Category O

In the representation theory of semisimple Lie algebras, Category O (or category $\mathcal{O}$ ) 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 $\mathfrak{n}:=\bigoplus_{\alpha\in\Phi^+} \mathfrak{g}_\alpha$ a nilpotent subalgebra.

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

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

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

$M$ is finitely generated
$M=\bigoplus_{\lambda\in\mathfrak{h}^*} M_\lambda$
$M$ is locally $\mathfrak{n}$ -finite. That is, 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.

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 under taking 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.

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.

{{Citation | last1=Humphreys | first1=James E. | author1-link=James E. Humphreys | title=Representations of semisimple Lie algebras in the BGG category O | publisher=AMS | year=2008 | isbn=978-0-8218-4678-0 | url=http://www.math.umass.edu/~jeh/bgg/main.pdf | url-status=dead | archiveurl=https://web.archive.org/web/20120321142849/http://www.math.umass.edu/~jeh/bgg/main.pdf | archivedate=2012-03-21 }}