Category O

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.


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

  1. is finitely generated
  2. 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.


  • 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


  • Humphreys, James E. (2008), Representations of semisimple Lie algebras in the BGG category O (PDF), AMS, ISBN 978-0-8218-4678-0, archived from the original (PDF) on 2012-03-21
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