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On isotropy group of locally finite derivations on $\mathbb{K}[X,Y]$

Luis Cid, Marcelo Veloso

Abstract

In this paper, we study the isotropy groups of locally finite derivations of the polynomial ring $\mathbb{K}[X,Y]$, using Van den Essen's classification of locally finite derivations in two variables. We compare the isotropy group of a locally finite derivation with that of its associated exponential automorphism, showing that they coincide in the locally nilpotent case, whereas they may differ when the semisimple part is nontrivial. We also prove that every nonzero locally finite derivation has a nontrivial isotropy group.

On isotropy group of locally finite derivations on $\mathbb{K}[X,Y]$

Abstract

In this paper, we study the isotropy groups of locally finite derivations of the polynomial ring , using Van den Essen's classification of locally finite derivations in two variables. We compare the isotropy group of a locally finite derivation with that of its associated exponential automorphism, showing that they coincide in the locally nilpotent case, whereas they may differ when the semisimple part is nontrivial. We also prove that every nonzero locally finite derivation has a nontrivial isotropy group.
Paper Structure (9 sections, 14 theorems, 66 equations)

This paper contains 9 sections, 14 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Locally finite endomorphisms and derivations
  4. Jordan decomposition and the exponential map
  5. Isotropy groups of locally finite derivations
  6. The linear case
  7. Isotropy groups for exponential automorphisms
  8. The exponential map for locally finite derivations
  9. Isotropy of $D$ and of $\exp(D)$

Key Result

Lemma 2.1

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

Theorems & Definitions (38)

  • Lemma 2.1: Corollary 4.7, V92
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 28 more