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154
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 110 (9 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
Anyons in an exactly solved model and beyond
, 2005
"... A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge f ..."
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Cited by 87 (2 self)
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A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are nonAbelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and nonAbelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.
CATEGORY THEORY FOR CONFORMAL BOUNDARY CONDITIONS
, 2001
"... We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur ind ..."
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Cited by 75 (18 self)
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We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the description of boundary conditions in twodimensional conformal field theory and present illustrative examples. We show that module categories give rise to NIMreps of the fusion rules, and discuss a possible relation with the representation theory of vertex operator algebras.
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 62 (26 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
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 61 (9 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
Higherdimensional algebra II: 2Hilbert spaces
"... A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also ..."
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Cited by 60 (13 self)
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A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also define monoidal, braided monoidal, and symmetric monoidal versions of 2Hilbert spaces, which we call 2H*algebras, braided 2H*algebras, and symmetric 2H*algebras, and we describe the relation between these and tangles in 2, 3, and 4 dimensions, respectively. We prove a generalized DoplicherRoberts theorem stating that every symmetric 2H*algebra is equivalent to the category Rep(G) of continuous unitary finitedimensional representations of some compact supergroupoid G. The equivalence is given by a categorified version of the Gelfand transform; we also construct a categorified version of the Fourier transform when G is a compact abelian group. Finally, we characterize Rep(G) by its universal properties when G is a compact classical group. For example, Rep(U(n)) is the free connected symmetric 2H*algebra on one even object of dimension n. 1
Representations of compact quantum groups and subfactors
 J. Reine Angew. Math
, 1999
"... Abstract: We associate Popa systems ( = standard invariants of subfactors, cf. [P3],[P4]) to the finite dimensional representations of compact quantum groups. We characterise the systems arising in this way: these are the ones which can be “represented ” on finite dimensional Hilbert spaces. This is ..."
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Cited by 55 (19 self)
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Abstract: We associate Popa systems ( = standard invariants of subfactors, cf. [P3],[P4]) to the finite dimensional representations of compact quantum groups. We characterise the systems arising in this way: these are the ones which can be “represented ” on finite dimensional Hilbert spaces. This is proved by an universal construction. We explicitely compute (in terms of some free products) the operation of going from representations of compact quantum groups to Popa systems and then back via the universal construction. This is related with our previous work [B2]. We prove a Kesten type result for the coamenability of compact quantum groups, which allows us to compare it with the amenability of subfactors.
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 50 (29 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
Liberation of orthogonal lie groups
 Adv. Math
"... Abstract. We show that under suitable assumptions, we have a onetoone correspondence between classical groups and free quantum groups, in the compact orthogonal case. We classify the groups under correspondence, with the result that there are exactly 6 of them: On, Sn, Hn, Bn, S ′ n, B ′ n. We inv ..."
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Cited by 49 (18 self)
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Abstract. We show that under suitable assumptions, we have a onetoone correspondence between classical groups and free quantum groups, in the compact orthogonal case. We classify the groups under correspondence, with the result that there are exactly 6 of them: On, Sn, Hn, Bn, S ′ n, B ′ n. We investigate the representation theory aspects of the correspondence, with the result that for On, Sn, Hn, Bn, this is compatible with the BercoviciPata bijection. Finally, we discuss some more general classification problems in the compact orthogonal case, notably with the construction of a new quantum group.