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On fusion categories
 Annals of Mathematics
"... Abstract. In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series ofC. For fusion categories with commutative Gr ..."
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Abstract. In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series ofC. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Müger. Dedicated to Leonid Vainerman on the occasion of his 60th birthday 1. introduction The theory of fusion categories arises in many areas of mathematics such as representation theory, quantum groups, operator algebras and topology. The representation categories of semisimple (quasi) Hopf algebras are important examples of fusion categories. Fusion categories have been studied extensively in the literature,
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
Integral transforms and Drinfeld centers in derived algebraic geometry
"... Compact objects are as necessary to this subject as air to breathe. R.W. Thomason to A. Neeman, [N3] Abstract. We study natural algebraic operations on categories arising in algebraic geometry and its homotopytheoretic generalization, derived algebraic geometry. We work with a broad class of derive ..."
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Compact objects are as necessary to this subject as air to breathe. R.W. Thomason to A. Neeman, [N3] Abstract. We study natural algebraic operations on categories arising in algebraic geometry and its homotopytheoretic generalization, derived algebraic geometry. We work with a broad class of derived stacks which we call stacks with air. The class of stacks with air includes in particular all quasicompact, separated derived schemes and (in characteristic zero) all quotients of quasiprojective or smooth derived schemes by affine algebraic groups, and is closed under derived fiber products. We show that the (enriched) derived categories of quasicoherent sheaves on stacks with air behave well under algebraic and geometric operations. Namely, we identify the derived category of a fiber product with the tensor product of the derived categories of the factors. We also identify functors between derived categories of sheaves with integral transforms (providing a generalization of a theorem of Toën [To1] for ordinary schemes over a ring). As a first application, for a stack Y with air, we calculate the Drinfeld center (or synonymously,
Fusion categories of rank 2
, 2003
"... Abstract. We classify semisimple rigid monoidal categories with two isomorphism classes of simple objects over the field of complex numbers. In the appendix written by P. Etingof it is proved that the number of semisimple Hopf algebras with a given finite number of irreducible representations is fin ..."
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Abstract. We classify semisimple rigid monoidal categories with two isomorphism classes of simple objects over the field of complex numbers. In the appendix written by P. Etingof it is proved that the number of semisimple Hopf algebras with a given finite number of irreducible representations is finite. 1.
Frobenius monads and pseudomonoids
 2CATEGORIES COMPANION 73
, 2004
"... Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalenc ..."
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Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to &quot;strongly separable &quot; Frobenius algebras and &quot;weak monoidal Morita equivalence&quot;. Wreath products of Frobenius algebras are discussed.
Correspondences of ribbon categories
, 2006
"... Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories ..."
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Cited by 14 (3 self)
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Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.
Grouptheoretical properties of nilpotent modular categories, eprint arXiv:0704.0195v2 [math.QA
"... Abstract. We characterize a natural class of modular categories of prime power FrobeniusPerron dimension as representation categories of twisted doubles of finite pgroups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects o ..."
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Cited by 13 (3 self)
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Abstract. We characterize a natural class of modular categories of prime power FrobeniusPerron dimension as representation categories of twisted doubles of finite pgroups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects ofC have integral FrobeniusPerron dimensions then C is grouptheoretical in the sense of [ENO]. As a consequence, we obtain that semisimple quasiHopf algebras of prime power dimension are grouptheoretical. Our arguments are based on a reconstruction of twisted group doubles from Lagrangian subcategories of modular categories (this is reminiscent to the characterization of doubles of quasiLie bialgebras in terms of Manin pairs given in [Dr]). 1. introduction In this paper we work over an algebraically closed field k of characteristic 0. By a fusion category we mean a klinear semisimple rigid tensor category C with finitely many isomorphism classes of simple objects, finite dimensional spaces of morphisms, and such that the unit object 1 of C is simple. We refer the reader to [ENO] for a general theory of such categories. A fusion category is pointed if all its simple objects are invertible. A pointed fusion category is equivalent to Vec ω G, i.e., the category of Ggraded vector spaces with the associativity constraint given by some cocycle ω ∈ Z 3 (G, k × ) (here G is a finite group). 1.1. Main results. Theorem 1.1. Any braided nilpotent fusion category has a unique decomposition into a tensor product of braided fusion categories whose FrobeniusPerron dimensions are powers of distinct primes. The notion of nilpotent fusion category was introduced in [GN]; we recall it in Subsection 2.2. Let us mention that the representation category Rep(G) of a finite group G is nilpotent if and only if G is nilpotent. It is also known that fusion categories of prime power FrobeniusPerron dimension are nilpotent [ENO]. On the other hand, Vec ω G is nilpotent for any G and ω. Therefore it is not true that any nilpotent fusion category is a tensor product of fusion categories of prime power dimensions.
Topological quantum field theories from compact Lie groups. arXiv:0905.0731
, 2009
"... Let G be a compact Lie group and BG a classifying space for G. Then a class in H n 1 BG; Z leads to an ndimensional topological quantum field theory (TQFT), at least for n�1,2,3. The theory for n�1is trivial, but we include it for completeness. The theory for n�2has some infinities if G is not a fi ..."
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Let G be a compact Lie group and BG a classifying space for G. Then a class in H n 1 BG; Z leads to an ndimensional topological quantum field theory (TQFT), at least for n�1,2,3. The theory for n�1is trivial, but we include it for completeness. The theory for n�2has some infinities if G is not a finite group; it is a topological limit of 2dimensional YangMills theory. The most direct analog for n�3 is an L 2 version of the topological quantum field theory based on the classical ChernSimons invariant, which is only partially defined. The TQFT constructed by Witten and ReshetikhinTuraev which goes by the name ‘ChernSimons theory ’ (sometimes ‘holomorphic ChernSimons theory ’ to distinguish it from the L 2 theory) is completely finite. The theories we construct here are extended, or multitiered, TQFTs which go all the way down to points. For the n�3 ChernSimons theory, which we term a ‘0123 theory ’ to emphasize the extension down to points, we only treat the cases where G is finite or G is a torus, the latter being one of the main novelties in this paper. In other words, for toral theories we provide an answer to the longstanding question: What does ChernSimons theory attach to a point? The answer is a bit subtle as ChernSimons is an anomalous field theory of oriented manifolds. 1 This framing anomaly was already flagged in Witten’s seminal paper [Wi]. Here we interpret the anomaly as an invertible