Connected Hopf algebras of dimension p2p2

Let H be a finite-dimensional connected Hopf algebra over an algebraically closed field k of characteristic p>0p>0. We provide the algebra structure of the associated graded Hopf algebra gr H. Then, we study the case when H is generated by a Hopf subalgebra K and another element and the case when H is cocommutative. When H is a restricted universal enveloping algebra, we give a specific basis for the second term of the Hochschild cohomology of the coalgebra H with coefficients in the trivial H-bicomodule k. Finally, we classify all connected Hopf algebras of dimension p2p2 over k.