Finite-dimensional pointed Hopf algebras with alternating groups are trivial

TL;DR

This paper proves that finite-dimensional pointed Hopf algebras with alternating or symmetric groups as their group of group-likes are trivial, by showing their associated Nichols algebras are infinite-dimensional, except for specific known cases.

Contribution

It establishes that all Nichols algebras over alternating groups are infinite-dimensional, leading to the classification that such Hopf algebras are essentially trivial, with some exceptions for symmetric groups.

Findings

Nichols algebras over A_m are infinite-dimensional for m>4

Most Nichols algebras over S_m are infinite-dimensional, except specific cases

Simple racks from symmetric groups mostly lead to infinite-dimensional Nichols algebras

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.