Pointed Hopf algebras of finite corepresentation type and their classifications

Let $k$ be an algebraically closed field. The main goal of this paper is to classify the finite-dimensional pointed Hopf algebras over $k$ of finite corepresentation type. To do so, we give a necessary and sufficient condition for a basic Hopf algebra over $k$ to be of finite representation type firstly. Explicitly, we prove that a basic Hopf algebra over $k$ is of finite representation type if and only if it is Nakayama. By this conclusion, we classify all finite-dimensional pointed Hopf algebras over $k$ of finite corepresentation type.