Isotropy Groups of $σ$-Derivations on the Quantum Plane

Let $\Bbbk$ be an algebraically closed field of characteristic zero, and let $\mathcal A_q=\Bbbk_q[x,y]$ be the quantum plane, with $q\in\Bbbk^\ast\setminus\{1\}$. For a fixed automorphism $σ$ of $\mathcal A_q$, we study $σ$-derivations and their isotropy groups under the conjugation action of the c