  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.
