Iterated Hopf Ore extensions in positive characteristic

TL;DR

This paper investigates iterated Hopf Ore extensions over algebraically closed fields of positive characteristic, revealing their polynomial identity properties, classification, and structural features, including their modules and centers.

Contribution

It classifies all 2-step IHOEs over k and explores their properties, extending the understanding of Hopf algebra structures in positive characteristic.

Findings

Every IHOE satisfies a polynomial identity with PI-degree a power of p.

All 2-step IHOEs are classified, generalizing unipotent algebraic groups.

2-step IHOEs have large Hopf centers and classified simple modules.

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.