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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 52 (6 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
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 30 (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
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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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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Cited by 20 (3 self)
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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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Cited by 6 (0 self)
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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.
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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Cited by 4 (0 self)
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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
LarsonSweedler theorem, grouplike elements, and invertible modules in weak Hopf algebras, preprint
, 2001
"... We extend the Larson–Sweedler theorem for weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a nondegenerate left integral. We establish the autonomous monoidal category of the modules of a weak Hopf algebra A and show the semisimplicity o ..."
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Cited by 3 (0 self)
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We extend the Larson–Sweedler theorem for weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a nondegenerate left integral. We establish the autonomous monoidal category of the modules of a weak Hopf algebra A and show the semisimplicity of the unit and the invertible modules of A. We also reveal the connection of these modules to left/right grouplike elements in the dual weak Hopf algebra Â.
Abelian categories of modules over a (lax) monoidal functor. Adv
 Department of Mathematics Kansas State University Manhattan, KS
, 2003
"... Abstract: In [CY98] Crane and Yetter introduced a deformation theory for monoidal categories. The related deformation theory for monoidal functors introduced by Yetter in [Yet98] is a proper generalization of Gerstenhaber’s deformation theory for associative algebras [Ger63, Ger64, GS88]. In the pre ..."
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Cited by 2 (1 self)
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Abstract: In [CY98] Crane and Yetter introduced a deformation theory for monoidal categories. The related deformation theory for monoidal functors introduced by Yetter in [Yet98] is a proper generalization of Gerstenhaber’s deformation theory for associative algebras [Ger63, Ger64, GS88]. In the present paper we solidify the analogy between lax monoidal functors and associative algebras by showing that under suitable conditions, categories of functors with an action of a lax monoidal functor are abelian categories. The deformation complex of a monoidal functor is generalized to an analogue of the Hochschild complex with coefficients in a bimodule, and the deformation complex of a monoidal natural transformation is shown to be a special case. It is shown further that the cohomology of a monoidal functor F with coefficients in an F, Fbimodule is given by right derived functors. 1 1