Iterated Hopf Ore extensions in positive characteristic

Brown, K. A. and Zhang, J. J. (2022) Iterated Hopf Ore extensions in positive characteristic. Journal of Noncommutative Geometry, 16(3), pp. 787-837. (doi: 10.4171/jncg/453)

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 (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 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.

• Item Type: Articles
• Additional Information: Funding: K. A. Brown was supported by Leverhulme Emeritus Fellowship EM-2017- 081-9 and J. J. Zhang by the US National Science Foundation (Nos. DMS-1700825 and DMS-2001015)
• Keywords: Geometry and Topology, Mathematical Physics, Algebra and Number Theory
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Brown, Professor Ken
• Authors: Brown, K. A., and Zhang, J. J.
• College/School: College of Social Sciences
• Journal Name: Journal of Noncommutative Geometry
• Publisher: European Mathematical Society
• ISSN: 1661-6952
• ISSN (Online): 1661-6960
• Published Online: 13 September 2022
• Copyright Holders: Copyright © 2022 European Mathematical Society
• First Published: First published in Journal of Noncommutative Geometry 16(3): 787-837
• Publisher Policy: Reproduced under a Creative Commons License