$D$-groups and the Dixmier–Moeglin equivalence

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 in the sense of Brown et al., “Connected Hopf algebras and iterated Ore extensions”, J. Pure Appl. Algebra 219:6 (2015).