# Simple $\mathbb{Z}$-graded domains of Gelfand-Kirillov dimension two

## TL;DR

This paper classifies simple $Z$-graded domains of Gelfand-Kirillov dimension two, linking their module categories to quasicoherent sheaves on quotient stacks and relating them to generalized Weyl algebras and classical translation principles.

## Contribution

It establishes an equivalence between module categories of these algebras and quasicoherent sheaves, and introduces a translation principle for their noncommutative schemes, connecting to classical Lie algebra theory.

## Key findings

- Category of modules equivalent to quasicoherent sheaves on a quotient stack
- Translation principle for noncommutative schemes of GWAs
- Connection to classical translation principle for $U(rak{sl}_2)$

## Abstract

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