On the classification of finite-dimensional pointed Hopf algebras

TL;DR

This paper classifies a specific class of finite-dimensional pointed Hopf algebras with abelian group-like elements, revealing their structure as deformations of generalized small quantum groups.

Contribution

It provides a complete classification of finite-dimensional pointed Hopf algebras with abelian group-like elements and prime divisors greater than 7, linking them to generalized small quantum groups.

Findings

Classified all such Hopf algebras under the given conditions.

Showed these algebras are deformations of generalized small quantum groups.

Provided an axiomatic framework for understanding these quantum groups.

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.