Let $k$ be an algebraically closed field of characteristic $p>0$. For a loop
$\circlearrowleft$, denote its path coalgebra by $k\circlearrowleft$. In this
paper, all the finite-dimensional commutative Hopf algebras over the sub
coalgebras of $k\circlearrowleft$ are given. As a direct consequence, all the
commutative infinitesimal groups $\mathcal{G}$ with
dim$_{k}$Lie$(\mathcal{G})=1$ are classified.

In this paper we investigate pointed Hopf algebras via quiver methods. We
classify all possible Hopf structures arising from minimal Hopf quivers, namely
basic cycles and the linear chain. This provides full local structure
information for general pointed Hopf algebras.