On pointed Hopf algebras associated with the symmetric groups

Andruskiewitsch, Nicolás; Fantino, Fernando; Zhang, Shouchuan

doi:10.1007/s00229-008-0237-0

Cited by 23 publications

(33 citation statements)

References 10 publications

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“…We now recall the notation on representations of the centralizer needed in the statement of Theorem 1.1. See [5,Sect. 2.2] for more details.…”

Section: Notations On Symmetric Groupsmentioning

confidence: 98%

“…Definition 2. 2 We shall say that a finite rack X collapses if for any finite faithful cocycle q (associated to any decomposition of X and of any degree n), dim B(X, q) = ∞.…”

Section: Cocyclesmentioning

confidence: 99%

“…2.2]; thus Q [1,1] A,T is denoted D (2) p . Therefore, the splitting technique includes (without having to resort to look for cocycles) the case of quasi-real orbits containing a dihedral subrack [3, Cor.…”

Section: Lemma 317mentioning

confidence: 99%

“…The case r = 1 of this Lemma is known (see for example[2, Prop. 2.4]) and it is used to kill the conjugacy class of involutions in A 4 .4.4 Proof of Theorem 1.1 Let σ ∈ A m ; if O A m σ is of type D, then O S m σ is of type D.Then Th.…”

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confidence: 96%

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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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“…We now recall the notation on representations of the centralizer needed in the statement of Theorem 1.1. See [5,Sect. 2.2] for more details.…”

Section: Notations On Symmetric Groupsmentioning

confidence: 98%

“…Definition 2. 2 We shall say that a finite rack X collapses if for any finite faithful cocycle q (associated to any decomposition of X and of any degree n), dim B(X, q) = ∞.…”

Section: Cocyclesmentioning

confidence: 99%

Section: Lemma 317mentioning

confidence: 99%

mentioning

confidence: 96%

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Finite-dimensional pointed Hopf algebras with alternating groups are trivial

Andruskiewitsch

Fantino

Graña

et al. 2010

Annali di Matematica

172

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“…Their results have appeared in a series of papers over the past 10 years; see for example [7][8][9] and capstone publication [10]. There are also some results for pointed Hopf algebras with nonabelian group of points; for a few examples see [3,4,30,31,51].…”

Section: Introductionmentioning

confidence: 96%

A Survey of Hopf Algebras of Low Dimension

Beattie

2008

Acta Appl Math

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This short survey is based on a talk on the classification of finite dimensional Hopf algebras given at Memorial University of Newfoundland in St. John's in May of 2008. This seminar was given in conjunction with the mini-course "Introduction to pointed Hopf algebras" offered by Professor Nicolás Andruskiewitsch which concentrated on classification methods and results for the pointed case. Here we will outline some low dimensional classifications for the general case.

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