On Isotropy Groups of Quantum Weyl Algebras and Jordanian Plane

TL;DR

This paper investigates the structure of automorphisms that commute with derivations in quantum Weyl algebras and the Jordanian plane, revealing subgroup classifications and conditions for automorphisms to belong to isotropy groups.

Contribution

It characterizes the isotropy groups of derivations in quantum Weyl algebras and the Jordanian plane, providing new subgroup classifications and criteria for automorphisms.

Findings

Isotropy group in Jordanian plane is generally a subgroup of ry_t.

Necessary and sufficient conditions for automorphisms in isotropy groups.

Characterization of automorphisms commuting with derivations in quantum Weyl algebras.

Abstract

Let $\delta$ be a derivation in a $K$-algebra $R$ and let $Aut_{\delta}(R)$ 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 $Aut_{\delta}(R)$ 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.