Category O


In the representation theory of semisimple Lie algebras, Category O is a category whose objects are certain representations of a semisimple Lie algebra and morphisms are homomorphisms of representations.

Introduction

Assume that is a 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
  1. is finitely generated
  3. 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