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", Journal of Pure and Applied Algebra, 219 (6), 2015].

Model theory and Kähler geometry

“…The assumption in Hrushovski's theorem of no nonconstant differential rational functions to the constants translates to the D-variety being "δ-rational" in the following terminology of [3]. The conclusion of Hrushovski's theorem, that of having only finitely many differential hypersurfaces, becomes the statement that (V, s) has only finitely many Dsubvarieties over k of codimension one, or expressed algebraically, that (k[V ], δ) has only finitely many height one prime differential ideals.…”

“…For example, if a point a ∈ V (k) is periodic but not fixed then it has finite but nontrivial orbit under φ, and that 2 That every complete non-locally-modular minimal type is nonorthogonal to a nonisolated minimal type over the empty set. 3 It is is possible to work with dominant rational self-maps instead but the statements would have to be modified somewhat, and we stick to this context for the sake of economy of exposition.…”

“…That is, δ is an a-coderivation on R. Geometrically this means that that (G, s) is not necessarily a D-group, but only an "a-twisted D-group". The extension of Theorem 5.1 from D-groups to a-twisted D-groups is one of the technically challenging aspects of [3], but the approach is the same: one shows that every D-subvariety of an a-twisted D-group over the constants is compound isotrivial (though now in at most five, rather than three, steps) and hence δ-rationality implies δ-local-closedness.…”

“…In the last five years a small body of work has arisen around a certain application of model theory to algebra. It appears in the papers [2,3,19,21] authored by, in various combinations, Jason Bell, Stéphane Launois, Omar León Sánchez, and myself. It is my intention in this article to survey that work, but also to trace its model-theoretic roots and to discuss other related recent developments.…”

On the Dixmier–Moeglin equivalence for Poisson–Hopf algebras

“…While we currently do not have an answer, in Remark 5.6(2) below we suggest how one could address this question. We note that in the differential-Hopf algebra context in a single derivation the cocommutative assumption can indeed be removed, see [2,Theorem 2.19], and also that Bell and Leung have asked a similar question in the noncommutative setting, see [3, Conjecture 1.3].…”