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

The first Weyl algebra, A1=k〈x,y〉/(xy−yx−1) is naturally Z-graded by letting deg⁡x=1 and deg⁡y=−1. Sue Sierra studied gr-A1, category of graded right A1-modules, computing its Picard group and classifying all rings graded equivalent to A1. 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 gr-A(f) is isomorphic to the Picard group of gr-A1.