Extreme modules over the Weyl algebra \(A_ n\)
We extend well-known results for Weyl algebras (Bernstein inequality, Bernstein filtration, Hilbert function, dimension and multiplicity of finitely generated modules, Bernstein class of modules, Stafford's theorems, global dimension, and Krull dimension) to the class of generalized Weyl algebras containing the Weyl algebras \(A_n\). We also present new results such as the simplicity of the tensor products of simple modules, the classification of certain classes of simple modules, e.g., of polynomial modules and modules with strong \(D\)-torsion (the latter serve as an analog of the weight modules of simple Lie algebras), the classification of extreme \(A_n\)-modules, and the theorem on the multiplicity of simple \(A_n\)-modules.
scientific article; zbMATH DE number 713548