In the representation theory of semisimple Lie algebras, Category O (or category ) is a category whose objects are certain representations of a semisimple Lie algebra and morphisms are homomorphisms of representations.
Introduction
Assume that is a (usually complex) semisimple Lie algebra with a Cartan subalgebra , is a root system and is a system of positive roots. Denote by the root space corresponding to a root and a nilpotent subalgebra.
If is a -module and , then is the weight space
Definition of category O
The objects of category are -modules such that
- is finitely generated
- is locally -finite. That is, for each , the -module generated by is finite-dimensional.
Morphisms of this category are the -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 -finite, i.e. if is an object and , then the subspace generated by under the action of the center of the universal enveloping algebra, is finite-dimensional.
Examples
- All finite-dimensional -modules and their -homomorphisms are in category O.
- Verma modules and generalized Verma modules and their -homomorphisms are in category O.
See also
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
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