Extension Fullness of the Categories of Gelfand-Zeitlin and Whittaker Modules

TL;DR

This paper proves that certain categories of modules related to Lie algebras are extension full within all modules, enabling the estimation of their global dimensions and advancing understanding of their homological properties.

Contribution

It establishes extension fullness of Gelfand-Zeitlin and Whittaker module categories in the broader module category, and uses this to analyze their global dimensions.

Findings

Categories are extension full in all modules

Global dimensions of these categories are estimated or determined

Advances understanding of homological properties of these modules

Abstract

We prove that the categories of Gelfand-Zeitlin modules of $\mathfrak{g}=\mathfrak{gl}_n$ and Whittaker modules associated with a semi-simple complex finite-dimensional algebra $\mathfrak{g}$ are extension full in the category of all $\mathfrak{g}$-modules. This is used to estimate and in some cases determine the global dimension of blocks of the categories of Gelfand-Zeitlin and Whittaker modules.