| Highest Weight Module |
Article Index for Highest |
Website Links For Weight |
Information AboutHighest Weight Module |
| CATEGORIES ABOUT HIGHEST WEIGHT MODULE | |
| lie algebras | |
| representation theory of lie groups | |
|
DEFINITION Let be a Representation of a Lie algebra and assume that a Cartan Subalgebra and a set of Positive Root s is chosen. is called ''highest weight module'', if it is generated by a Weight Vector that is annihilated by the action of all Positive Root spaces in . Note that this is something more special then a - Module with a Highest Weight . Similarly we can define a highest weight module for representation of a Lie Group resp. an Associative Algebra . PROPERTIES
a unique (up to isomorphism) Simple highest weight -module with highest weight which is denoted It can be shown that each highest weight module with highest weight is a Quotient of the Verma Module . This is just a restatement of ''universality property'' in the definition of a Verma module. A highest weight modules is a Weight Module , i.e. it is a direct sum of its Weight Space s. The Weight Space s in a Highest Weight Module are always finite dimensional. SEE ALSO |
|
|