On Isotropy Groups of Quantum Weyl Algebras and Jordanian Plane

Let $\delta$ be a derivation in a $K$-algebra $R$ and let $Au{t}_{\delta }(R)$ be the isotropy group with respect to the natural conjugation action of $Aut(R)$ of $K$-automorphisms on the set $Der(R)$ of $K$-derivations: that is, the subgroup of automorphisms that commute with the derivation. We explore the characterization of $Au{t}_{\delta }(R)$ for quantum Weyl algebras and we prove that in the case of the Jordanian plane, with the inner part defined by a monomial, it is in general a subgroup of ${\mathbb{Z}}_t$. Furthermore, we obtain a necessary and sufficient condition for an automorphism to be in the isotropy group of any inner derivation in the Jordanian Plane.