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On the Tame Isotropy Group of Locally Finite Derivations of K[X,Y]

Luis Cid, Marcelo Veloso

Abstract

Let K be an algebraically closed field of characteristic zero. We study the tame isotropy group Tame_D(K[X,Y]) of locally finite derivations of the polynomial ring K[X,Y], using Van den Essen's classification up to conjugation. For each normal form, we explicitly determine the corresponding tame isotropy group. We then compare Tame_D(K[X,Y]) with the tame isotropy group of the associated exponential automorphism exp(D), and prove that these groups always coincide. This stands in contrast to the behaviour of the full automorphism group, where such an equality may fail for derivations with a nontrivial semisimple part.

On the Tame Isotropy Group of Locally Finite Derivations of K[X,Y]

Abstract

Let K be an algebraically closed field of characteristic zero. We study the tame isotropy group Tame_D(K[X,Y]) of locally finite derivations of the polynomial ring K[X,Y], using Van den Essen's classification up to conjugation. For each normal form, we explicitly determine the corresponding tame isotropy group. We then compare Tame_D(K[X,Y]) with the tame isotropy group of the associated exponential automorphism exp(D), and prove that these groups always coincide. This stands in contrast to the behaviour of the full automorphism group, where such an equality may fail for derivations with a nontrivial semisimple part.

Paper Structure

This paper contains 12 sections, 16 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Generalities
  3. Locally finite derivations and their classification
  4. Jordan decomposition and the exponential automorphism
  5. The tame isotropy groups
  6. The linear derivations
  7. Tame isotropy groups for exponential automorphisms
  8. The triangular derivation $D=f(X)\frac{\partial}{\partial Y}$
  9. The derivation $D=\dfrac{\partial}{\partial X}+bY\dfrac{\partial}{\partial Y}$
  10. The derivation $D=aX\dfrac{\partial}{\partial X}+(amY+X^m)\dfrac{\partial}{\partial Y}$
  11. The linear derivations
  12. Comparison with the full automorphism group

Key Result

Lemma 2.2

Let $D\neq 0$ be a locally finite derivation on $\mathbb{K}[X,Y]$. Then there exists $\varphi\in\operatorname{Aut}(\mathbb{K}[X,Y])$ such that $\varphi D\varphi^{-1}$ is one of the following: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (33)

  • Example 2.1
  • Lemma 2.2: Corollary 4.7 of V92
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Theorem 3.2: Theorem 2.3 of freitas25
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • ...and 23 more