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

Definition of category O
The objects of category $$\mathcal O$$ are $$\mathfrak{g}$$-modules $$M$$ such that
 * 1) $$M$$ is finitely generated
 * 2) $$M=\bigoplus_{\lambda\in\mathfrak{h}^*} M_\lambda$$
 * 3) $$M$$ 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.

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 $$Z(\mathfrak{g})$$-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.