Isotropy group of Lotka-Volterra derivations

Himanshu Rewri; Surjeet Kour

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arXiv · 2412.20832

Isotropy group of Lotka-Volterra derivations

Himanshu Rewri·Surjeet Kour

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Abstract

In this paper, we study the isotropy group of Lotka-Volterra derivations of $K[x_{1},\cdots,x_{n}]$, i.e., a derivation $d$ of the form $d(x_{i})=x_{i}(x_{i-1}-C_{i}x_{i+1})$. If $n=3$ or $n \geq 5$, we have shown that the isotropy group of $d$ is finite. However, for $n=4$, it is observed that the isotropy group of $d$ need not be finite. Indeed, for $C_{i}=-1$, we observed an infinite collection of automorphisms in the isotropy group of $d$. Moreover, for $n \geq 3, ~~\text{and}~~C_{i}=1$, we have shown that the isotropy group of $d$ is isomorphic to the dihedral group of order $2n$.

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BibTeX RIS

Himanshu Rewri, Surjeet Kour. 2024-12-30. Isotropy group of Lotka-Volterra derivations. https://arxiv.org/abs/2412.20832

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