Category O

In the representation theory of semisimple Lie algebras, Category O (or category $\mathcal$) is a category whose objects are certain representations of a semisimple Lie algebra and morphisms are homomorphisms of representations.

Introduction

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

If $M$ is a $\mathfrak$-module and $\lambda\in\mathfrak^*$, then $M_\lambda$ is the weight space
:$M_\lambda=\.$

Definition of category O

The objects of category $\mathcal O$ are $\mathfrak$-modules $M$ such that
# $M$ is finitely generated
# $M=\bigoplus_ M_\lambda$
# $M$ is locally $\mathfrak$-finite. That is, for each $v \in M$, the $\mathfrak$-module generated by $v$ is finite-dimensional.

Morphisms of this category are the $\mathfrak$-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 under taking submodules, quotients and finite direct sums.
*Objects in O are $Z(\mathfrak)$-finite, i.e. if $M$ is an object and $v\in M$, then the subspace $Z(\mathfrak) v\subseteq M$ generated by $v$ under the action of the center of the universal enveloping algebra, is finite-dimensional.

Examples

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

See also

*Highest-weight module
*Universal enveloping algebra
* Highest-weight category

References

*{{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

Representation theory of Lie algebras