Highest weight module

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Highest weight modules is an important class of representations of Lie algebras, resp. Lie groups.

[edit] Definition

Let V be a representation of a Lie algebra \mathfrak{g} and assume that a Cartan subalgebra \mathfrak{h} and a set of positive roots is chosen. V is called highest weight module, if it is generated by a weight vector v\in V that is annihilated by the action of all positive root spaces in \mathfrak{g}.

Note that this is something more special then a \mathfrak{g}-module with a highest weight.

Similarly we can define a highest weight module for representation of a Lie group resp. an associative algebra.

[edit] Properties

For each weight \lambda\in\mathfrak{h}^*, there exists a unique irreducible highest weight \mathfrak{g}-module with highest weight λ.

It can be shown that each highest weight module with highest weight λ is a quotient of the Verma module M(λ).

Highest weight modules are weight modules, i.e. they are direct sum of weight spaces.

The weight spaces in highest weight modules are always finite dimensional.

[edit] See also

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