D-groups and the Dixmier–Moeglin equivalence

Bell, Jason; León Sánchez, Omar; Moosa, Rahim

doi:10.2140/ant.2018.12.343

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$D$-groups and the Dixmier–Moeglin equivalence

Jason Bell, Omar León Sánchez and Rahim Moosa

Vol. 12 (2018), No. 2, 343–378

DOI: 10.2140/ant.2018.12.343

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).

Keywords

$D$-groups, model theory of differentially closed fields, Dixmier–Moeglin equivalence, Hopf Ore extensions

Mathematical Subject Classification 2010

Primary: 03C98

Secondary: 12H05, 16S36, 16T05

Milestones

Received: 30 November 2016

Revised: 29 September 2017

Accepted: 30 October 2017

Published: 13 May 2018

Authors

Jason Bell
Department of Pure Mathematics University of Waterloo Waterloo Canada

Omar León Sánchez
School of Mathematics University of Manchester Manchester United Kingdom

Rahim Moosa
Department of Pure Mathematics University of Waterloo Waterloo Canada

Vol. 12, No. 2, 2018