On Isotropy Groups of Quantum Plane

TL;DR

This paper classifies the automorphisms of the Quantum Plane that commute with a fixed derivation, revealing how the structure of these isotropy groups varies with the derivation and whether q is a root of unity.

Contribution

It provides a detailed description of isotropy groups of derivations on the Quantum Plane, including conditions for trivial, finite, or infinite groups, and characterizes finite subgroups explicitly.

Findings

Isotropy groups are trivial, finite, or infinite depending on the derivation and q.

Explicit structure of finite isotropy groups when q is a root of unity.

Identification of which finite subgroups can occur as isotropy groups.

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.