On Isotropy Groups of Quantum Weyl Algebras and Jordanian Plane
We study isotropy groups of $σ$-derivations of the quantum Weyl algebra and of ordinary derivations of the Jordanian plane. For the quantum Weyl algebra $A_q^1(\Bbbk)$, with $q$ not a root of unity, we use Brzezinski's classification to decompose every $σ$-derivation into inner and non-inner stable components. This yields an intersection formula for the isotropy group of an arbitrary $σ$-derivation and leads to explicit arithmetic descriptions. For the Jordanian plane $Λ_2(\Bbbk)$, we give a necessary and sufficient condition for an automorphism to belong to the isotropy group of an inner derivation. We compute the isotropy groups of monomial inner derivations and of locally nilpotent derivations. These examples show that isotropy groups in the Jordanian plane may contain large triangular subgroups, unlike the quantum Weyl algebra. In this way, isotropy groups provide a natural invariant that reflects the structural difference between the Jordanian plane and the quantum Weyl algebra.