Ozone groups and centers of two-parameter quantum Heisenberg algebras
Abstract
Let p,q be roots of unity with pq≠1, and put ℓ=|⟨ p,q⟩| and t=ord(pq). Up to isomorphism, the two-parameter quantum Heisenberg algebras ℋp,q are exactly the noetherian PI graded down-up algebras, so everything here holds for that whole class. The ozone group is μℓ×μℓ/t, generated by conjugation by three normal elements, and the center is recovered from the fixed ring of the ozone group as the invariants of a cyclic group of order t. We give a presentation of the center as a hypersurface over a cyclic quotient surface singularity. It is regular exactly when $1$ is a characteristic root, which disproves a conjecture of Chan, Gaddis, Won and Zhang, and Gorenstein exactly when pm/h=qn/h, where m=ord(p), n=ord(q) and h=gcd(m,n); for p=ζ5, q=ζ52 it is not. We also decide when the fixed ring of the ozone group is Gorenstein and when ℋp,q is Calabi--Yau, and show that the algebras with regular center have non-affine automorphisms and nonzero locally nilpotent derivations.
Related
- Isomorphic classification of the coordinate algebras of the two-parameter braided quantum groups SL2.
- The Knot Invariant Associated to Two-Parameter Quantum Algebras II
- Quantum groups of Lie colour algebras fulfilling the Cartan-Weyl paradigm
- Quasi-quantum groups arising from cotensor coquasi-Hopf algebras
- Generalized Ricci Solitons on Heisenberg and Oscillator Algebras
BibTeX
@misc{gomez2026ozone,
title = {{Ozone groups and centers of two-parameter quantum Heisenberg algebras}},
author = {James Gómez and Héctor Suárez and Helbert Venegas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31781},
eprint = {2610.31781},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31781},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Ozone groups and centers of two-parameter quantum Heisenberg algebras
AU - James Gómez
AU - Héctor Suárez
AU - Helbert Venegas
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31781
UR - https://arxiv.org/abs/2610.31781
AB - Let p,q be roots of unity with pq≠1, and put ℓ=|⟨ p,q⟩| and t=ord(pq). Up to isomorphism, the two-parameter quantum Heisenberg algebras ℋ_p,q are exactly the noetherian PI graded down-up algebras, so everything here holds for that whole class. The ozone group is μ_ℓ×μ_ℓ/t, generated by conjugation by three normal elements, and the center is recovered from the fixed ring of the ozone group as the invariants of a cyclic group of order t. We give a presentation of the center as a hypersurface over a cyclic quotient surface singularity. It is regular exactly when $1$ is a characteristic root, which disproves a conjecture of Chan, Gaddis, Won and Zhang, and Gorenstein exactly when p^m/h=q^n/h, where m=ord(p), n=ord(q) and h=gcd(m,n); for p=ζ_5, q=ζ_5^2 it is not. We also decide when the fixed ring of the ozone group is Gorenstein and when ℋ_p,q is Calabi--Yau, and show that the algebras with regular center have non-affine automorphisms and nonzero locally nilpotent derivations.
ER -