Non-affine Hopf algebra domains of Gelfand-Kirillov dimension two

TL;DR

This paper classifies certain non-affine Hopf algebras over algebraically closed fields of characteristic zero, focusing on those with Gelfand-Kirillov dimension two and non-trivial extension groups, expanding previous affine classifications.

Contribution

It provides a complete classification of non-affine Hopf algebra domains of Gelfand-Kirillov dimension two with non-zero Ext^1, extending earlier affine case results.

Findings

Classification of all non-affine Hopf algebra domains of GK dimension two

Identification of conditions for non-trivial Ext^1 groups

Extension of affine classification results to non-affine cases

Abstract

We classify all non-affine Hopf algebras $H$ over an algebraically closed field $k$ of characteristic zero that are integral domains of Gelfand-Kirillov dimension two and satisfy the condition $\text{Ext}^1_H(k, k) \neq 0$. The affine ones were classified by the authors in 2010.