Theorem of the highest weight

In representation theory, a branch of mathematics, the theorem of the highest weight classifies the irreducible representations of a complex semisimple Lie algebra ${\displaystyle {\mathfrak {g}}}$.^[1][2] There is a closely related theorem classifying the irreducible representations of a connected compact Lie group ${\displaystyle K}$.^[3] The theorem states that there is a bijection

${\displaystyle \lambda \mapsto [V^{\lambda }]}$

from the set of "dominant integral elements" to the set of equivalence classes of irreducible representations of ${\displaystyle {\mathfrak {g}}}$ or ${\displaystyle K}$. The difference between the two results is in the precise notion of "integral" in the definition of a dominant integral element. If ${\displaystyle K}$ is simply connected, this distinction disappears.

The theorem was originally proved by Élie Cartan in his 1913 paper.^[4] The version of the theorem for a compact Lie group is due to Hermann Weyl. The theorem is one of the key pieces of representation theory of semisimple Lie algebras.