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

Abstract

Let p, q, r be distinct prime numbers and k an algebraically closed field of characteristic p. We study the classification of pointed Hopf algebras over k of dimension p 2 q, pq 2 and pqr. We obtain a complete classification of pointed Hopf algebras of dimension pq, pqr, p 2 q, 2q 2 and 4p. We also classify all pointed Hopf algebras of dimension pq 2 whose diagrams are Nichols algebras and show that pointed Hopf algebras of dimension pq 2 whose diagrams are not Nichols algebras must be Hopf extensions of Taft algebras by restricted universal enveloping algebras of dimension p. In particular, we obtain many new non-commutative and non-cocommutative finite-dimensional pointed Hopf algebras.