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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,
Weakly grouptheoretical and solvable fusion categories
"... To Izrail Moiseevich Gelfand on his 95th birthday with admiration ..."
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Cited by 59 (8 self)
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To Izrail Moiseevich Gelfand on his 95th birthday with admiration
On nonsemisimple fusion rules and tensor categories
, 2006
"... Category theoretic aspects of nonrational conformal field theories are discussed. We consider the case that the category C of chiral sectors is a finite tensor category, i.e. a rigid monoidal category whose class of objects has certain finiteness properties. Besides the simple objects, the indecom ..."
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Cited by 30 (0 self)
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Category theoretic aspects of nonrational conformal field theories are discussed. We consider the case that the category C of chiral sectors is a finite tensor category, i.e. a rigid monoidal category whose class of objects has certain finiteness properties. Besides the simple objects, the indecomposable projective objects of C are of particular interest. The fusion rules of C can be blockdiagonalized. A conjectural connection between the blockdiagonalization and modular transformations of characters of modules over vertex algebras is exemplified with the case of the (1,p) minimal models.
Grouptheoretical properties of nilpotent modular categories
"... 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 ..."
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Cited by 30 (5 self)
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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]).
FUSION CATEGORIES AND HOMOTOPY THEORY
, 2009
"... 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 pa ..."
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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
Cofiniteness conditions, projective covers and the logarithmic tensor product theory
, 2007
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A FINITENESS PROPERTY FOR BRAIDED FUSION CATEGORIES
"... We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations factor o ..."
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Cited by 19 (9 self)
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We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations factor over a finite group, and suggest that categories of integral FrobeniusPerron dimension are precisely those with property F.
From boundary to bulk in logarithmic CFT
, 2007
"... The analogue of the chargeconjugation modular invariant for rational logarithmic conformal field theories is constructed. This is done by reconstructing the bulk spectrum from a simple boundary condition (the analogue of the Cardy ‘identity brane’). We apply the general method to the c1,p triplet m ..."
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Cited by 16 (1 self)
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The analogue of the chargeconjugation modular invariant for rational logarithmic conformal field theories is constructed. This is done by reconstructing the bulk spectrum from a simple boundary condition (the analogue of the Cardy ‘identity brane’). We apply the general method to the c1,p triplet models and reproduce the previously known bulk theory for p = 2 at c = −2. For general p we verify that the resulting partition functions are modular invariant. We also construct the complete set of 2p boundary states, and confirm that the identity brane from which we started indeed exists. As a byproduct we obtain a logarithmic version of the Verlinde formula for the c1,p triplet models.