Highest-weight theorem - Imported Knowlpedia Content

Fix a complex semisimple Lie algebra $\mathfrak g$ , a Cartan subalgebra $\mathfrak h$ , and a choice of positive roots (hence simple roots and fundamental weights).

Theorem (Highest-weight classification, Lie algebra form).

Every finite-dimensional irreducible $\mathfrak g$ -module is a highest-weight representation with a unique highest weight $\lambda\in\mathfrak h^*$ .
A weight $\lambda$ occurs as the highest weight of a finite-dimensional irreducible module if and only if $\lambda$ is dominant integral (i.e. it pairs with all simple coroots to give nonnegative integers).
For each dominant integral $\lambda$ , there exists (up to isomorphism) a unique finite-dimensional irreducible $\mathfrak g$ -module $V(\lambda)$ with highest weight $\lambda$ .

Remarks

Group form (compact groups). If $G$ is a compact connected Lie group with maximal torus $T$ (see maximal tori), then irreducible unitary representations of $G$ are classified by the dominant weights that lie in the character lattice $X^*(T)$ . This lattice depends on the global form of $G$ ; not every dominant integral weight of the complexified Lie algebra integrates to every group with that Lie algebra.

Context. This theorem is the conceptual reason that objects like fundamental representations (highest weights equal to fundamental weights) play a foundational role: they correspond to the vertices of the Dynkin diagram and generate the dominant cone.