On the isotropy group of a simple derivation

Abstract

Let R=K[X1,…,Xn] be a polynomial ring in n variables over a field K of characteristic zero and d a K-derivation of R. Consider the isotropy group of d: Aut(R)d:={ρ∈AutK(R)|ρdρ−1=d}. In his doctoral thesis ([1]), Baltazar proved that if d is a simple Shamsuddin derivation of K[X1,X2], 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 ([13]) generalized this result to certain derivations of K[X1,X2] proving that, under certain hypothesis, a derivation of K[X1,X2] 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[X1,…,Xn] is trivial. We also calculate other isotropy groups of (not necessarily simple) derivations of K[X1,X2].