Kummer-Witt-Jackson algebras

TL;DR

This paper introduces the Jackson algebra, a non-commutative polynomial identity algebra, exploring its ring-theoretic and geometric properties, and showing it parametrizes Brauer classes over extensions of the base.

Contribution

It constructs and analyzes a new example of a polynomial identity algebra, the Jackson algebra, from ring-theoretic and geometric perspectives, with implications for non-commutative geometry.

Findings

Jackson algebra is a non-commutative family of central simple algebras

It parametrizes Brauer classes over extensions of the base

Provides a foundation for future studies in non-commutative arithmetic geometry

Abstract

This paper is concerned with the construction of a small, but non-trivial, example of a polynomial identity algebra, which we call the \emph{Jackson algebra}, that will be used in sequels to this paper to study non-commutative arithmetic geometry. In this paper this algebra is studied from a ring-theoretic and geometric viewpoint. Among other things it turns out that this algebra is a "non-commutative family" of central simple algebras and thus parametrises Brauer classes over extensions of the base.