On the presentation of pointed Hopf algebras

TL;DR

This paper provides a generator-and-relations presentation for a class of pointed Hopf algebras, including generalized quantum groups and Nichols algebras, enhancing their structural understanding and classification.

Contribution

It introduces a new presentation framework for pointed Hopf algebras generated by skew-primitive elements and abelian group-like elements, extending the understanding of their structure.

Findings

Provides a presentation for pointed Hopf algebras with skew-primitive generators

Derives an analog presentation for Nichols algebras of diagonal type

Enhances classification methods for quantum groups and related structures

Abstract

We give a presentation in terms of generators and relations of Hopf algebras generated by skew-primitive elements and abelian group of group-like elements with action given via characters. This class of pointed Hopf algebras has shown great importance in the classification theory and can be seen as generalized quantum groups. As a consequence we get an analog presentation of Nichols algebras of diagonal type.