The Grothendieck and Picard groups of finite rank torsion free $\mathfrak{sl}(2)$-modules

TL;DR

This paper investigates the structure of finite rank torsion free and rational rak{sl}(2)-modules, establishing their Grothendieck and Picard groups, and revealing new insights into their invariants and classifications.

Contribution

It provides a complete description of the Grothendieck and Picard groups for these module categories, linking simple modules via rationalization and introducing new results about their invariants.

Findings

Rationalization induces an isomorphism between simple modules

Complete description of Grothendieck and Picard groups

New results on invariants of rak{sl}(2)-modules

Abstract

The classification problem for simple $\mathfrak{sl}(2)$-modules leads in a natural way to the study of the category of finite rank torsion free $\mathfrak{sl}(2)$-modules and its subcategory of rational $\mathfrak{sl}(2)$-modules. We prove that the rationalization functor induces an identification between the isomorphism classes of simple modules of these categories. This raises the question of what is the precise relationship between other invariants associated with them. We give a complete solution to this problem for the Grothendieck and Picard groups, obtaining along theway several new results regarding these categories that are interesting in their own right.