Tensor Categories

Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor

doi:10.1090/surv/205

Cited by 836 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%

“…Let A be a fusion category. Recall that the objects of the center Z(A) of A are pairs (Z, γ), where Z is an object of A and γ = {γ X : X ⊗ Z → Z ⊗ X} is a natural isomorphism satisfying certain coherence axioms, see, e.g., [EGNO,7.13]. The fusion category Z(A) has a canonical braiding and there is an induction homomorphism…”

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%

Section: Preliminariesmentioning

confidence: 99%

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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“…It is known that any finite tensor category C is equivalent to the representation category of some finite dimensional weak quasi-Hopf algebra over k, see [12,Proposition 2.7]. The existence of such an equivalence provides that every indecomposable non-projective object in C is always the ending term of an almost split sequence in C. More precisely, if F defines an equivalence between C and Rep(H) for some finite dimensional weak quasi-Hopf algebra H over k, then for any indecomposable non-projective objet Z in C, F(Z) is also an indecomposable non-projective representation in Rep(H).…”

Section: Almost Split Sequencesmentioning

confidence: 99%

A Criterion for the Jacobson Semisimplicity of the Green Ring of a Finite Tensor Category

Wang

Li²,

Zhang³

2017

Glasgow Math. J.

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Abstract. This paper deals with the Green ring G(C) of a finite tensor category C with finitely many isomorphism classes of indecomposable objects over an algebraically closed field. The first part of this paper deals with the question of when the Green ring G(C), or the Green algebra G(C) ⊗ ‫ޚ‬ K over a field K, is Jacobson semisimple (namely, has zero Jacobson radical). It turns out that G(C) ⊗ ‫ޚ‬ K is Jacobson semisimple if and only if the Casimir number of C is not zero in K. For the Green ring G(C) itself, G(C) is Jacobson semisimple if and only if the Casimir number of C is not zero. The second part of this paper focuses on the case where C = Rep(kG) for a cyclic group G of order p over a field k of characteristic p. In this case, the Casimir number of C is computable and is shown to be 2p 2 . This leads to a complete description of the Jacobson radical of the Green algebra G(C) ⊗ ‫ޚ‬ K over any field K.

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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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Tensor Categories

Cited by 836 publications

References 0 publications

On the Brauer–Picard groups of fusion categories

On the Brauer–Picard groups of fusion categories

A Criterion for the Jacobson Semisimplicity of the Green Ring of a Finite Tensor Category

Two and Three Dimensional Calculus

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