Finite Dimensional Hopf Algebras Over the Dual Group Algebra of the Symmetric Group in Three Letters

Andruskiewitsch, Nicolás; Vay, Cristian

doi:10.1080/00927872.2011.616429

Cited by 33 publications

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“…This classification was completed by Angiono and Garcia Iglesias in [AG19], where it is also shown that all Hopf algebras whose coradical is an abelian group algebra are cocycle deformations of their associated graded Hopf algebras. For more classification results of non-semisimple Hopf algebras in which cocycle deformations appear, see [AAIMV14], [FGM19], [GarM15], [GruM] and [AV11].…”

Section: Introductionmentioning

confidence: 99%

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Hopf cocycle deformations and invariant theory

Meir

2019

Math. Z.

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For a given finite dimensional Hopf algebra H we describe the set of all equivalence classes of cocycle deformations of H as an affine variety, using methods of geometric invariant theory. We show how our results specialize to the Universal Coefficients Theorem in the case of a group algebra, and we also give examples from other families of Hopf algebras, including dual group algebras and Bosonizations of Nichols algebras. In particular, we use the methods developed here to classify the cocycle deformations of a dual pointed Hopf algebra associated to the symmetric group on three letters. We also give an example of a cocycle deformation over a dual group algebra, which has only rational invariants, but which is not definable over the rational field. This differs from the case of group algebras, in which every 2-cocycle is equivalent to one which is definable by its invariants.

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Section: Introductionmentioning

confidence: 99%

“…The first case appears in [IM11]. For the second case, the resulting double-twisted Hopf algebras α H α −1 appear in [AV11].…”

Section: Introductionmentioning

confidence: 99%

Hopf cocycle deformations and invariant theory

Meir

2019

Math. Z.

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“…We recall that a Hopf algebra A is said to be pointed if A (0) = kG(A) and copointed when A (0) = k G for some non-abelian group G. Our main result is the following. Finite-dimensional copointed Hopf algebras over k S 3 are classified in [AV1].…”

Section: (C ⊗ Id)(id ⊗C)(c ⊗ Id) = (Id ⊗C)(c ⊗ Id)(id ⊗C)mentioning

confidence: 99%

Copointed Hopf algebras overS4

Iglesias

Vay

2018

Journal of Pure and Applied Algebra

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“…Nonpointed Hopf algebras with Chevalley property known to exist [AV1,AV2]. 9 dim p 3 q, p 2 q 2 , p 2 qr: For dimensions 88, 100, 84, 90, the classification of the pointed Hopf algebras was completed for those with coradical a group algebra of order a power of 2 in [Ni] and [Gr1].…”

Section: Open Casesmentioning

confidence: 99%

Classifying Hopf algebras of a given dimension

Beattie

García²

2013

Contemporary Mathematics

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Abstract. Classifying all Hopf algebras of a given finite dimension over C is a challenging problem which remains open even for many small dimensions, not least because few general approaches to the problem are known. [ChNg]. In this paper, we add to the classification tools in [BG] and apply our results to Hopf algebras of dimension rpq and 8p where p, q, r are distinct primes. At the end of this paper we summarize in a table the status of the classification for dimensions up to 100 to date.

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Finite Dimensional Hopf Algebras Over the Dual Group Algebra of the Symmetric Group in Three Letters

Abstract: We classify finite-dimensional Hopf algebras whose coradical is isomorphic to the algebra of functions on 3 . We describe a new infinite family of Hopf algebras of dimension 72.

Cited by 33 publications

References 10 publications

Hopf cocycle deformations and invariant theory

Hopf cocycle deformations and invariant theory

Copointed Hopf algebras overS4

Classifying Hopf algebras of a given dimension

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