The noncommutative schemes of generalized Weyl algebras

Abstract

The first Weyl algebra over $k$, $A_1=k⟨x,y⟩/(xy-yx-1)$ admits a natural $\mathbb{Z}$-grading by letting $\mathrm{deg}x=1$ and $\mathrm{deg}y=-1$. Paul Smith showed that $\mathrm{gr}-A_1$ is equivalent to the category of quasicoherent sheaves on a certain quotient stack. Using autoequivalences of $\mathrm{gr}-A_1$, Smith constructed a commutative ring $C$, graded by finite subsets of the integers. He then showed $\mathrm{gr}-A_1\equiv \mathrm{gr}-(C,{\mathbb{Z}}_{\mathrm{fin}})$. In this paper, we generalize results of Smith by using autoequivalences of a graded module category to construct rings with equivalent graded module categories. For certain generalized Weyl algebras, we use autoequivalences defined in a companion paper so that these constructions yield commutative rings.

Revised version

References in corpus (3)

• G-algebras, twistings, and equivalences of graded categories
• Rings graded equivalent to the Weyl algebra
• The Picard group of the graded module category of a generalized Weyl algebra

Cited by in corpus (3)

• Growth of generalized Weyl algebras over polynomial algebras and Laurent polynomial algebras
• The Picard group of the graded module category of a generalized Weyl algebra
• Simple $\mathbb{Z}$-graded domains of Gelfand-Kirillov dimension two