Non-affine Hopf algebra domains of Gelfand-Kirillov dimension two
  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. 
K.R. Goodearl and J.J. Zhang
math.RA