Finitely semisimple spherical categories and modular categories are self-dual

Pfeiffer, Hendryk

doi:10.1016/j.aim.2009.03.002

Cited by 9 publications

(15 citation statements)

References 24 publications

(55 reference statements)

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“…The second main result of the paper generalizes Pfeiffer's theorem [Pfe09] on self-duality of pivotal fusion categories. As a special case of theorem 6.5, any fusion category is the representation category of a weak Hopf algebra which is self-dual.…”

Section: Introductionmentioning

confidence: 78%

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Module categories of finite Hopf algebroids, and self-duality

Schauenburg¹

2016

Trans. Amer. Math. Soc.

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International audienceWe characterize the module categories of suitably finite Hopf algebroids (more precisely, $X_R$-bialgebras in the sense of Takeuchi (1977) that are Hopf and finite in the sense of a work by the author (2000)) as those $k$-linear abelian monoidal categories that are module categories of some algebra, and admit dual objects for "sufficiently many" of their objects. Then we proceed to show that in many situations the Hopf algebroid can be chosen to be self-dual, in a sense to be made precise. This generalizes a result of Pfeiffer for pivotal fusion categories and the weak Hopf algebras associated to them

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Section: Introductionmentioning

confidence: 78%

“…The following corollary was proved by Pfeiffer [Pfe09] under the additional hypothesis that the category in question is spherical.…”

Section: Denoting Again F the Category Equivalencementioning

confidence: 87%

Module categories of finite Hopf algebroids, and self-duality

Schauenburg¹

2016

Trans. Amer. Math. Soc.

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“…For example, the category of G-graded rational vector spaces is a complete rational form of the category of G-graded complex vector spaces, while the center of the former category is only an incomplete rational form of the center of the latter. Second, any fusion category is the category of representations of a weak Hopf algebra A [19,8,9,23] and if that weak Hopf algebra has an rational form A k then A k − mod is, in general, an incomplete rational form for A − mod. Third, the notion of incomplete rational form arises naturally in the context of planar algebras.…”

Section: Fusion Categories and Fields Of Definitionmentioning

confidence: 99%

Non-cyclotomic fusion categories

Morrison

Snyder

2012

Trans. Amer. Math. Soc.

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Etingof, Nikshych and Ostrik ask in [7, §2] if every fusion category can be completely defined over a cyclotomic field. We show that this is not the case: in particular one of the fusion categories coming from the Haagerup subfactor [1] and one coming from the newly constructed extended Haagerup subfactor [2] can not be completely defined over a cyclotomic field. On the other hand, we show that the Drinfel'd center of the even part of the Haagerup subfactor is completely defined over a cyclotomic field. We identify the minimal field of definition for each of these fusion categories, compute the Galois groups, and identify their Galois conjugates.

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“…If C is symmetric monoidal, H is cotriangular. Further structure and properties of C such as a pivotal structure, a ribbon structure, or the properties that a pivotal category C be spherical or that a ribbon category C be modular, can be translated into additional structure and properties of H = coend(C, ω) as well [6,16].…”

Section: Tannaka-kreǐn Reconstructionmentioning

confidence: 99%

“…We use the same basis of ωM = Hom( V , V ⊗M ) as in [6,16] for j, j 0 , j 1 , ℓ, ℓ 0 , ℓ 1 ∈ I and j 1 = j 0 ± 1, ℓ 1 = ℓ 0 ± 1. We refer to the explanations preceding Theorem 3.8 for the bases used on the right hand side.…”

Section: The Fundamental Surjectionmentioning

confidence: 99%

Fusion categories in terms of graphs and relations

Pfeiffer¹

2011

Quantum Topol.

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Every fusion category C that is k-linear over a suitable field k, is the category of finite-dimensional comodules of a Weak Hopf Algebra H. This Weak Hopf Algebra is finite-dimensional, cosemisimple and has commutative bases. It arises as the universal coend with respect to the long canonical functor ω : C → Vect k . We show that H is a quotient H = H[G]/I of a Weak Bialgebra H[G] which has a combinatorial description in terms of a finite directed graph G that depends on the choice of a generator M of C and on the fusion coefficients of C. The algebra underlying H[G] is the path algebra of the quiver G × G, and so the composability of paths in G parameterizes the truncation of the tensor product of C. The ideal I is generated by two types of relations. The first type enforces that the tensor powers of the generator M have the appropriate endomorphism algebras, thus providing a Schur-Weyl dual description of C. If C is braided, this includes relations of the form 'RT T = T T R' where R contains the coefficients of the braiding on ωM ⊗ ωM , a generalization of the construction of Faddeev-Reshetikhin-Takhtajan to Weak Bialgebras. The second type of relations removes a suitable set of group-like elements in order to make the category of finite-dimensional comodules equivalent to C over all tensor powers of the generator M . As examples, we treat the modular categories associated with U q (sl 2 ).Proposition 4.12. Let C, M and G be as in Theorem 3.8 and E = (E (n) ) n∈AE 0 be an endomorphism system for C with respect to M .

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Finitely semisimple spherical categories and modular categories are self-dual

Cited by 9 publications

References 24 publications

Module categories of finite Hopf algebroids, and self-duality

Module categories of finite Hopf algebroids, and self-duality

Non-cyclotomic fusion categories

Fusion categories in terms of graphs and relations

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