Iterated Hopf Ore extensions in positive characteristic

Iterated Hopf Ore extensions (IHOEs) over an algebraically closed base field $\Bbbk$ of positive characteristic $p$ are studied. We show that every IHOE satisfies a polynomial identity (PI), with PI-degree power $p$, and it is filtered deformation commutative ring. classify all $2$-step IHOEs $\Bbbk$, thus generalising the classification $2$-dimensional connected unipotent algebraic groups $\Bbbk$. Further properties described: for example their simple modules classified, shown to possess large center hence analog restricted enveloping algebra Lie $\Bbbk$-algebra. As one number questions listed, we propose such may exist