Let be a semi-simple Lie algebra, a Cartan subalgebra, and a Borel subalgebra. We work over a field . Given a weight , let be the 1-d dimensional -module on which acts by multiplication by , and the positive root spaces act trivially. Now, the Verma module of the weight is the -module induced from , i.e.
We say a -module is a highest weight module if it has a weight and a non-zero vector with for any in a positive root space and such that is generated as a -module by . The Verma module is a highest weight module and we fix a generator .
The most important property of Verma modules is that they are universal amongst highest weight modules, in the following sense. If is a highest weight module generated by which has weight then there is a unique surjective homomorphism which sends to . That is, all highest weight modules with highest weight are quotients of . Also, has a unique maximal submodule, so there is a unique irreducible representation with highest weight . If is dominant and integral then this module is finite dimensional.
|Date of creation||2013-03-22 13:12:13|
|Last modified on||2013-03-22 13:12:13|
|Last modified by||owenjonesuk (12024)|
|Defines||highest weight module|