Editing Weight (representation theory) (section)

===Highest-weight module===
A representation (not necessarily finite dimensional) ''V'' of <math>\mathfrak g</math> is called ''highest-weight module'' if it is generated by a weight vector ''v'' ∈ ''V'' that is annihilated by the action of all [[positive root]] spaces in <math>\mathfrak g</math>. Every irreducible <math>\mathfrak g</math>-module with a highest weight is necessarily a highest-weight module, but in the infinite-dimensional case, a highest weight module need not be irreducible. 
For each <math>\lambda\in\mathfrak h^*</math>—not necessarily dominant or integral—there exists a unique (up to isomorphism) [[irreducible (representation theory)|simple]] highest-weight <math>\mathfrak g</math>-module with highest weight λ, which is denoted ''L''(λ), but this module is infinite dimensional unless λ is dominant integral. It can be shown that each highest weight module with highest weight λ is a [[quotient module|quotient]] of the [[Verma module]] ''M''(λ). This is just a restatement of ''universality property'' in the definition of a Verma module.

Every ''finite-dimensional'' highest weight module is irreducible.<ref>This follows from (the proof of) Proposition 6.13 in {{harvnb|Hall|2015}} together with the general result on complete reducibility of finite-dimensional representations of semisimple Lie algebras</ref>