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.

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

a nilpotent subalgebra.

is the weight space The objects of category

such that Morphisms of this category are the