Editing Weight (representation theory) (section)

===Theorem of the highest weight===
{{main|Theorem on the highest weights}}
A weight <math>\lambda</math> of a representation <math>V</math> of <math>\mathfrak g</math> is called a '''highest weight''' if every other weight of <math>V</math> is lower than <math>\lambda</math>.

The theory [[Lie algebra representation#Classifying finite-dimensional representations of Lie algebras|classifying the finite-dimensional irreducible representations]] of <math>\mathfrak g</math> is by means of a "theorem of the highest weight." The theorem says that<ref>{{harvnb|Hall|2015}} Theorems 9.4 and 9.5</ref>
:(1) every irreducible (finite-dimensional) representation has a highest weight, 
:(2) the highest weight is always a dominant, algebraically integral element, 
:(3) two irreducible representations with the same highest weight are isomorphic, and 
:(4) every dominant, algebraically integral element is the highest weight of an irreducible representation. 
The last point is the most difficult one; the representations may be constructed using [[Verma module]]s.