Group-graded Ore extensions and graded simplicity
Abstract
We study Ore extensions R[t;σ,δ] of rings graded by an arbitrary group Γ. The grading of R extends with t of degree γ if and only if σ is compatible with conjugation by γ and δ with left translation by γ. The extended grading is then unique. For nonabelian Γ the degree of t need not commute with the degrees of R, and this twist cannot always be removed. If R is gr-simple and σ is an automorphism, then R[t;σ,δ] is gr-simple if and only if it has no homogeneous normal element that is monic of positive t-degree. The quantized Weyl algebra shows that the graded forms of the classical necessary conditions do not suffice. Over a group algebra kG, these extensions correspond to actions of G on the affine line, and gr-simplicity means that the action has no finite orbit. In characteristic zero this is the nonvanishing of a class in H1(G,kχ), and simplicity forces G to be metabelian. We prove the analogous results for Poisson polynomial rings and relate the two settings by semiclassical limits.
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BibTeX
@misc{mohamed2026groupgraded,
title = {{Group-graded Ore extensions and graded simplicity}},
author = {Yassine Ait Mohamed},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.38131},
eprint = {2609.38131},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38131},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Group-graded Ore extensions and graded simplicity
AU - Yassine Ait Mohamed
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2609.38131
UR - https://arxiv.org/abs/2609.38131
AB - We study Ore extensions R[t;σ,δ] of rings graded by an arbitrary group Γ. The grading of R extends with t of degree γ if and only if σ is compatible with conjugation by γ and δ with left translation by γ. The extended grading is then unique. For nonabelian Γ the degree of t need not commute with the degrees of R, and this twist cannot always be removed. If R is gr-simple and σ is an automorphism, then R[t;σ,δ] is gr-simple if and only if it has no homogeneous normal element that is monic of positive t-degree. The quantized Weyl algebra shows that the graded forms of the classical necessary conditions do not suffice. Over a group algebra kG, these extensions correspond to actions of G on the affine line, and gr-simplicity means that the action has no finite orbit. In characteristic zero this is the nonvanishing of a class in H^1(G,k_χ), and simplicity forces G to be metabelian. We prove the analogous results for Poisson polynomial rings and relate the two settings by semiclassical limits.
ER -