Let $A=KQ/I$ be a finite dimensional quiver algebra with admissible relations $I$. $A$ can be made into a restricted Lie algebra over a field of characteristic $p$ via $[x,y]=xy-yx$ and $x^{p}=x^p$. The restricted enveloping algebra of $A$ is then defined as $u(A):=T_K(L)/I(L)$, with $T_K(L)$ the tensor algebra over $K$ and $I(L)$ the ideal generated by $[x,y]-x \otimes y + y \otimes x$ for all $x,y \in A$ and $x^{p}-x^p$ for all $x \in A$.
This makes $u(A)$ a cocommutative Hopf algebra.
Question: Can quiver and relations of $u(A)$ explicitly be described in terms of $A$? Maybe this works at least in the case of $A=KQ$ being hereditary.
For example let $A=KQ$ be the hereditary algebra of Dynkin type $A_2$ over a field with $p$ elements. The algebra $u(A)$ is given by $C=K<x,y,z>/(z-xz+zx,-xy+yx,-z-yz+zy,x-x^p,y-y^p,z^p)$ and should be a $p^3$ dimensional algebra with Center isomorphic to $K^p$ and $C/rad(C)$ being $p^2$ dimensional. Not sure what exactly the quiver and relations are.
