Category O

Last updated February 25, 2021

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.

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References

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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