# 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

1. $M$ is finitely generated
2. $M=\oplus _{\lambda \in {\mathfrak {h}}^{*}}M_{\lambda }$ 3. $M$ 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.