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11
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
On Group Theoretical Hopf Algebras and Exact Factorizations of Finite Groups
 J. of Algebra
, 2003
"... Abstract. We show that a semisimple Hopf algebra A is group theoretical if and only if its Drinfeld double is a twisting of the DijkgraafPasquierRoche quasiHopf algebra D ω (Σ), for some finite group Σ and some ω ∈ Z 3 (Σ, k ×). We show that semisimple Hopf algebras obtained as bicrossed products ..."
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Cited by 11 (4 self)
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Abstract. We show that a semisimple Hopf algebra A is group theoretical if and only if its Drinfeld double is a twisting of the DijkgraafPasquierRoche quasiHopf algebra D ω (Σ), for some finite group Σ and some ω ∈ Z 3 (Σ, k ×). We show that semisimple Hopf algebras obtained as bicrossed products from an exact factorization of a finite group Σ are group theoretical. We also describe their Drinfeld double as a twisting of D ω (Σ), for an appropriate 3cocycle ω coming from the Kac exact sequence. 1.
Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups
 Comm. Math. Phys
"... Abstract. We classify Lagrangian subcategories of the representation category of a twisted quantum double D ω (G), where G is a finite group and ω is a 3cocycle on it. In view of results of [DGNO] this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of ..."
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Cited by 7 (5 self)
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Abstract. We classify Lagrangian subcategories of the representation category of a twisted quantum double D ω (G), where G is a finite group and ω is a 3cocycle on it. In view of results of [DGNO] this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups. We also establish a canonical bijection between
FrobeniusSchur indicators and exponents of spherical categories
 Adv. Math
"... Abstract. We obtain two formulae for the higher FrobeniusSchur indicators: one for a spherical fusion category in terms of the twist of its center and the other one for a modular tensor category in terms of its twist. The first one is a categorical generalization of an analogous result by Kashina, ..."
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Cited by 5 (2 self)
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Abstract. We obtain two formulae for the higher FrobeniusSchur indicators: one for a spherical fusion category in terms of the twist of its center and the other one for a modular tensor category in terms of its twist. The first one is a categorical generalization of an analogous result by Kashina, Sommerhäuser, and Zhu for Hopf algebras, and the second one extends Bantay’s 2nd indicator formula for a conformal field theory to higher degree. These formulae imply the sequence of higher indicators of an object in these categories is periodic. We define the notion of FrobeniusSchur (FS)exponent of a pivotal category to be the global period of all these sequences of higher indicators, and we prove that the FSexponent of a spherical fusion category is equal to the order of the twist of its center. Consequently, the FSexponent of a spherical fusion category is a multiple of its exponent by a factor not greater than 2. As applications of these results, we prove that the FSexponent of a semisimple quasiHopf algebra H has the same set of prime divisors as of dim(H) and it divides dim(H) 4. In addition, if H is a grouptheoretic quasiHopf algebra, the FSexponent of H divides dim(H) 2, and this upper bound is shown to be tight. 1.
On the Gauge Equivalence of Twisted Quantum Doubles of Elementary Abelian and ExtraSpecial 2Groups
, 2006
"... We establish explicit gauge equivalences, independent of all choices of cocycle, between the quantum doubles of certain pairs of finite groups G,H twisted by 3cocycles which are inflated from a 3cocycle on a common quotient. We give two main applications: if G is an extraspecial 2group of width ..."
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Cited by 3 (0 self)
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We establish explicit gauge equivalences, independent of all choices of cocycle, between the quantum doubles of certain pairs of finite groups G,H twisted by 3cocycles which are inflated from a 3cocycle on a common quotient. We give two main applications: if G is an extraspecial 2group of width at least 2, we show that the quantum double of G twisted by a 3cocycle ω is gauge equivalent to a twisted quantum double of an elementary abelian 2group if, and only if, ω 2 is trivial; we discuss the gauge equivalence classes of twisted quantum doubles of groups of order 8. 1
AN EMBEDDING THEOREM FOR ABELIAN MONOIDAL CATEGORIES
, 2000
"... We show that, with some technical conditions, an abelian category can be embedded into the category of bimodules over a ring. The case of semisimple rigid monoidal categories is studied in more detail. ..."
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Cited by 3 (1 self)
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We show that, with some technical conditions, an abelian category can be embedded into the category of bimodules over a ring. The case of semisimple rigid monoidal categories is studied in more detail.
TANNAKAKREIN DUALITY FOR HOPF ALGEBROIDS
, 2007
"... We show that a Hopf algebroid can be reconstructed from a monoidal ..."
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Cited by 2 (0 self)
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We show that a Hopf algebroid can be reconstructed from a monoidal
Hopf Algebra Extensions and Monoidal Categories
, 2002
"... Tannaka reconstruction provides a close link between monoidal categories and (quasi)Hopf algebras. We discuss some applications of the ideas of Tannaka reconstruction to the theory of Hopf algebra extensions, based on the following construction: For certain inclusions of a Hopf algebra into a coq ..."
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Cited by 1 (1 self)
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Tannaka reconstruction provides a close link between monoidal categories and (quasi)Hopf algebras. We discuss some applications of the ideas of Tannaka reconstruction to the theory of Hopf algebra extensions, based on the following construction: For certain inclusions of a Hopf algebra into a coquasibialgebra one can consider a natural monoidal category consisting of Hopf modules, and one can reconstruct a new coquasibialgebra from that monoidal category.
FUSION CATEGORIES AND HOMOTOPY THEORY
, 909
"... Abstract. We apply the yoga of classical homotopy theory to classification problems of Gextensions of fusion and braided fusion categories, where G is a finite group. Namely, we reduce such problems to classification (up to homotopy) of maps from BG to classifiying spaces of certain higher groupoid ..."
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Abstract. We apply the yoga of classical homotopy theory to classification problems of Gextensions of fusion and braided fusion categories, where G is a finite group. Namely, we reduce such problems to classification (up to homotopy) of maps from BG to classifiying spaces of certain higher groupoids. In particular, to every fusion category C we attach the 3groupoid BrPic(C) of invertible Cbimodule categories, called the BrauerPicard groupoid of C, such that equivalence classes of Gextensions of C are in bijection with homotopy classes of maps from BG to the classifying space of BrPic(C). This gives rise to an explicit description of both the obstructions to existence of extensions and the data parametrizing them; we work these out both topologically and algebraically. One of the central results of the paper is that the 2truncation of BrPic(C) is canonically the 2groupoid of braided autoequivalences of the Drinfeld center Z(C) of C. In particular, this implies that the
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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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.