Category O Explained

In the representation theory of semisimple Lie algebras, Category O (or category

l{O} ) is a category whose objects are certain representations of a semisimple Lie algebra and morphisms are homomorphisms of representations.

Introduction

Assume that

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

If

M is a ak{g} -module and λ \in ak{h} ^{*} , then M _{λ} is the weight spaceM _{λ=\{v} \in M : \forall h \in ak{h}h ⋅ v = λ (h)v\}.

Definition of category O

The objects of category

l O are ak{g} -modules M such thatM is finitely generatedM = o plus ^{*}} _{λ \in ak{h} M _{λ} M is locally ak{n} -finite. That is, for each v \in M , the ak{n} -module generated by v is finite-dimensional.

Morphisms of this category are the

ak{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( ak{g}) -finite, i.e. if M is an object and v\in M , then the subspace Z( ak{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

ak{g} -modules and their ak{g} -homomorphisms are in category O.

• Verma modules and generalized Verma modules and their

ak{g} -homomorphisms are in category O.

See also

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