On Isotropy Groups of Quantum Weyl Algebras and Jordanian Plane

Let be a derivation in a K-algebra R and let Aut_{} (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 Aut_{} (R) for quantum Weyl algebras and we prove that in the case of the Jordanian plane it is always a finite cyclic group. Furthermore, we obtain an arithmetic characterization that allows us to trivially demonstrate that these algebras are not isomorphic.