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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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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
TANNAKA–KREĬN DUALITY FOR COMPACT QUANTUM HOMOGENEOUS SPACES. I. GENERAL THEORY
, 2013
"... An ergodic action of a compact quantum group G on an operator algebra A can be interpreted as a quantum homogeneous space for G. Such an action gives rise to the category of finite equivariant Hilbert modules over A, which has a module structure over the tensor category Rep(G) of finitedimensional ..."
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Cited by 11 (7 self)
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An ergodic action of a compact quantum group G on an operator algebra A can be interpreted as a quantum homogeneous space for G. Such an action gives rise to the category of finite equivariant Hilbert modules over A, which has a module structure over the tensor category Rep(G) of finitedimensional representations of G. We show that there is a onetoone correspondence between the quantum Ghomogeneous spaces up to equivariant Morita equivalence, and indecomposable module C∗categories over Rep(G) up to natural equivalence. This gives a global approach to the duality theory for ergodic actions as developed by C. Pinzari and J. Roberts.
Tensor categories: A selective guided tour
, 2008
"... These are the – only lightly edited – lecture notes for a short course on tensor categories. The coverage in these notes is relatively nontechnical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something int ..."
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Cited by 8 (1 self)
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These are the – only lightly edited – lecture notes for a short course on tensor categories. The coverage in these notes is relatively nontechnical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something interesting in them. Once the basic definitions are given, the focus is mainly on klinear categories with finite dimensional homspaces. Connections with quantum groups and low dimensional topology are pointed out, but these notes have no pretension to cover the latter subjects at any depth. Essentially, these notes should be considered as annotations to the extensive bibliography. We also recommend the recent review [33], which covers less ground in a deeper way.
Fiber functors on TemperleyLieb categories
, 2004
"... Abstract. Fiber functors on TemperleyLieb categories are determined with the help of classification results on nondegenerate bilinear forms. The case of unitary fiber functors is also investigated. 1. ..."
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Abstract. Fiber functors on TemperleyLieb categories are determined with the help of classification results on nondegenerate bilinear forms. The case of unitary fiber functors is also investigated. 1.
Bundles of C*categories and duality
, 2008
"... We introduce the notions of multiplier C*category and continuous bundle of C*categories, as the categorical analogues of the corresponding C*algebraic notions. Every symmetric tensor C*category with conjugates is a continuous bundle of C*categories, with base space the spectrum of the C*algebr ..."
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We introduce the notions of multiplier C*category and continuous bundle of C*categories, as the categorical analogues of the corresponding C*algebraic notions. Every symmetric tensor C*category with conjugates is a continuous bundle of C*categories, with base space the spectrum of the C*algebra associated with the identity object. We classify tensor C*categories with fibre the dual of a compact Lie group in terms of suitable principal bundles, and give a cohomological obstruction to embed such tensor categories into the one of vector bundles. If such a cohomological obstruction vanishes, then the given tensor category is the dual of a (nonunique) group bundle. This also provides a classification for certain C*algebra bundles, with fibres fixedpoint algebras of Od.
Monoids, Embedding Functors and Quantum Groups
, 2008
"... We show that the left regular representation πl of a discrete quantum group (A,∆) has the absorbing property and forms a monoid (πl, ˜m, ˜η) in the representation category Rep(A,∆). Next we show that an absorbing monoid in an abstract tensor ∗category C gives rise to an embedding functor (or fiber ..."
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We show that the left regular representation πl of a discrete quantum group (A,∆) has the absorbing property and forms a monoid (πl, ˜m, ˜η) in the representation category Rep(A,∆). Next we show that an absorbing monoid in an abstract tensor ∗category C gives rise to an embedding functor (or fiber functor) E: C → VectC, and we identify conditions on the monoid, satisfied by (πl, ˜m, ˜η), implying that E is ∗preserving. As is wellknown, from an embedding functor E: C → Hilb the generalized Tannaka theorem produces a
c © World Scientific Publishing Company MONOIDS, EMBEDDING FUNCTORS AND QUANTUM GROUPS
, 2007
"... We show that the left regular representation πl of a discrete quantum group (A,∆) has the absorbing property and forms a monoid (πl, m̃, η̃) in the representation category Rep(A,∆). Next we show that an absorbing monoid in an abstract tensor ∗category C gives rise to an embedding functor (or fiber ..."
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We show that the left regular representation πl of a discrete quantum group (A,∆) has the absorbing property and forms a monoid (πl, m̃, η̃) in the representation category Rep(A,∆). Next we show that an absorbing monoid in an abstract tensor ∗category C gives rise to an embedding functor (or fiber functor) E: C → VectC, and we identify conditions on the monoid, satisfied by (πl, m̃, η̃), implying that E is ∗preserving. As is wellknown, from an embedding functor E: C → Hilb the generalized Tannaka theorem produces a discrete quantum group (A,∆) such that C Repf (A,∆). Thus, for a C∗tensor category C with conjugates and irreducible unit the following are equivalent: (1) C is equivalent to the representation category of a discrete quantum group (A,∆), (2) C admits an absorbing monoid, (3) there exists a ∗preserving embedding functor E: C → Hilb.
Contents
, 2006
"... This is an appendix to “Algebraic Quantum Field Theory ” by Hans Halvorson, to appear in J. Butterfield & J. Earman (eds.): Handbook of the Philosophy of Physics. Its aim is to give a proof of Theorem 2.18, first proved by S. Doplicher and J.E. Roberts in 1989, according to which every symmetric ..."
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This is an appendix to “Algebraic Quantum Field Theory ” by Hans Halvorson, to appear in J. Butterfield & J. Earman (eds.): Handbook of the Philosophy of Physics. Its aim is to give a proof of Theorem 2.18, first proved by S. Doplicher and J.E. Roberts in 1989, according to which every symmetric tensor ∗category with conjugates, direct sums, subobjects and End1 = C is equivalent to the category of finite dimensional unitary representations of a uniquely determined compact supergroup. Our approach is based on a modern formulation of Tannaka’s theorem (1939) and a simplified approach to Deligne’s characterization of tannakian categories (1991).
Theory
, 2009
"... We discuss finite local extensions of quantum field theories in low space time dimensions in connection with categorical structures and the question of modular invariants in conformal field theory, also touching upon purely mathematical ramifications. ..."
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We discuss finite local extensions of quantum field theories in low space time dimensions in connection with categorical structures and the question of modular invariants in conformal field theory, also touching upon purely mathematical ramifications.
FIBER FUNCTORS ON TEMPERLEYLIEB CATEGORIES
, 2004
"... Abstract. Fiber functors on TemperleyLieb categories are investigated with the help of classification results on nondegenerate bilinear forms. The case of unitary fiber functors is also considered. 1. ..."
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Abstract. Fiber functors on TemperleyLieb categories are investigated with the help of classification results on nondegenerate bilinear forms. The case of unitary fiber functors is also considered. 1.