On the classification 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.

38 pages. The determination of all isomorphisms between the Hopf algebras presented here is given. The title is changed. Third version: several improvements on the exposition mainly following suggestions of the referee