Category O

Category O (or category $\mathcal{O}$ ) is a mathematical object in representation theory of semisimple Lie algebras. It is a category whose objects are certain representations of a semisimple Lie algebra and morphisms are homomorphisms of representations.

Introduction

Assume that $\mathfrak{g}$ is a (usually complex) semisimple Lie algebra with a Cartan subalgebra $\mathfrak{h}$ , $\Phi$ is a root system and $\Phi^+$ is a system of positive roots. Denote by $\mathfrak{g}_\alpha$ the root space corresponding to a root $\alpha\in\Phi$ and $\mathfrak{n}:=\oplus_{\alpha\in\Phi^+} \mathfrak{g}_\alpha$ a nilpotent subalgebra.

If $M$ is a $\mathfrak{g}$ -module and $\lambda\in\mathfrak{h}^*$ , then $M_\lambda$ is the weight space

M_\lambda=\{v\in M;\,\, \forall\,h\in\mathfrak{h}\,\,h\cdot v=\lambda(h)v\}.

Definition of category O

The objects of category O are $\mathfrak{g}$ -modules $M$ such that

$M$ is finitely generated
$M=\oplus_{\lambda\in\mathfrak{h}^*} M_\lambda$
$M$ is locally $\mathfrak{n}$ -finite, i.e. for each $v\in M$ , the $\mathfrak{n}$ -module generated by $v$ is finite-dimensional.

Morphisms of this category are the $\mathfrak{g}$ -homomorphisms of these modules.

Basic properties

Each module in a category O has finite-dimensional weight spaces.
Each module in category O is a Noetherian module.
O is an abelian category
O has enough projectives and injectives.
O is closed to submodules, quotients and finite direct sums
Objects in O are $Z(\mathfrak{g})$ -finite, i.e. if $M$ is an object and $v\in M$ , then the subspace $Z(\mathfrak{g}) v\subseteq M$ generated by $v$ under the action of the center of the universal enveloping algebra, is finite-dimensional.

Examples

All finite-dimensional $\mathfrak{g}$ -modules and their $\mathfrak{g}$ -homomorphisms are in category O.
Verma modules and generalized Verma modules and their $\mathfrak{g}$ -homomorphisms are in category O.

References

Humphreys, James E. (2008), Representations of semisimple Lie algebras in the BGG category O, AMS, ISBN 978-0-8218-4678-0