Iterated Hopf Ore extensions in positive characteristic

Abstract

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