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.
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
The objects of category are -modules such that
Morphisms of this category are the -homomorphisms of these modules.
This section needs expansion. You can help by
adding to it. (September 2011) |
This section needs expansion. You can help by
adding to it. (September 2011) |
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.
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
The objects of category are -modules such that
Morphisms of this category are the -homomorphisms of these modules.
This section needs expansion. You can help by
adding to it. (September 2011) |
This section needs expansion. You can help by
adding to it. (September 2011) |