Finite-dimensional pointed Hopf algebras with alternating groups are trivial

N. ANDRUSKIEWITSCH, F. FANTINO, M. GRAÑA, L. VENDRAMIN

ANNALI DI MATEMATICA PURA ED APPLICATA

SPRINGER HEIDELBERG

Lugar: Berlin; Año: 2010 vol. 190 p. 225 - 245

0373-3114

It is shown that Nichols algebras over alternating groups Am, m>= 5, are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes isomorphic to Am is isomorphic to the group algebra. In a similar fashion, it is shown that the Nichols algebras over the symmetric groups Sm 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 S5. 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.