The Picard group of the graded module category of a generalized Weyl algebra

TL;DR

This paper extends the understanding of the Picard group of graded module categories from the first Weyl algebra to certain generalized Weyl algebras, revealing structural similarities.

Contribution

It generalizes Sierra's results by computing the Picard group of the graded module category of specific generalized Weyl algebras with quadratic polynomial base rings.

Findings

Picard group of gr-A(f) is isomorphic to that of gr-A_1

Classification of rings graded equivalent to generalized Weyl algebras

Extension of Sierra's results to a broader class of algebras

Abstract

The first Weyl algebra, $A_1 = k \langle x, y\rangle/(xy-yx - 1)$ is naturally $\mathbb{Z}$-graded by letting $\operatorname{deg} x = 1$ and $\operatorname{deg} y = -1$. Sue Sierra studied $\operatorname{gr}- A_1$, category of graded right $A_1$-modules, computing its Picard group and classifying all rings graded equivalent to $A_1$. In this paper, we generalize these results by studying the graded module category of certain generalized Weyl algebras. We show that for a generalized Weyl algebra $A(f)$ with base ring $k[z]$ defined by a quadratic polynomial $f$, the Picard group of $\operatorname{gr}- A(f)$ is isomorphic to the Picard group of $\operatorname{gr}- A_1$. In a companion paper, we use these results to construct commutative rings which are graded equivalent to generalized Weyl algebras.