# Category O

Jump to navigation
Jump to search

**Category O** (or **category **) is a mathematical object in representation theory of semisimple Lie algebras. It 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 O are -modules such that

- is finitely generated
- is locally -finite, i.e. 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 to 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

## References

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}