Ozone groups and centers of two-parameter quantum Heisenberg algebras

Abstract

Let $p,q$ be roots of unity with $pq\ne 1$, and put $\ell =|⟨p,q⟩|$ 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.