Commutative Hopf structures over a loop

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.