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arXiv · 0812.4628

Finite-dimensional pointed Hopf algebras with alternating groups are trivial

N. Andruskiewitsch·F. Fantino·M. Graña·L. Vendramin
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Abstract

It is shown that Nichols algebras over alternating groups A_m, m>4, 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 [FK], 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 of (X,q) is infinite dimensional, for q an arbitrary cocycle. arXiv:0904.3978 is included here.

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N. Andruskiewitsch, F. Fantino, M. Graña, L. Vendramin. 2008-12-25. Finite-dimensional pointed Hopf algebras with alternating groups are trivial. https://doi.org/10.1007/s10231-010-0147-0

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