Subcategories of Module Categories via Restricted Yoneda Embeddings

TL;DR

This paper introduces a framework using restricted Yoneda embeddings to generate and analyze interesting subcategories of module categories over associative algebras, connecting to various well-known module categories.

Contribution

It develops a new method to produce and understand subcategories of module categories via restricted Yoneda embeddings, encompassing many classical examples.

Findings

Identifies subcategories where the restricted Yoneda embedding yields categorical equivalences.

Provides a unified perspective on categories like weight modules and Harish-Chandra modules.

Connects subcategories of modules over associative algebras to modules over Mickelsson step algebras.

Abstract

We propose a framework for producing interesting subcategories of the category ${}_A\mathsf{Mod}$ 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 ${}_A\mathsf{Mod}$ with a restriction to certain subcategories $\mathcal{B}\subset {}_A\mathsf{Mod}$, 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 ${}_A\mathsf{Mod}$ 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 $\Gamma$ of $A$. In other special cases the equivalence involves modules over the Mickelsson step algebra associated to a reductive pair of Lie algebras.