arXiv:2610.31781v1 Announce Type: new Abstract: Let $p,q$ be roots of unity with $pq\neq1$, and put $\ell=|\langle p,q\rangle|$ and $t=\ord(pq)$. Up to isomorphism, the two-parameter quantum Heisenberg algebras $\mathcal H_{p,q}$ are exactly the noetherian PI graded down-up algebras, so everything here holds for that whole class. The ozone group is $\mu_\ell\times\mu_{\ell/t}$, 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 $p=\zeta_5$, $q=\zeta_5^{2}$ it is not. We also decide when the fixed ring of the ozone group is Gorenstein and when $\mathcal H_{p,q}$ is Calabi--Yau, and show that the algebras with regular center have non-affine automorphisms and nonzero locally nilpotent derivations.
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