title: Category O
text: # Category O
  
  In the representation theory of semisimple Lie algebras, Category O (or category ${\mathcal {O}}$) is a category whose objects are certain representations of a semisimple Lie algebra and morphisms are homomorphisms of representations.
  
  ## Introduction
  
  Assume that ${\mathfrak {g}}$ is a (usually complex) semisimple Lie algebra with a Cartan subalgebra ${\mathfrak {h}}$, $\Phi$ is a root system and $\Phi ^{+}$ is a system of positive roots. Denote by ${\mathfrak {g}}_{\alpha }$ the root space corresponding to a root $\alpha \in \Phi$ and ${\mathfrak {n}}:=\bigoplus _{\alpha \in \Phi ^{+}}{\mathfrak {g}}_{\alpha }$ a nilpotent subalgebra.
  
  If $M$ is a ${\mathfrak {g}}$-module and $\lambda \in {\mathfrak {h}}^{*}$, then $M_{\lambda }$ is the weight space
  
      $M_{\lambda }=\{v\in M:\forall h\in {\mathfrak {h}}\,\,h\cdot v=\lambda (h)v\}.$
  
  ## Definition of category O
  
  The objects of category ${\mathcal {O}}$ are ${\mathfrak {g}}$-modules $M$ such that
  
  * $M$ is finitely generated
  * $M=\bigoplus _{\lambda \in {\mathfrak {h}}^{*}}M_{\lambda }$
  * $M$ is locally ${\mathfrak {n}}$-finite. That is, for each $v\in M$, the ${\mathfrak {n}}$-module generated by $v$ is finite-dimensional.
  
  Morphisms of this category are the ${\mathfrak {g}}$-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 under taking submodules, quotients and finite direct sums.
  * Objects in O are $Z({\mathfrak {g}})$-finite, i.e. if $M$ is an object and $v\in M$, then the subspace $Z({\mathfrak {g}})v\subseteq M$ generated by $v$ under the action of the center of the universal enveloping algebra, is finite-dimensional.
  
  ## Examples
  
  * All finite-dimensional ${\mathfrak {g}}$-modules and their ${\mathfrak {g}}$-homomorphisms are in category O.
  * Verma modules and generalized Verma modules and their ${\mathfrak {g}}$-homomorphisms are in category O.
  