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

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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

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The objects of category   are  -modules   such that

  1.   is finitely generated
  2.  
  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

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  • 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

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  • 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

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References

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  • 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