Iterated Hopf Ore extensions in positive characteristic

Abstract

Iterated Hopf Ore extensions (IHOEs) over an algebraically closed base field k of positive characteristic p are studied. We show that every IHOE over k satisfies a polynomial identity, 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 k, thus generalising the classification of 2-dimensional connected unipotent algebraic groups over 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 k-algebra. As one of a number of questions listed, we propose that such a restricted Hopf algebra may exist for every IHOE over k.