Quantum projective planes and Beilinson algebras of $3$-dimensional quantum polynomial algebras for Type S'

Let $A=\mathcal{A}(E,\sigma )$ be a $3$-dimensional quantum polynomial algebra where $E$ is ${\mathbb{P}}^2$ or a cubic divisor in ${\mathbb{P}}^2$, and $\sigma \in {\mathrm{A}\mathrm{u}\mathrm{t}}_{k}E$. Artin-Tate-Van den Bergh proved that $A$ is finite over its center if and only if the order $\mathrm{\mid }\sigma \mathrm{\mid }$ of $\sigma$ is finite. As a categorical analogy of their result, the author and Mori showed that the following conditions are equivalent; (1) $\mathrm{\mid }{\nu }^{\ast }{\sigma }^3\mathrm{\mid }<\mathrm{\infty }$, where $\nu$ is the Nakayama automorphism of $A$. (2) The norm $\mathrm{\parallel }\sigma \mathrm{\parallel }$ of $\sigma$ is finite. (3) The quantum projective plane ${\mathsf{P}\mathsf{r}\mathsf{o}\mathsf{j}}_{\mathrm{n}\mathrm{c}}A$ is finite over its center. In this paper, we will prove for Type S’ algebra $A$ that the following conditions are equivalent; (1) ${\mathsf{P}\mathsf{r}\mathsf{o}\mathsf{j}}_{\mathrm{n}\mathrm{c}}A$ is finite over its center. (2) The Beilinson algebra $\mathrm{\nabla }A$ of $A$ is $2$-representation tame. (3) The isomorphism classes of simple $2$-regular modules over $\mathrm{\nabla }A$ are parametrized by ${\mathbb{P}}^2$.