D-groups and the Dixmier-Moeglin equivalence

TL;DR

This paper establishes a differential-algebraic analogue of the Dixmier-Moeglin equivalence for D-groups over constants, linking model theory and algebraic structures, with applications to Hopf algebras and Ore extensions.

Contribution

It introduces a differential-algebraic analogue of the Dixmier-Moeglin equivalence and proves it for D-groups over constants, connecting model theory with algebraic structures.

Findings

D-groups over constants satisfy the differential-algebraic Dixmier-Moeglin equivalence.

Ore extensions extending Hopf algebra structures satisfy the classical Dixmier-Moeglin equivalence.

All such Ore extensions are Hopf Ore extensions.

Abstract

A differential-algebraic geometric analogue of the Dixmier-Moeglin equivalence is articulated, and proven to hold for $D$-groups over the constants. The model theory of differentially closed fields of characteristic zero, in particular the notion of analysability in the constants, plays a central role. As an application it is shown that if $R$ is a commutative affine Hopf algebra over a field of characteristic zero, and $A$ is an Ore extension to which the Hopf algebra structure extends, then $A$ satisfies the classical Dixmier-Moeglin equivalence. Along the way it is shown that all such $A$ are Hopf Ore extensions