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

The first Weyl algebra, $A_1=k⟨x,y⟩\mathrm{/}(xy-yx-1)$ is naturally $\mathbb{Z}$-graded by letting $\mathrm{deg}x=1$ and $\mathrm{deg}y=-1$. Sue Sierra studied $\mathrm{gr}-A_1$, 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 $\mathrm{gr}-A(f)$ is isomorphic to the Picard group of $\mathrm{gr}-A_1$. In a companion paper, we use these results to construct commutative rings which are graded equivalent to generalized Weyl algebras.