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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
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 31 (7 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 10 (4 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
Concurrent Kleene Algebra
"... A concurrent Kleene algebra offers, next to choice and iteration, operators for sequential and concurrent composition, related by an inequational form of the exchange law. We show applicability of the algebra to a partiallyordered trace model of program execution semantics and demonstrate its usefu ..."
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Cited by 10 (2 self)
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A concurrent Kleene algebra offers, next to choice and iteration, operators for sequential and concurrent composition, related by an inequational form of the exchange law. We show applicability of the algebra to a partiallyordered trace model of program execution semantics and demonstrate its usefulness by validating familiar proof rules for sequential programs (Hoare triples) and for concurrent ones (Jones’s rely/guarantee calculus). This involves an algebraic notion of invariants; for these the exchange inequation strengthens to an equational distributivity law. Most of our reasoning has been checked by computer.
Representations of algebraic quantum groups and reconstruction theorems for tensor categories
"... We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the TannakaKrein reconstruction problem. We show that every concrete semisimple tensor ∗category with conjugates is equivalent to the category of finite dimensional nondegene ..."
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Cited by 9 (4 self)
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We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the TannakaKrein reconstruction problem. We show that every concrete semisimple tensor ∗category with conjugates is equivalent to the category of finite dimensional nondegenerate ∗representations of a discrete algebraic quantum group. Working in the selfdual framework of algebraic quantum groups, we then relate this to earlier results of S. L. Woronowicz and S. Yamagami. We establish the relation between braidings and Rmatrices in this context. Our approach emphasizes the role of the natural transformations of the embedding functor. Thanks to the semisimplicity of our categories and the emphasis on representations rather than corepresentations, our proof is more direct and conceptual than previous reconstructions. As a special case, we reprove the classical TannakaKrein result for compact groups. It is only here that analytic aspects enter, otherwise we proceed in a purely algebraic way. In particular, the existence of a Haar functional is reduced to a well known general result concerning discrete multiplier Hopf ∗algebras. 1
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 8 (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
A FEW LOCALISATION THEOREMS
"... Given a functor T: C → D carrying a class of morphisms ..."
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Cited by 6 (2 self)
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Given a functor T: C → D carrying a class of morphisms
Quark State Confinement as a Consequence of the Extension of the Boson/Fermion Recoupling to SU(3) Colour 2003
 J. Phys. A: Math. Gen
"... Abstract. The Bose–Fermi recoupling of particles arising from the Z2–grading of the irreducible representations of SU(2) is responsible for the Pauli exclusion principle. We demonstrate from fundamental physical assumptions how to extend this to gradings, other than the Z2 grading, arising from othe ..."
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Abstract. The Bose–Fermi recoupling of particles arising from the Z2–grading of the irreducible representations of SU(2) is responsible for the Pauli exclusion principle. We demonstrate from fundamental physical assumptions how to extend this to gradings, other than the Z2 grading, arising from other groups. This requires non–associative recouplings where phase factors arise due to rebracketing of states. In particular, we consider recouplings for the Z3–grading of SU(3) colour and demonstrate that all the recouplings graded by triality leading to the Pauli exclusion principle demand quark state confinement. Note that quark state confinement asserts that only ensembles of triality zero are possible, as distinct from spatial confinement where particles are confined to a small region of space by a confining force such as given by the dynamics of QCD. PACS numbers: 02.10.Ws, 02.20.Mp, 05.30.Ch, 12.38.AwQuark State Confinement as a Consequence of the Extension of the Bose–Fermi Recoupling to SU(3) Colour2