Classification of the simple modules of the quantum Weyl algebra and the quantum plane

Abstract

for all α, β ∈ D and i, j ∈ Z and some map (a “2-cocycle”) c : Z×Z→ Z(D), the centre of D. Throughout the paper a “module” means a left module. Nontrivial examples of R are generalized Weyl algebras. Let D be a ring, σ ∈Aut (D), a a central element of D. The generalized Weyl algebra (GWA) A = D(σ, a) (of degree 1) is the ring generated by D and by two indeterminates X = v1 and Y = v−1 subject to the relations [Bav 1]: