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PreprintCC BY-SA 4.0arXiv

Isotropy Groups of σ-Derivations on the Quantum Plane

  • R. Baltazar
  • R. Cavalheiro
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Abstract

Let Bbbk be an algebraically closed field of characteristic zero, and let 𝒜q=Bbbkq[x,y] be the quantum plane, with q∈Bbbk∗∖{1}. For a fixed automorphism σ of 𝒜q, we study σ-derivations and their isotropy groups under the conjugation action of the centralizer of σ in Aut(𝒜q). For q≠±1, we use Jordan's classification of skew derivations for toric automorphisms, which generalizes the description of Almulhem and Brzeziński for the quantum plane, to describe these groups by character equations on the torus (Bbbk∗)2, obtaining an arithmetic criterion for finiteness. In this case, we also prove that, for every fixed σ, every finite subgroup of the torus occurs, up to isomorphism, as the isotropy group of a σ-derivation, including when q is a root of unity. For q=−1, we classify the σ-derivations for both toric and flip automorphisms σ and describe their isotropy groups. In particular, we explicitly determine the isotropy groups of ordinary derivations of 𝒜−1.

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Cite

BibTeXRIS
BibTeX
@misc{baltazar2026isotropy,
  title = {{Isotropy Groups of $σ$-Derivations on the Quantum Plane}},
  author = {R. Baltazar and R. Cavalheiro},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.20516},
  eprint = {2605.20516},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.20516},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Isotropy Groups of σ-Derivations on the Quantum Plane
AU  - R. Baltazar
AU  - R. Cavalheiro
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2605.20516
UR  - https://arxiv.org/abs/2605.20516
AB  - Let Bbbk be an algebraically closed field of characteristic zero, and let 𝒜_q=Bbbk_q[x,y] be the quantum plane, with q∈Bbbk^∗∖{1}. For a fixed automorphism σ of 𝒜_q, we study σ-derivations and their isotropy groups under the conjugation action of the centralizer of σ in Aut(𝒜_q). For q≠±1, we use Jordan's classification of skew derivations for toric automorphisms, which generalizes the description of Almulhem and Brzeziński for the quantum plane, to describe these groups by character equations on the torus (Bbbk^∗)^2, obtaining an arithmetic criterion for finiteness. In this case, we also prove that, for every fixed σ, every finite subgroup of the torus occurs, up to isomorphism, as the isotropy group of a σ-derivation, including when q is a root of unity. For q=−1, we classify the σ-derivations for both toric and flip automorphisms σ and describe their isotropy groups. In particular, we explicitly determine the isotropy groups of ordinary derivations of 𝒜_−1.
ER  -