Vol. 12, No. 2, 2018

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
$D$-groups and the Dixmier–Moeglin equivalence

Jason Bell, Omar León Sánchez and Rahim Moosa

Vol. 12 (2018), No. 2, 343–378
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