  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
$U(\mathfrak{sl}_2)$.

Rings and Algebras