The isotropy group of a derivation on a Danielewski-type algebra

2025/10/08 by Abdessamad Ahouita, Ahouita, Abdessamad, Rene Baltazar +5
Computer Science · Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Holomorphic and Operator Theory #Matrix Theory and Algorithms #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2510.07059

openalex publication_date 2025/10/08 · openalex created_date 2025/10/13 · openalex updated_date 2026/07/28

Abstract

Given an algebraically closed field k of characteristic zero, we consider in this paper k-algebras of the form A_c,q=k[x,y,z]/(c(x)z-q(x,y)), where c(x)∈ k[x] is a polynomial of degree at least two and q(x,y)∈ k[x,y] is a quasi-monic polynomial of degree at least two with respect to y. We give a complete description of the k-automorphism group of A_c,q as an abstract group. Moreover, for every non-locally nilpotent k-derivation δ of A_c,q we prove that the isotropy group of δ is a linear algebraic group of dimension at most three.

Citations

• Isotropy subgroups of some almost rigid domains
• A characterization of local nilpotence for dimension two polynomial derivations
• On Isotropy Group of Danielewski Surfaces
• On the Automorphism Group of a Polynomial Differential Ring in Two Variables
• On the geometry of the automorphism groups of affine varieties
• Classification(s) of Danielewski hypersurfaces

Related

• On the Isotropy Groups of Non-Invertible Simple Derivations
• On the isotropy group of a simple derivation
• The group of automorphisms of the Lie algebra of derivations of a polynomial algebra
• Isomorphism problems and groups of automorphisms for Ore extensions K[x][y; δ] (zero characteristic)
• On automorphisms of Danielewski surfaces