Weight of a representation

Definition

Let $\mathfrak g$ be a finite-dimensional complex semisimple Lie algebra (see semisimple Lie algebras ) and let $\mathfrak h\subset \mathfrak g$ be a Cartan subalgebra . For a representation $\rho:\mathfrak g\to \mathfrak{gl}(V)$, a linear functional $\lambda\in \mathfrak h^\ast$ is called a weight of $V$ if the corresponding weight space

is nonzero.

Equivalently, $\lambda$ is a weight if there exists $0\neq v\in V$ such that every $H\in\mathfrak h$ acts on $v$ by the scalar $\lambda(H)$ (so $v$ is a simultaneous eigenvector for the commuting family $\rho(\mathfrak h)$).

Context

Weights organize the representation theory of semisimple Lie algebras: the set of weights (with multiplicities $\dim V_\lambda$) encodes much of $V$, and irreducible representations are classified by their highest weight (see the highest-weight theorem ). The ambient space where weights live is explained in weights in the dual Cartan .