# Category O

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

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

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

## Definition of category O

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

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

Morphisms of this category are the ${\displaystyle {\mathfrak {g}}}$-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 ${\displaystyle Z({\mathfrak {g}})}$-finite, i.e. if ${\displaystyle M}$ is an object and ${\displaystyle v\in M}$, then the subspace ${\displaystyle Z({\mathfrak {g}})v\subseteq M}$ generated by ${\displaystyle v}$ under the action of the center of the universal enveloping algebra, is finite-dimensional.

## Examples

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