Category O

In the representation theory of semisimple Lie algebras, Category O (or category $𝒪$) is a category whose objects are certain representations of a semisimple Lie algebra and morphisms are homomorphisms of representations.

Introduction

Assume that $𝔤$ is a (usually complex) semisimple Lie algebra with a Cartan subalgebra $𝔥$, $\mathrm{\Phi }$ is a root system and ${\mathrm{\Phi }}^{+}$ is a system of positive roots. Denote by ${𝔤}_{\alpha }$ the root space corresponding to a root $\alpha \in \mathrm{\Phi }$ and $𝔫:=\underset{\alpha \in {\mathrm{\Phi }}^{+}}{⨁}{𝔤}_{\alpha }$ a nilpotent subalgebra.

If $M$ is a $𝔤$-module and $\lambda \in {𝔥}^{*}$, then $M_{\lambda }$ is the weight space

$M_{\lambda }=\{v\in M:\mathrm{\forall }h\in 𝔥h\cdot v=\lambda (h)v\}.$

Definition of category O

The objects of category $𝒪$ are $𝔤$-modules $M$ such that

1. $M$ is finitely generated
2. $M=\underset{\lambda \in {𝔥}^{*}}{⨁}{M}_{\lambda }$
3. $M$ is locally $𝔫$-finite. That is, for each $v\in M$, the $𝔫$-module generated by $v$ is finite-dimensional.

Morphisms of this category are the $𝔤$-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 $Z(𝔤)$-finite, i.e. if $M$ is an object and $v\in M$, then the subspace $Z(𝔤)v\subseteq M$ generated by $v$ under the action of the center of the universal enveloping algebra, is finite-dimensional.

Examples

• All finite-dimensional $𝔤$-modules and their $𝔤$-homomorphisms are in category O.
• Verma modules and generalized Verma modules and their $𝔤$-homomorphisms are in category O.

See also

• Highest-weight module
• Universal enveloping algebra
• Highest-weight category

References

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