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 charactersitic zero and d a K-derivation of R. Consider the isotropy group if d: Aut(R)d :=\ ∈ AutK(R)|\; d -1=d\. In his doctoral thesis, 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 generalized this result to an arbitrary derivation of K[X1,X2] proving that 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] and prove that they are finite cyclic groups.