On Isotropy Groups of Quantum Weyl Algebras and Jordanian Plane
Adriano de Santana, Rene Baltazar, Robson Vinciguerra, Wilian de Araujo
TL;DR
The paper investigates Aut_δ(R), the isotropy group of a derivation δ under conjugation by K-automorphisms, for two noncommutative algebras: the quantum Weyl algebra and the Jordanian plane. It provides explicit classifications of skew derivations and their innerness, and derives arithmetic criteria (in terms of gcds of exponents) that determine the isotropy groups, which are shown to be finite cyclic subgroups in many cases. In the Jordanian plane, the authors show that Aut_δ(Λ_2(K)) often lies in a finite cyclic subgroup determined by the degree of the associated ψ'(x), while also revealing large non-algebraic isotropy subgroups in certain inner-derivation scenarios and proving non-isomorphism with the first Weyl algebra. The work culminates in open questions about isotropy for Ore extensions and highlights the nuanced interplay between automorphisms and derivations in these quantum and noncommutative settings.
Abstract
Let $δ$ be a derivation in a $K$-algebra $R$ and let $Aut_δ(R)$ 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 $Aut_δ(R)$ 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.