highest weight module
Highest weight modules is an important class of representations of Lie algebras, resp. Lie groups.
Definition
Let V be a representation of a Lie algebra
and assume that a
Cartan subalgebra 
and a set of positive roots is chosen. V 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
For each weight 
, there exists a unique (up to
isomorphism) simple highest weight -module with highest weight λ, which is denoted L(λ).
It can be shown that each highest weight module with highest weight λ is a quotient of the Verma module M(λ). 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 spaces.
The weight spaces in a highest weight module are always finite dimensional.
See also
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