Classification of connected Hopf algebras of dimension p3p3 I

Let p be a prime, and k be an algebraically closed field of characteristic p  . In this paper, we provide the classification of connected Hopf algebras of dimension p3p3, except for the case when the primitive space of the Hopf algebra is a two-dimensional abelian restricted Lie algebra. Each isomorphism class is presented by generators x, y, z with relations and Hopf algebra structures. Let μ   be the multiplicative group of (p2+p−1)(p2+p−1)-th roots of unity. When the primitive space is one-dimensional and p   is odd, there is an infinite family of isomorphism classes, which is naturally parameterized by Ak1/μ.