On the Tame Isotropy Group of Locally Finite Derivations of K[X,Y]

TL;DR

This paper investigates the structure of the tame isotropy group of locally finite derivations of the polynomial ring in two variables over an algebraically closed field, revealing that it coincides with the group of automorphisms fixing the exponential automorphism.

Contribution

It explicitly determines the tame isotropy groups for each normal form of derivations and proves their equality with the groups fixing the exponential automorphism.

Findings

Tame isotropy groups are explicitly computed for each normal form.

These groups always coincide with the automorphisms fixing the exponential automorphism.

The equality contrasts with the behavior of the full automorphism group.

Abstract

Let K be an algebraically closed field of characteristic zero. We study the tame isotropy group Tame_D(K[X,Y]) of locally finite derivations of the polynomial ring K[X,Y], using Van den Essen's classification up to conjugation. For each normal form, we explicitly determine the corresponding tame isotropy group. We then compare Tame_D(K[X,Y]) with the tame isotropy group of the associated exponential automorphism exp(D), and prove that these groups always coincide. This stands in contrast to the behaviour of the full automorphism group, where such an equality may fail for derivations with a nontrivial semisimple part.