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 than 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 (up to isomorphism) simple highest weight $\mathfrak{g}$ -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.