Category O
In the representation theory of semisimple Lie algebras, Category O (or category [math]\displaystyle{ \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]\displaystyle{ \mathfrak{g} }[/math] is a (usually complex) semisimple Lie algebra with a Cartan subalgebra [math]\displaystyle{ \mathfrak{h} }[/math], [math]\displaystyle{ \Phi }[/math] is a root system and [math]\displaystyle{ \Phi^+ }[/math] is a system of positive roots. Denote by [math]\displaystyle{ \mathfrak{g}_\alpha }[/math] the root space corresponding to a root [math]\displaystyle{ \alpha\in\Phi }[/math] and [math]\displaystyle{ \mathfrak{n}:=\bigoplus_{\alpha\in\Phi^+} \mathfrak{g}_\alpha }[/math] a nilpotent subalgebra.
If [math]\displaystyle{ M }[/math] is a [math]\displaystyle{ \mathfrak{g} }[/math]-module and [math]\displaystyle{ \lambda\in\mathfrak{h}^* }[/math], then [math]\displaystyle{ M_\lambda }[/math] is the weight space
- [math]\displaystyle{ 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]\displaystyle{ \mathcal O }[/math] are [math]\displaystyle{ \mathfrak{g} }[/math]-modules [math]\displaystyle{ M }[/math] such that
- [math]\displaystyle{ M }[/math] is finitely generated
- [math]\displaystyle{ M=\bigoplus_{\lambda\in\mathfrak{h}^*} M_\lambda }[/math]
- [math]\displaystyle{ M }[/math] is locally [math]\displaystyle{ \mathfrak{n} }[/math]-finite. That is, for each [math]\displaystyle{ v \in M }[/math], the [math]\displaystyle{ \mathfrak{n} }[/math]-module generated by [math]\displaystyle{ v }[/math] is finite-dimensional.
Morphisms of this category are the [math]\displaystyle{ \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]\displaystyle{ Z(\mathfrak{g}) }[/math]-finite, i.e. if [math]\displaystyle{ M }[/math] is an object and [math]\displaystyle{ v\in M }[/math], then the subspace [math]\displaystyle{ Z(\mathfrak{g}) v\subseteq M }[/math] generated by [math]\displaystyle{ v }[/math] under the action of the center of the universal enveloping algebra, is finite-dimensional.
Examples
- All finite-dimensional [math]\displaystyle{ \mathfrak{g} }[/math]-modules and their [math]\displaystyle{ \mathfrak{g} }[/math]-homomorphisms are in category O.
- Verma modules and generalized Verma modules and their [math]\displaystyle{ \mathfrak{g} }[/math]-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, AMS, ISBN 978-0-8218-4678-0, archived from the original on 2012-03-21, https://web.archive.org/web/20120321142849/http://www.math.umass.edu/~jeh/bgg/main.pdf
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