Hopf algebras with the dual Chevalley property of discrete corepresentation type

We try to classify Hopf algebras with the dual Chevalley property of discrete corepresentation type over an algebraically closed field $\mathbb{k}$ with characteristic 0. For such Hopf algebra $H$, we characterize the link quiver of $H$ and determine the structures of the link-indecomposable component $H_{(1)}$ containing $\mathbb{k}1$. Besides, we construct an infinite-dimensional non-pointed non-cosemisimple link-indecomposable Hopf algebra $H(e_{\pm 1},f_{\pm 1},u,v)$ with the dual Chevalley property of discrete corepresentation type.