On Noetherian affine prime regular Hopf algebras of Gelfand-Kirillov dimension 1

Let $k$ be an algebraically closed field. In 2007, D.-M. Lu, Q.-S. Wu, and J. J. Zhang asked the following question: Besides the group algebras $k\mathbb{Z},\;k\mathbb{D}$ and infinite dimensional prime Taft algebras, are there other noetherian affine prime regular Hopf algebras of GK-dimension 1? In this paper, we give a new one. Another problem posed by Lu, Wu, and Zhang can also be resolved by this example. Assuming $H$ is a noetherian affine prime regular Hopf algebra of GK-dimension 1, we show that gr $H:= \bigoplus _{s\geq 0}J_{iq}^{s}/J_{iq}^{s+1}$, as a Hopf algebra, is isomorphic to an infinite dimensional prime Taft algebra. This gives a characterization of infinite dimensional prime Taft algebras.