Subcategories of Module Categories via Restricted Yoneda Embeddings

We propose a framework for producing interesting subcategories of the category AMod of left A-modules, where A is an associative algebra over a field k. The construction is based on the composition, Y, of the Yoneda embedding of AMod with a restriction to certain subcategories B AMod, typically consisting of cyclic modules. We describe the subcategories on which Y provides an equivalence of categories. This also provides a way to understand the subcategories of AMod that arise this way. Many well-known categories are obtained in this way, including categories of weight modules and Harish-Chandra modules with respect to a subalgebra Γ of A. In other special cases the equivalence involves modules over the Mickelsson step algebra associated to a reductive pair of Lie algebras.