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

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. 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.

Let ${\displaystyle {\mathfrak {g}}}$ be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra ${\displaystyle {\mathfrak {h}}}$. Let ${\displaystyle R}$ be the associated root system. We then say that an element ${\displaystyle \lambda \in {\mathfrak {h}}^{*}}$ is integral if

is an integer for each root ${\displaystyle \alpha }$. Next, we choose a set ${\displaystyle R^{+}}$ of positive roots and we say that an element ${\displaystyle \lambda \in {\mathfrak {h}}^{*}}$ is dominant if ${\displaystyle \langle \lambda ,\alpha \rangle \geq 0}$ for all ${\displaystyle \alpha \in R^{+}}$. An element ${\displaystyle \lambda \in {\mathfrak {h}}^{*}}$ is dominant integral if it is both dominant and integral. Finally, if ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ are in ${\displaystyle {\mathfrak {h}}^{*}}$, we say that ${\displaystyle \lambda }$ is higher than ${\displaystyle \mu }$ if ${\displaystyle \lambda -\mu }$ is expressible as a linear combination of positive roots with non-negative real coefficients.

A weight ${\displaystyle \lambda }$ of a representation ${\displaystyle V}$ of ${\displaystyle {\mathfrak {g}}}$ is then called a highest weight if ${\displaystyle \lambda }$ is higher than every other weight ${\displaystyle \mu }$ of ${\displaystyle V}$.

The theorem of the highest weight then states:

The most difficult part is the last one; the construction of a finite-dimensional irreducible representation with a prescribed highest weight.