On the isotropy group of a simple derivation

Abstract

Let R = K[X 1 , . . . , 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 := {ρ ∈ Aut K (R)| ρdρ −1 = d}. In his doctoral thesis ([1]), 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 ([12]) 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 , . . . , Xn] 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.

On Shamsuddin derivations and the isotropy groups

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In the paper, we give an affirmative answer to the conjecture in [2]. We prove that a Shamsuddin derivation D is simple if and only if Aut(K[x, y 1 , .

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“…In [2], L.N.Bertoncello and D.Levcovitz have proved that the isotropy group of simple Shamsuddin derivations is trivial. They also conjectured that if the isotropy group of a Shamsuddin derivation is trivial, then the Shamsuddin derivation is simple.…”