On isotropy group of locally finite derivations on $\mathbb{K}[X,Y]$

TL;DR

This paper investigates the symmetry groups of certain derivations on polynomial rings in two variables, revealing conditions under which these groups coincide or differ, and establishing that all nonzero derivations have nontrivial symmetry groups.

Contribution

It provides a detailed comparison of isotropy groups of locally finite derivations and their exponential automorphisms, extending understanding of their structure and properties.

Findings

Isotropy groups coincide for locally nilpotent derivations.

Isotropy groups may differ when the semisimple part is nontrivial.

Every nonzero locally finite derivation has a nontrivial isotropy group.

Abstract

In this paper, we study the isotropy groups of locally finite derivations of the polynomial ring $\mathbb{K}[X,Y]$, using Van den Essen's classification of locally finite derivations in two variables. We compare the isotropy group of a locally finite derivation with that of its associated exponential automorphism, showing that they coincide in the locally nilpotent case, whereas they may differ when the semisimple part is nontrivial. We also prove that every nonzero locally finite derivation has a nontrivial isotropy group.