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35
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
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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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 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
On Galois extensions of braided tensor categories, mapping class group representations and simple current extensions
 In preparation
"... We show that the author’s notion of Galois extensions of braided tensor categories [22], see also [3], gives rise to braided crossed Gcategories, recently introduced for the purposes of 3manifold topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G non ..."
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Cited by 25 (6 self)
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We show that the author’s notion of Galois extensions of braided tensor categories [22], see also [3], gives rise to braided crossed Gcategories, recently introduced for the purposes of 3manifold topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G nontrivial objects of grade g exist in C ⋊ S. 1
Unoriented topological quantum field theory and link homology
"... We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply.1C1/–dimensional unoriented topological quantum field theories to BarNatan’s geometric formalism to define new theories for stable equivalence classes. 57M25, 57R56; 81T40 1 ..."
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Cited by 19 (1 self)
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We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply.1C1/–dimensional unoriented topological quantum field theories to BarNatan’s geometric formalism to define new theories for stable equivalence classes. 57M25, 57R56; 81T40 1
Conformal Field Theory and DoplicherRoberts Reconstruction
 In
"... Abstract. After a brief review of recent rigorous results concerning the representation theory of rational chiral conformal field theories (RCQFTs) we focus on pairs (A, F) of conformal field theories, where F has a finite group G of global symmetries and A is the fixpoint theory. The comparison of ..."
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Cited by 18 (5 self)
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Abstract. After a brief review of recent rigorous results concerning the representation theory of rational chiral conformal field theories (RCQFTs) we focus on pairs (A, F) of conformal field theories, where F has a finite group G of global symmetries and A is the fixpoint theory. The comparison of the representation categories of A and F is strongly intertwined with various issues related to braided tensor categories. We
Conformal Orbifold Theories and Braided Crossed GCategories
, 2004
"... The aim of the paper is twofold. First, we show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category G−LocA of twisted representations. This category is a braided crossed Gcategory in the sense of Turaev [60]. Its degree zero subcategor ..."
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Cited by 16 (3 self)
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The aim of the paper is twofold. First, we show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category G−LocA of twisted representations. This category is a braided crossed Gcategory in the sense of Turaev [60]. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of Gmanifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT ❀ modular category ❀ 3manifold invariant. Secondly, we study the relation between G−LocA and the braided (in the usual sense) representation category Rep AG of the orbifold theory AG. We prove the equivalence Rep AG ≃ (G−LocA) G, which is a rigorous implementation of the insight that one needs to take the twisted representations of A into account in order to determine Rep AG. In the opposite direction we have G−LocA ≃ Rep AG ⋊ S, where S ⊂ Rep AG is the full
On the Rozansky–Witten weight systems
"... Ideas of Rozansky and Witten, as developed by Kapranov, show that a complex symplectic manifold X gives rise to Vassiliev weight systems. In this paper we ..."
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Cited by 16 (4 self)
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Ideas of Rozansky and Witten, as developed by Kapranov, show that a complex symplectic manifold X gives rise to Vassiliev weight systems. In this paper we