Finite-dimensional Hopf Algebras over the Smallest Non-pointed Basic Hopf Algebra

Let K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal{K}$$\end{document} be the smallest dimensional non-pointed basic Hopf algebra. We determine all finite-dimensional Nichols algebras in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}<^>{\cal{K}}_{\cal{K}}\cal{Y}\cal{D}$$\end{document} and compute the liftings. The result shows that there are some new Nichols algebras of non-diagonal type and new finite-dimensional Hopf algebras without the dual Chevalley property.