Pointed Hopf algebras of dimension $p^2q$ in characteristic $p$

Abstract

Let $\mathds{k}$ be an algebraically closed field of characteristic $p$. We give the complete classification of pointed Hopf algebras over $\mathds{k}$ of dimension $p^2q$ for a prime number $q$. The result shows that there are finitely many isomorphism classes, including 10 classes that are not generated by group-like elements and skew-primitive elements. In particular, there are many new examples of finite-dimensional pointed Hopf algebras.