On the isotropy group of a simple derivation

TL;DR

This paper extends the understanding of the isotropy groups of simple derivations in polynomial rings, proving triviality in higher dimensions and classifying finite cyclic groups for certain non-simple derivations.

Contribution

It generalizes previous results by proving the triviality of isotropy groups for simple Shamsuddin derivations in any dimension and classifies isotropy groups of some non-simple derivations.

Findings

Isotropy group of a simple Shamsuddin derivation in $K[X_1,...,X_n]$ is trivial.

Certain non-simple derivations of $K[X_1,X_2]$ have finite cyclic isotropy groups.

The results extend known classifications from two variables to higher dimensions.

Abstract

Let $R=K[X_1,\dots, X_n]$ be a polynomial ring in $n$ variables over a field $K$ of charactersitic zero and $d$ a $K$-derivation of $R$. Consider the isotropy group if $d$: $ \text{Aut}(R)_d :=\{\rho \in \text{Aut}_K(R)|\; \rho d \rho^{-1}=d\}$. In his doctoral thesis, Baltazar proved that if $d$ is a simple Shamsuddin derivation of $K[X_1,X_2]$, then its isotropy group is trivial. He also gave an example of a non-simple derivation whose isotropy group is infinite. Recently, Mendes and Pan generalized this result to an arbitrary derivation of $K[X_1,X_2]$ proving that a derivation of $K[X_1,X_2]$ is simple if, and only if, its isotropy group is trivial. In this paper, we prove that the isotropy group of a simple Shamsuddin derivation of the polynomial ring $R=K[X_1,\dots, X_n]$ is trivial. We also calculate other isotropy groups of (not necessarily simple) derivations of $K[X_1,X_2]$ and prove that they are finite cyclic groups.