On the Structure of Weak Hopf Algebras

Nikshych, Dmitri

doi:10.1016/s0001-8708(02)92081-5

Cited by 42 publications

(39 citation statements)

References 12 publications

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“…Sufficient conditions for semisimplicity. The next proposition is a refinement of [N,Prop. 6.4] and is analogous to the Larson-Radford theorem [LR1] for usual Hopf algebras which says that if A is a finite dimensional Hopf algebra with Tr(S 2 | A ) = 0 then A is semisimple and cosemisimple.…”

Section: Proposition 49 Let a Be A Finite-dimensional Weak Hopf Algmentioning

confidence: 98%

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On fusion categories

2005

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Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any (not necessarily hermitian) modular category is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion category. At the end of the paper we generalize some of these results to positive characteristic.

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Section: Proposition 49 Let a Be A Finite-dimensional Weak Hopf Algmentioning

confidence: 98%

“…Obviously, A is an ordinary Hopf algebra if and only if A min = k1. Minimal weak Hopf algebras over k, i.e., those for which A = A min , were completely classified in [N,Prop. 3.4].…”

Section: Remark 43 a Weak Hopf Algebra Is A Hopf Algebra If And Onlmentioning

confidence: 99%

On fusion categories

2005

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“…A more recent proof, that is in the spirit of the one we present below, appears in [18]. Generalizations of the formula from the case of Hopf algebras to other situations, braided Hopf algebras, bF algebras -braided and classical, quasi Hopf algebras, weak Hopf algebras, Hopf algebras over rings, and even for the very general case of finite tensor categories, can be found in the following references: [1], [2], [5], [8], [9], [10] and [16].…”

Section: Observation 1 It Is Worth Noticing That One Can Construct Bmentioning

confidence: 99%

Radford's Formula for Bifrobenius Algebras and Applications

Santos

Haim

2008

Communications in Algebra

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Abstract. In a biFrobenius algebra H, in particular in the case that H is a finite dimensional Hopf algebra, the antipode S : H → H can be decomposed as S =tc•c φ where c φ : H → H * and tc : H * → H are the Frobenius and coFrobenius isomorphisms. We use this decomposition to present an easy proof of Radford's formula for S 4 . Then, in the case that the map S satisfies the additional condition that S ⋆ id = id ⋆ S = uε, we prove the trace formula: tr(S 2 ) = ε(t)φ(1) . We finish by applying the above results to study the semisimplicity and cosemisimplicity of H.

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“…So the Lemma holds for f = id W . In this case u describes a deformation in the sense of [11,Remark 3.7]. Such (left) deformed WBA's have identical underlying left bialgebroids, so deformations should be interpreted as weak left automorphisms.…”

Section: Weak Automorphisms Twists and Half Grouplike Elementsmentioning

confidence: 99%

Galois actions by finite quantum groupoids

Szlachányi¹

2002

Locally Compact Quantum Groups and Groupoids

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Proposing a certain category of bialgebroid maps we show that the balanced depth 2 extensions appear as they were the finitary Galois extensions in the context of quantum groupoid actions, i.e., actions by finite bialgebroids, weak bialgebras or weak Hopf algebras. We comment on deformation of weak bialgebras, on half grouplike elements, on uniqueness of weak Hopf algebra reconstructions and discuss the example of separable field extensions.For extensions of rings, algebras or C * -algebras the notion of depth 2, introduced originally for von Neumann factors by A. Ocneanu, has many features that makes it the analogue of Galois extension of fields. The extension N ⊂ M of k-algebras is called of depth 2 if the canonical N -M bimodule X = N M M and M -N bimoduleX = M M N satisfy: X ⊗X ⊗ X is a direct summand in a finite direct sum of copies of X andX ⊗ X ⊗X is a direct summand in a finite direct sum ofX's. The right module M N is called balanced if End E M ∼ = N where E = End M N . For any balanced depth 2 extension the endomorphism ring A = End N M N carries a bialgebroid structure which is finite projective over the centralizer, or relative commutant, R = C M (N ) both as a left and as a right module. Moreover the canonical 0

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On the Structure of Weak Hopf Algebras

Cited by 42 publications

References 12 publications

On fusion categories

On fusion categories

Radford's Formula for Bifrobenius Algebras and Applications

Galois actions by finite quantum groupoids

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