Tensoring with infinite-dimensional modules in $\scr O_0$

TL;DR

This paper investigates how tensoring with infinite-dimensional modules induces various endofunctors on the principal block of category O for semisimple Lie algebras, revealing their structures and generalizations.

Contribution

It introduces and analyzes tensoring functors with infinite-dimensional modules in category O, establishing their properties, adjoints, and monad structures, and extends results to parabolic subcategories.

Findings

Tensoring with infinite-dimensional modules induces faithful endofunctors on .

These functors preserve tilting, projective, or injective modules depending on their exactness.

Explicit computations are provided for the  for  in .

Abstract

We show that the principal block $\scr O_0$ of the BGG category $\scr O$ for a semisimple Lie algebra $\germ g$ acts faithfully on itself via exact endofunctors which preserve tilting modules, via right exact endofunctors which preserve projective modules and via left exact endofunctors which preserve injective modules. The origin of all these functors is tensoring with arbitrary (not necessarily finite-dimensional) modules in the category $\scr O$. We study such functors, describe their adjoints and show that they give rise to a natural (co)monad structure on $\scr O_0$. Furthermore, all this generalises to parabolic subcategories of $\scr O_0$. As an example, we present some explicit computations for the algebra $\germ{sl}_3$.