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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
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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StateSum Invariants of 4Manifolds
 J. Knot Theory Ram
, 1997
"... Abstract: We provide, with proofs, a complete description of the authors ’ construction of statesum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda’s surgery ..."
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Cited by 37 (6 self)
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Abstract: We provide, with proofs, a complete description of the authors ’ construction of statesum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda’s surgery invariants [Br1,Br2] using techniques developed in the case of the semisimple subquotient of Rep(Uq(sl2)) (q a principal 4r th root of unity) by Roberts [Ro1]. We briefly discuss the generalizations to invariants of 4manifolds equipped with 2dimensional (co)homology classes introduced by Yetter [Y6] and Roberts [Ro2], which are the subject of the sequel. 1 1
Weak Hopf Algebras II: Representation theory, dimensions, and the Markov trace
 J. Algebra
"... If A is a weak C∗Hopf algebra then the category of finite dimensional unitary representations of A is a monoidal C∗category with monoidal unit being the GNS representation Dε associated to the counit ε. This category has isomorphic left dual and right dual objects which leads, as usual, to the not ..."
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If A is a weak C∗Hopf algebra then the category of finite dimensional unitary representations of A is a monoidal C∗category with monoidal unit being the GNS representation Dε associated to the counit ε. This category has isomorphic left dual and right dual objects which leads, as usual, to the notion of dimension function. However, if ε is not pure the dimension function is matrix valued with rows and columns labelled by the irreducibles contained in Dε. This happens precisely when the inclusions AL ⊂ A and AR ⊂ A are not connected. Still there exists a trace on A which is the Markov trace for both inclusions. We derive two numerical invariants for each C∗WHA of trivial hypercenter. These are the common indices I and δ, of the Haar, respectively Markov conditional expectations of either one of the inclusions AL/R ⊂ A and ÂL/R ⊂ Â. In generic cases I> δ. In the special case of weak Kac algebras we show that I = δ is an integer. Submitted to J. Algebra
LarsonSweedler theorem and the role of grouplike elements in weak Hopf algebras
 QA/0111045. Matematiska Institutionen, Göteborg University, S412 96 Göteborg, Sweden Email address: lkadison@c2i.net
"... We extend the Larson–Sweedler theorem [10] to weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a nondegenerate left integral. We show that the category of modules over a weak Hopf algebra is autonomous monoidal with semisimple unit and i ..."
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We extend the Larson–Sweedler theorem [10] to weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a nondegenerate left integral. We show that the category of modules over a weak Hopf algebra is autonomous monoidal with semisimple unit and invertible modules. We also reveal the connection of invertible modules to left and right grouplike elements in the dual weak Hopf algebra. Defining distinguished left and right grouplike elements we derive the Radford formula [15] for the fourth power of the antipode in a weak Hopf algebra and prove that the order of the antipode is finite up to an inner automorphism by a grouplike element in the trivial subalgebra A T of the underlying weak Hopf algebra A.
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.
Braided deformations of monoidal categories and vassiliev invariants
 Higher Category Theory
, 1998
"... It is the purpose of this paper to provide an exposition of a cohomological deformation theory for braided monoidal categories, an exposition and proof of a very general theorem of the form “all quantum invariants are Vassiliev invariants, ” and to relate to the existence of the Vassiliev ..."
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It is the purpose of this paper to provide an exposition of a cohomological deformation theory for braided monoidal categories, an exposition and proof of a very general theorem of the form “all quantum invariants are Vassiliev invariants, ” and to relate to the existence of the Vassiliev
Deformations of (Bi)tensor Categories
"... In [1], a new approach was suggested to the construction of four dimensional Topological Quantum Field Theories (TQFTs), proceeding from a new algebraic structure called a Hopf category. In [2], it was argued that a well behaved 4d TQFT would in fact contain such a category at least formally. An app ..."
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In [1], a new approach was suggested to the construction of four dimensional Topological Quantum Field Theories (TQFTs), proceeding from a new algebraic structure called a Hopf category. In [2], it was argued that a well behaved 4d TQFT would in fact contain such a category at least formally. An approach to construction of Hopf categories was also outlined in [1], making use of the