relations between the combinatorial structure of subfactors, solvable lattice models, (rational) conformal field theory, and topological quantum field theory. Key words: conformal field theory, modular automorphism group, modular invariant, orbifold construction, statistical mec ..."
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Cited by 1 (0 self)
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Abstract. We discuss relations between the combinatorial structure of subfactors, solvable lattice models, (rational) conformal field theory, and topological quantum field theory. Key words: conformal field theory, modular automorphism group, modular invariant, orbifold construction, statistical mechanics, subfactors, topological invariant 1.
MaxPlanckInstitut für Mathematik
, 1998
"... We show that a leftrigid monoidal C ∗category with irreducible monoidal unit is also a sovereign and spherical category. Defining a FrobeniusSchur type indicator we obtain selection rules for the fusion coefficients of irreducible objects. As a main result we prove S4invariance of 6jsymbols in ..."
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We show that a leftrigid monoidal C ∗category with irreducible monoidal unit is also a sovereign and spherical category. Defining a FrobeniusSchur type indicator we obtain selection rules for the fusion coefficients of irreducible objects. As a main result we prove S4invariance of 6jsymbols in such a category. 1 1