Isotropy Groups of $σ$-Derivations on the Quantum Plane

Abstract

Let k be an algebraically closed field of characteristic zero and let k_q[x,y] be the quantum plane. We study sigma-derivations of k_q[x,y] and their isotropy groups under the conjugation action of automorphisms. For q\neq\pm1, we use Jordan's recent classification of skew derivations for toric automorphisms, which generalizes the description of Almulhem and Brzeziński for the quantum plane. Using this classification, we determine the isotropy groups of arbitrary sigma-derivations. These groups are described by character equations on the torus k^2, reducing the problem to arithmetic conditions. We recover the ordinary derivation case when sigma=id and exhibit new phenomena for nontrivial sigma-derivations, including cases where q is a root of unity. We also analyze the singular case q=-1. In this setting, we classify the sigma-derivations and describe the corresponding isotropy groups. In particular, for sigma=id, we obtain an explicit description of the isotropy groups of ordinary derivations of k_q[x,y], completing the singular case left open in previous work \cite{SBVA}.