Finite quantum groups over abelian groups of prime exponent

Andruskiewitsch, Nicolás; Schneider, Hans‐Jürgen

doi:10.1016/s0012-9593(01)01082-5

Cited by 48 publications

(72 citation statements)

References 33 publications

(29 reference statements)

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“…It suffices to show that u (D) and u(D) are quasi-isomorphic. Now, from the proof of Theorem 5.17. in [AS3] we see that u(D) is isomorphic to (U ⊗ B) σ /K + , where U, B and K are Hopf algebras to be described shortly. σ is a 2-cocycle constructed from the linking datum.…”

Section: Introductionmentioning

confidence: 94%

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Pointed Hopf Algebras and Quasi-isomorphisms

Didt

2005

Algebr Represent Theor

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We show that a large class of finite dimensional pointed Hopf algebras is quasi-isomorphic to their associated graded version coming from the coradical filtration, i.e. they are 2-cocycle deformations of the latter. This supports a slightly specialized form of a conjecture in [M]. IntroductionRecently there has been a lot of progress in determining the structure of pointed Hopf algebras. This has led to a discovery of whole new classes of such Hopf algebras and to some important classification results. See the survey article [AS4] for an introduction and references. A lot of these classes contain infinitely many non-isomorphic Hopf algebras of the same dimension, thus disproving an old conjecture of Kaplansky. Masuoka showed in [M], and in a private note, that for certain of these new families the Hopf algebras are all 2-cocycle deformations of each other. This led him to weaken Kaplansky's conjecture, stating that up to quasi-isomorphisms there should only be a finite number of Hopf algebras of a given dimension. This was disproved in [EG] for a family of Hopf algebras of dimension 32. However, our results support Masuoka's conjecture in a really big class of * This work will be part of the author's Ph.D. thesis written under the supervision of Professor H.-J. Schneider. The author is a member of the Graduiertenkolleg "Mathematik im Bereich ihrer Wechselwirkung mit der Physik" at Munich University.

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Section: Introductionmentioning

confidence: 94%

“…Vertices with λ ij = 0 are called linked. In [AS3,Lemma 5.6. ] it is shown that a vertex can not be linked to two different vertices.…”

Section: Introductionmentioning

confidence: 99%

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Pointed Hopf Algebras and Quasi-isomorphisms

Didt

2005

Algebr Represent Theor

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“…Since A 3 is abelian, finite dimensional Nichols algebras over it are classified, there are 25 of them; this can be deduced from [8,Th. 1.3], [10,Th. 1.8].…”

Section: Theorem 15 If M ≥ 5 Then the Alternating Group A M Collapsesmentioning

confidence: 99%

Finite-dimensional pointed Hopf algebras with alternating groups are trivial

Andruskiewitsch

Fantino

Graña

et al. 2010

Annali di Matematica

172

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It is shown that Nichols algebras over alternating groups A m (m ≥ 5) are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes isomorphic to A m is isomorphic to the group algebra. In a similar fashion, it is shown that the Nichols algebras over the symmetric groups S m are all infinite-dimensional, except maybe those related to the transpositions considered in Fomin and Kirillov (Progr Math 172:146-182, 1999), and the class of type (2, 3) in S 5 . We also show that any simple rack X arising from a symmetric group, with the exception of a small list, collapse, in the sense that the Nichols algebra B(X, q) is infinite dimensional, q an arbitrary cocycle.

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“…The question of the classification of all co-Frobenius Hopf algebras whose coradical is a Hopf subalgebra arises naturally. The Lifting method [2][3][4][5] seems to be the pertinent tool to deal with this very hard problem.…”

Section: Example 31 Letmentioning

confidence: 99%