Classification and Isotropy of $\sigma$-Derivations on the Quantum Plane

TL;DR

This paper classifies sigma-derivations on the quantum plane, analyzes their isotropy groups, and explores special cases including roots of unity and the singular case q=-1, extending prior classifications.

Contribution

It extends the classification of sigma-derivations on the quantum plane to include non-inner parts and analyzes their isotropy groups, especially in special cases.

Findings

Classified all sigma-derivations for q ≠ ±1.

Determined isotropy groups via character equations on the algebraic torus.

Described automorphism groups and sigma-derivations at q=-1.

Abstract

We study sigma-derivations of the quantum plane and their isotropy groups under the conjugation action of automorphisms. For the case where q is different from plus or minus one, we classify all sigma-derivations for an arbitrary automorphism of the quantum plane. This classification decomposes each sigma-derivation into an inner part and explicit non-inner families, extending the classification of Almulhem and Brzezinski 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 algebraic torus, reducing the problem to arithmetic conditions. We recover the ordinary derivation case when sigma is the identity, and we exhibit new phenomena for nontrivial sigma-derivations, including cases where q is a root of unity. We also analyze the singular case q equals minus one. In this setting, the automorphism group contains both toric and flip automorphisms. We classify the corresponding sigma-derivations and describe their isotropy groups. In particular, when sigma is the identity, we obtain an explicit description of the isotropy groups of ordinary derivations of the quantum plane at q equals minus one, completing the singular case left open in previous work.