On the isotropy of differential Ore extensions

TL;DR

This paper investigates the automorphism group actions on derivations of differential Ore extensions, providing explicit descriptions and criteria for isotropy in both square-free and singular cases.

Contribution

It explicitly describes the automorphism group of differential Ore extensions and characterizes the isotropy groups of derivations, including new phenomena in singular cases.

Findings

Explicit description of automorphism groups for deg(h) >= 1

Criteria for isotropy of derivations in square-free case

New phenomena in isotropy behavior in singular case

Abstract

Let Ah = k[x][t; d] be the differential Ore extension. We study the action of the automorphism group of Ah on the derivations of Ah and explicitly describe, using Nowicki's decomposition of the derivations of Ah, the isotropy groups of this action. More precisely, we first obtain an explicit description of the automorphism group of Ah for deg(h) >= 1. Then we determine the isotropy groups of derivations of the form D = ad_w + Delta_s(x), which exhaust all derivations in the square-free case, that is, when gcd(h,h') = 1. In the singular case, where gcd(h,h') is not equal to 1 and special derivations of type EH appear, we show that the isotropy problem is governed by a suitable localization and by the element w* = w + psi^(-1)H, where psi = gcd(h,h'). This yields a general criterion for the isotropy of a derivation of the form D = ad_w + EH + Delta_s(x). Finally, we provide explicit examples illustrating the new phenomena that arise in this setting.