Categorical Morita Equivalence for Group-Theoretical Categories

Naidu, Deepak

doi:10.1080/00927870701511996

Cited by 58 publications

(90 citation statements)

References 13 publications

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“…It suffices to show that isomorphisms i f;g , j f;g defined by equations (30) and (31) do not depend on the choice of equivalences (28) is isomorphic to the identity as a D-bimodule functor; (3) natural isomorphisms˛f ;g;h (27) satisfying identity (29).…”

Section: An Action Determined By Amentioning

confidence: 99%

Fusion categories and homotopy theory

Etingof¹,

Nikshych²,

Ostrik³

2010

Quantum Topol.

293

666

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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 classifying spaces of certain higher groupoids. In particular, to every fusion category C we attach the 3-groupoid BrPic.C / of invertible C -bimodule categories, called the Brauer-Picard groupoid of C , such that equivalence classes of G-extensions 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 article is that the 2-truncation of BrPic.C / is canonically equivalent to the 2-groupoid of braided auto-equivalences of the Drinfeld center Z.C/ of C. In particular, this implies that the Brauer-Picard group BrPic.C / (i.e., the group of equivalence classes of invertible C -bimodule categories) is naturally isomorphic to the group of braided auto-equivalences of Z.C /. Thus, if C D Vec A , where A is a finite abelian group, then BrPic.C / is the orthogonal group O.A˚A /. This allows one to obtain a rather explicit classification of extensions in this case; in particular, in the case G D Z 2 , we re-derive (without computations) the classical result of Tambara andYamagami. Moreover, we explicitly describe the category of all .Vec A 1 ; Vec A 2 /-bimodule categories (not necessarily invertible ones) by showing that it is equivalent to the hyperbolic part of the category of Lagrangian correspondences.Mathematics Subject Classification (2010). 18D10, 55S35.

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Section: An Action Determined By Amentioning

confidence: 99%

Fusion categories and homotopy theory

Etingof¹,

Nikshych²,

Ostrik³

2010

Quantum Topol.

293

666

View full text Add to dashboard Cite

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“…So if we find the necessary and sufficient condition for the existence of such U ⊆ G × G and ϕ ∈ H 2 (U, C × ) we can answer the question of whether Z(G) and Z(G ) are equivalent. This question was first answered by Naidu and Nikshych [11,12] based on the classification of Lagrangian subcategories of Z(G). Here we state the necessary and sufficient conditions of Davydov which is more appropriate for us.…”

Section: When Are Z(g) and Z(g ) Equivalent?mentioning

confidence: 99%

The Quantum Double Model with Boundary: Condensations and Symmetries

Beigi

Whalen

2011

Commun. Math. Phys.

163

275

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Associated to every finite group, Kitaev has defined the quantum double model for every orientable surface without boundary. In this paper, we define boundaries for this model and characterize condensations; that is, we find all quasi-particle excitations (anyons) which disappear when they move to the boundary. We then consider two phases of the quantum double model corresponding to two groups with a domain wall between them, and study the tunneling of anyons from one phase to the other. Using this framework we discuss the necessary and sufficient conditions when two different groups give the same anyon types. As an application we show that in the quantum double model for S3 (the permutation group over three letters) there is a chargeon and a fluxion which are not distinguishable. This group is indeed a special case of groups of the form of the semidirect product of the additive and multiplicative groups of a finite field, for all of which we prove a similar symmetry.

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“…Here ν is a certain 2-cocycle in Z 2 (H\G, H) coming from the G-invariance of µ and ̟ is a certain 3-cocycle on H ⋊ ν (H\G) depending on ν and on the exact sequence 1 → H → G → H\G → 1 (see [Na,Theorem 5.8] for precise definitions). The one corresponding to r 2 is equivalent to Rep(Z/2Z×Z/2Z×Z/2Z).…”

Section: Proof Let Us First Show That Altmentioning

confidence: 99%

Lagrangian Subcategories and Braided Tensor Equivalences of Twisted Quantum Doubles of Finite Groups

Naidu

Nikshych

2008

Commun. Math. Phys.

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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 3-cocycle 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 Lagrangian subcategories of Rep(D ω (G)) and module categories over the category Vec ω G of twisted G-graded vector spaces such that the dual tensor category is pointed. This can be viewed as a quantum version of V. Drinfeld's characterization of homogeneous spaces of a Poisson-Lie group in terms of Lagrangian subalgebras of the double of its Lie bialgebra [Dr]. As a consequence, we obtain that two group-theoretical fusion categories are weakly Morita equivalent if and only if their centers are equivalent as braided tensor categories.

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Categorical Morita Equivalence for Group-Theoretical Categories

Cited by 58 publications

References 13 publications

Fusion categories and homotopy theory

Fusion categories and homotopy theory

The Quantum Double Model with Boundary: Condensations and Symmetries

Lagrangian Subcategories and Braided Tensor Equivalences of Twisted Quantum Doubles of Finite Groups

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