Let $k$ be an algebraically closed field and $A$ a $\mathbb{Z}$-graded finitely generated simple $k$-algebra which is a domain of Gelfand-Kirillov dimension 2. We show that the category of $\mathbb{Z}$-graded right $A$-modules is equivalent to the category of quasicoherent sheaves on a certain quotient stack. The theory of these simple algebras is closely related to that of a class of generalized Weyl algebras (GWAs). We prove a translation principle for the noncommutative schemes of these GWAs, shedding new light on the classical translation principle for the infinite-dimensional primitive quotients of $U(\mathfrak{sl}_2)$.

中文翻译：

---

Gelfand-Kirillov 维二的简单 Z 分级域

令 $k$ 是代数闭域，$A$ 是 $\mathbb{Z}$-graded 有限生成简单 $k$-algebra，它是 Gelfand-Kirillov 维 2 的域。我们证明 $\ mathbb{Z}$-graded right $A$-modules 等价于某个商栈上的准相干层的范畴。这些简单代数的理论与一类广义外尔代数 (GWA) 的理论密切相关。我们证明了这些 GWA 的非交换方案的翻译原理，为 $U(\mathfrak{sl}_2)$ 的无限维原始商的经典翻译原理提供了新的思路。