On Isotropy Groups of Quantum Weyl Algebras and Jordanian Plane

Let δ be a derivation in a K-algebra R and let Autδ(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 Autδ(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.