Isotropy group of Lotka-Volterra derivations

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 $n\ge 5$, 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 $n\ge 3,\text{and}{C}_i=1$, we have shown that the isotropy group of $d$ is isomorphic to the dihedral group of order $2n$.