Category O
In the representation theory of semisimple Lie algebras, Category O (or category ) is a category whose objects are certain representations of a semisimple Lie algebra and morphisms are homomorphisms of representations. Introduction 
Assume that  is a (usually complex) semisimple Lie algebra with a Cartan subalgebra
,  is a root system and  is a system of positive roots. Denote by 
the root space corresponding to a root  and  a nilpotent subalgebra. If  is a -module and , then  is the weight space

Definition of category O 
The objects of category  are -modules  such that
  is finitely generated
 
  is locally -finite. That is, for each , the -module generated by  is finite-dimensional. Morphisms of this category are the -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 -finite, i.e. if  is an object and , then the subspace  generated by  under the action of the center of the universal enveloping algebra, is finite-dimensional. Examples 

 All finite-dimensional -modules and their -homomorphisms are in category O.
 Verma modules and generalized Verma modules and their -homomorphisms are in category O. See also 
Highest-weight module
Universal enveloping algebra
Highest-weight category

References 

Representation theory of Lie algebras