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