Category O

In the representation theory of semisimple Lie algebras, Category O (or category formula_1) is a category whose objects are certain representations of a semisimple Lie algebra and morphisms are homomorphisms of representations.
Introduction.
Assume that formula_2 is a (usually complex) semisimple Lie algebra with a Cartan subalgebra
formula_3, formula_4 is a root system and formula_5 is a system of positive roots. Denote by formula_6
the root space corresponding to a root formula_7 and formula_8 a nilpotent subalgebra.
If formula_9 is a formula_2-module and formula_11, then formula_12 is the weight space
Definition of category O.
The objects of category formula_14 are formula_2-modules formula_9 such that
Morphisms of this category are the formula_2-homomorphisms of these modules.