On the classification of of finite-dimensional pointed Hopf algebras

Abstract

We classify finite-dimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elements G (A) is abelian such that all prime divisors of the order of G.(A) are > 7. Since these Hopf algebras turn out to be deformations of a natural class of generalized small quantum groups, our result can be read as an axiomatic description of generalized small quantum groups.