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.