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

Abstract

[EN] The classification problem for simple sl(2)-modules leads in a natural way to the study of the category of finite rank torsion free sl(2)-modules and its subcategory of rational 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 the way several new results regarding these categories that are interesting in their own right.