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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.19295 |
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| _version_ | 1866908535797841920 |
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| author | De Santana, Adriano Baltazar, Rene Vinciguerra, Robson De Araujo, Wilian |
| author_facet | De Santana, Adriano Baltazar, Rene Vinciguerra, Robson De Araujo, Wilian |
| contents | This paper investigates the isotropy groups of derivations on the Quantum Plane $\Bbbk_q[x, y]$, defined by the relation $yx = qxy$, where $q \in \Bbbk^*$, with $q^2\neq 1$. The main goal is to determine the automorphisms of the Quantum Plane that commutes with a fixed derivation $δ$. We describe conditions under which the isotropy group $\text{Aut}_δ(A)$ is trivial, finite, or infinite, depending on the structure of $δ$ and whether $q$ is a root of unity: additionally, we present the structure of the group in the finite case. A key tool is the analysis of polynomial equations of the form $μ_1^a μ_2^b = 1$, arising from monomials in the inner part of $δ$. We also make explicit which finite subgroups of $Aut(\Bbbk_q[x, y])$ are isotropy groups of some derivation: either $q$ root of unity or not. Techniques from algebraic geometry, such as intersection multiplicity, are also employed in the classification of the finite case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19295 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Isotropy Groups of Quantum Plane De Santana, Adriano Baltazar, Rene Vinciguerra, Robson De Araujo, Wilian Rings and Algebras This paper investigates the isotropy groups of derivations on the Quantum Plane $\Bbbk_q[x, y]$, defined by the relation $yx = qxy$, where $q \in \Bbbk^*$, with $q^2\neq 1$. The main goal is to determine the automorphisms of the Quantum Plane that commutes with a fixed derivation $δ$. We describe conditions under which the isotropy group $\text{Aut}_δ(A)$ is trivial, finite, or infinite, depending on the structure of $δ$ and whether $q$ is a root of unity: additionally, we present the structure of the group in the finite case. A key tool is the analysis of polynomial equations of the form $μ_1^a μ_2^b = 1$, arising from monomials in the inner part of $δ$. We also make explicit which finite subgroups of $Aut(\Bbbk_q[x, y])$ are isotropy groups of some derivation: either $q$ root of unity or not. Techniques from algebraic geometry, such as intersection multiplicity, are also employed in the classification of the finite case. |
| title | On Isotropy Groups of Quantum Plane |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2505.19295 |