On the Isotropy Groups of Non-Invertible Simple Derivations

TL;DR

This paper studies the symmetry groups of certain non-invertible derivations on polynomial rings, establishing conditions under which these groups resemble translation groups, thus advancing understanding of their algebraic structure.

Contribution

It provides a systematic analysis of isotropy groups for non-invertible simple derivations and identifies conditions for their conjugacy to translation subgroups.

Findings

Isotropy groups can be conjugate to translation groups under certain conditions.

Extension of derivations preserves specific symmetry properties.

Provides criteria for the structure of symmetry groups of non-invertible derivations.

Abstract

Let $k$ be a field of characteristic zero, and let $i$ and $n$ be positive integers with $i\geq 2$ and $n>i$. Consider a non-invertible $k$-derivation $d_i$ of the polynomial ring $k[x_1,\ldots,x_i]$. Let $d_n$ be an extension of $d_i$ to a derivation of $k[x_1,\ldots, x_n]$ such that $d_n(x_j)\in k[x_{j-1}]\setminus k$ for each $j$ with $i+1 \leq j\leq n$. In this article, we undertake a systematic study of the isotropy groups associated with such non-invertible derivations. We establish sufficient conditions on $d_i$ under which the isotropy group of the non-invertible simple derivation $d_n$ is conjugate to a subgroup of translations.