The ozone groups of the algebras $B_q(f)$

Abstract

Let $q$ be a primitive $n$-th root of unity, $n>1$, and let $f$ be a nonzero polynomial such that $n\nmid(j+1)$ for every $j\in\supp(f)$. Set $e=\gcd(n,\{j+1:j\in\supp(f)\})$. We show that $\Oz(B_q(f))\cong\mu_e\times\mu_e$: the defining relations are homogeneous for a $\mathbb{Z}/e\times\mathbb{Z}/e$ grading, the center sits in degree zero, and the ozone group is the character group of that grading. The determination of the ozone group only requires the central elements $u^n$, $v^n$, and $\Omega$. The regular normal elements modulo the center form the same group, generated by $u^{n/e}$ and $v^{n/e}$, so every normal element is central exactly when $e=1$. For $f=t^2$ we recover a computation of Chan, Gaddis, Won and Zhang, and for $e>1$ we obtain an infinite family of Calabi--Yau algebras with nontrivial ozone group.