Category O

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

Introduction

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

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

${\displaystyle M_{\lambda }=\{v\in M:\forall h\in {\mathfrak {h}}\,\,h\cdot v=\lambda (h)v\}.}$

Definition of category O

The objects of category ${\displaystyle {\mathcal {O}}}$ are ${\displaystyle {\mathfrak {g}}}$-modules ${\displaystyle M}$ such that

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

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

Examples

• All finite-dimensional ${\displaystyle {\mathfrak {g}}}$-modules and their ${\displaystyle {\mathfrak {g}}}$-homomorphisms are in category O.
• Verma modules and generalized Verma modules and their ${\displaystyle {\mathfrak {g}}}$-homomorphisms are in category O.

See also

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

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