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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.20832 |
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| _version_ | 1866915084936151040 |
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| author | Rewri, Himanshu Kour, Surjeet |
| author_facet | Rewri, Himanshu Kour, Surjeet |
| contents | 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 $2n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20832 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Isotropy group of Lotka-Volterra derivations Rewri, Himanshu Kour, Surjeet Rings and Algebras 13N15, 13P05 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 $2n$. |
| title | Isotropy group of Lotka-Volterra derivations |
| topic | Rings and Algebras 13N15, 13P05 |
| url | https://arxiv.org/abs/2412.20832 |