Connected Hopf Algebras of Dimension $p^2$

Let $H$ be a finite-dimensional connected Hopf algebra over an algebraically closed field $\field$ of characteristic $p>0$. 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 $\field$. Finally, we classify all connected Hopf algebras of dimension $p^2$ over $\field$.