Category O

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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

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See also


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