Finite duals of affine prime regular Hopf algebras of GK-dimension one

This paper is an attempt to construct a special kind of Hopf pairing $⟨-,-⟩:H^{\bullet }\otimes H\to$. Specifically, $H^{\bullet }$ and $H$ should be both affine, noetherian and of the same GK-dimension. In addition, some properties of them would be dual to each other. We test the ideas in two steps for all the affine prime regular Hopf algebras $H$ of GK-dimension one: 1) We compute the finite duals $H^{\circ }$ of them, which are given by generators and relations; 2) the Hopf pairings desired are determined by choosing certain Hopf subalgebras $H^{\bullet }$ of $H^{\circ }$, where $⟨-,-⟩$ becomes the evaluation.