On the Isotropy Groups of Non-Invertible Simple Derivations
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arXiv
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| Format: | Preprint |
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2025
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| author | Mishra, Sumit Chandra Mondal, Dibyendu Shukla, Pankaj |
| author_facet | Mishra, Sumit Chandra Mondal, Dibyendu Shukla, Pankaj |
| contents | Let $k$ be a field of characteristic zero, and let $i$ and $n$ be positive integers with $i\geq 2$ and $n>i$. Consider a non-invertible $k$-derivation $d_i$ of the polynomial ring $k[x_1,\ldots,x_i]$. Let $d_n$ be an extension of $d_i$ to a derivation of $k[x_1,\ldots, x_n]$ such that $d_n(x_j)\in k[x_{j-1}]\setminus k$ for each $j$ with $i+1 \leq j\leq n$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15525 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Isotropy Groups of Non-Invertible Simple Derivations Mishra, Sumit Chandra Mondal, Dibyendu Shukla, Pankaj Commutative Algebra 13N15, 12H05, 13P05 Let $k$ be a field of characteristic zero, and let $i$ and $n$ be positive integers with $i\geq 2$ and $n>i$. Consider a non-invertible $k$-derivation $d_i$ of the polynomial ring $k[x_1,\ldots,x_i]$. Let $d_n$ be an extension of $d_i$ to a derivation of $k[x_1,\ldots, x_n]$ such that $d_n(x_j)\in k[x_{j-1}]\setminus k$ for each $j$ with $i+1 \leq j\leq n$. 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. |
| title | On the Isotropy Groups of Non-Invertible Simple Derivations |
| topic | Commutative Algebra 13N15, 12H05, 13P05 |
| url | https://arxiv.org/abs/2507.15525 |