On the Isotropy Groups of Non-Invertible Simple Derivations

2025/07/21 by Mishra, Sumit Chandra, Mondal, Dibyendu, Shukla, Pankaj
#12H05 #13N15 #13P05 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.15525

Abstract

Let k be a field of characteristic zero, and let i and n be positive integers with i≥ 2 and n>i. Consider a non-invertible k-derivation d_i of the polynomial ring k[x₁,…,x_i]. Let d_n be an extension of d_i to a derivation of k[x₁,…, x_n] such that d_n(x_j)∈ k[x_j-1]∖ k for each j with i+1 ≤ j≤ 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.

Citations

• On polynomial automorphisms commuting with a simple derivation
• On Isotropy Group of Danielewski Surfaces
• On Shamsuddin derivations and the isotropy groups
• On the isotropy group of a simple derivation
• On plane polynomial automorphisms commuting with simple derivations

Related

• On the isotropy group of a simple derivation
• A Class of simple derivations of polynomial ring k[x₁,x₂, … ,x_n]
• On the tame isotropy group of a derivation
• The isotropy group of a derivation on a Danielewski-type algebra
• Isotropy group of Lotka-Volterra derivations