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arXiv · 2505.19295

On Isotropy Groups of Quantum Plane

Adriano De Santana·Rene Baltazar·Robson Vinciguerra·Wilian De Araujo
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Abstract

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 $\delta$. We describe conditions under which the isotropy group $\text{Aut}_\delta(A)$ is trivial, finite, or infinite, depending on the structure of $\delta$ 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 $\mu_1^a \mu_2^b = 1$, arising from monomials in the inner part of $\delta$. 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.

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Adriano De Santana, Rene Baltazar, Robson Vinciguerra, Wilian De Araujo. 2025-05-25. On Isotropy Groups of Quantum Plane. https://arxiv.org/abs/2505.19295

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