On braided fusion categories I

Drinfeld, Vladimir; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor

doi:10.1007/s00029-010-0017-z

Cited by 308 publications

(671 citation statements)

References 46 publications

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“…By a fusion category we mean a semisimple rigid tensor category with finitely many isomorphism classes of simple objects and finite dimensional spaces of morphisms. For basic results of the theory of fusion categories see [EGNO,DGNO2].…”

Section: Preliminariesmentioning

confidence: 99%

“…Let p be a prime integer. By a fusion p-category we mean a fusion category whose Frobenius-Perron dimension is p n for some integer n. Such categories were characterized in [DGNO1] (see also [EGNO,Section 9.4]). Namely, any such category A which is integral (i.e., such that every object of A has an integral Frobenius-Perron dimension) is group-theoretical, i.e., there is a p-group G and ω ∈ H 3 (G, k × ) such that A is categorically Morita equivalent to C(G, ω).…”

Section: Preliminariesmentioning

confidence: 99%

“…Fusion p-categories were classified in [DGNO1] (see also [EGNO,Section 9.14]). Namely, it was shown that any integral fusion p-category A is group-theoretical, i.e., it is categorically Morita equivalent to a pointed fusion category.…”

Section: Introductionmentioning

confidence: 99%

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On the Brauer–Picard groups of fusion categories

Marshall

Nikshych

2017

Math. Z.

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Abstract. We develop methods of computation of the Brauer-Picard groups of fusion categories and apply them to compute such groups for several classes of fusion categories of prime power dimension: representation categories of elementary abelian groups with twisted associativity, extra special p-groups, and the Kac-Paljutkin Hopf algebra. We conclude that many finite groups of Lie type occur as composition factors of the Brauer-Picard groups of pointed fusion categories.

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

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On the Brauer–Picard groups of fusion categories

Marshall

Nikshych

2017

Math. Z.

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“…The category Ꮿ is said to be nondegenerate if Ꮿ = Vec (the fusion category generated by the unit object). If Ꮿ is a premodular category, that is, if it has a twist, then it is nondegenerate if and only if it is modular [Beliakova and Blanchet 2001;Müger 2003b;Drinfeld et al 2010]. …”

Section: Preliminariesmentioning

confidence: 99%

“…Turaev's notion [2000; 2008] of a crossed category (short for braided group-crossed category) has attracted much attention recently [Drinfeld et al 2010;Kirillov 2001a;2001b;Müger 2004;. Roughly, a crossed category consists of a group G, a G-graded tensor category Ꮿ, an action g → T g of G on Ꮿ by tensor autoequivalences, and G-braidings c(X, Y ) : X ⊗ Y ∼ − → T g (Y ) ⊗ X for X, Y ∈ Ꮿ, satisfying certain compatibility conditions.…”

Section: Introductionmentioning

confidence: 99%

Crossed pointed categories and their equivariantizations

Naidu¹

2010

Pacific J. Math.

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We propose a notion, quasiabelian third cohomology of crossed modules, which generalizes Eilenberg and Mac Lane's abelian and Ospel's quasiabelian cohomology. We classify crossed pointed categories in terms of it. We apply the process of equivariantization to the latter to obtain braided fusion categories, which may be viewed as generalizations of the categories of modules over twisted Drinfeld doubles of finite groups. As a consequence, we obtain a description of all braided group-theoretical categories. We give a criterion for these categories to be modular. We describe the quasitriangular quasi-Hopf algebras underlying these categories. IntroductionTuraev's notion [2000; 2008] of a crossed category (short for braided group-crossed category) has attracted much attention recently [Drinfeld et al. 2010;Kirillov 2001a;2001b;Müger 2004;. Roughly, a crossed category consists of a group G, a G-graded tensor category Ꮿ, an action g → T g of G on Ꮿ by tensor autoequivalences, and G-braidings c(X, Y ) : X ⊗ Y ∼ − → T g (Y ) ⊗ X for X, Y ∈ Ꮿ, satisfying certain compatibility conditions. Crossed categories are known to arise in various contexts; for instance, Müger [2004] showed that Galois extensions of braided tensor categories have a natural structure of crossed categories. In [2005], Müger established a connection between 1-dimensional quantum field theories and crossed categories. Kirillov [2001b] showed that crossed categories arise in the theory of vertex operator algebras.A fusion category is said to be pointed if all its simple objects are invertible. One of the goals of this paper is to classify all crossed pointed categories. From [Joyal and Street 1993], it is known that braided pointed categories are classified by Eilenberg and Mac Lane's abelian cohomology H 3 ab (A, ‫ދ‬ × ), where A is a finite abelian group. On the other hand, certain crossed pointed categories in which the group action is strict were described by Turaev [2000;2008] in terms of Ospel's quasiabelian cohomology H 3 qa (G, ‫ދ‬ × ), where G is a (not necessarily abelian) finite MSC2000: primary 18D10; secondary 16W30.

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Two and Three Dimensional Calculus

Dyke

2018

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Quantum field theory has various projective characteristics which are captured by what are called anomalies. This paper explores this idea in the context of fully-extended three-dimensional topological quantum field theories (TQFTs).Given a three-dimensional TQFT (valued in the Morita 3-category of fusion categories), the anomaly identified herein is an obstruction to gauging a naturally occurring orthogonal group of symmetries, i.e. we study 't Hooft anomalies. In other words, the orthogonal group almost acts: There is a lack of coherence at the top level. This lack of coherence is captured by a "higher (central) extension" of the orthogonal group, obtained via a modification of the obstruction theory of Etingof-Nikshych-Ostrik-Meir [ENO10]. This extension tautologically acts on the given TQFT/fusion category, and this precisely classifies a projective (equivalently anomalous) TQFT. We explain the sense in which this is an analogue of the classical spin representation. This is an instance of a phenomenon emphasized by Freed [Fre23]: Quantum theory is projective.In the appendices we establish a general relationship between the language of projectivity/anomalies and the language of topological symmetries. We also identify a universal anomaly associated with any theory which is appropriately "simple". JACKSON VAN DYKE 3.7. Module structures on 3d TQFTs 31 Appendix A. TQFT and category theory 32 A.1. The Cobordism Hypothesis 32 A.2. Relative theories 34 A.3. Invertible theories 35 Appendix B. Topological symmetry 35 B.1. TQFTs associated to π-finite spaces 35 B.2. Module structures 38 B.3. Reduction of topological symmetry 39 Appendix C. Anomalies 41 C.1. Definitions 42 C.2. The anomaly associated to a projectivity class 43 C.3. Projective theories 45 C.4. Anomalies and sandwiches 46 C.5. A universal anomaly 47 References 48

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On braided fusion categories I

Cited by 308 publications

References 46 publications

On the Brauer–Picard groups of fusion categories

On the Brauer–Picard groups of fusion categories

Crossed pointed categories and their equivariantizations

Two and Three Dimensional Calculus

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