On Isotropy Groups of Quantum Plane

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.