Knowlpedia

Weight space

The simultaneous eigenspace of a representation corresponding to a weight on an abelian subalgebra.

Let g\mathfrak g be a complex Lie algebra, let h⊆g\mathfrak h\subseteq\mathfrak g be an abelian subalgebra, and let ρ:g→gl(V)\rho:\mathfrak g\to\mathfrak{gl}(V) be a representation. For λ∈h∗\lambda\in\mathfrak h^\ast, the λ\lambda-weight space is

Vλ={v∈V∣ρ(H)v=λ(H)v for all H∈h}.V_\lambda=\{v\in V\mid \rho(H)v=\lambda(H)v\ \text{for all }H\in\mathfrak h\}.

If Vλ≠0V_\lambda\neq 0, then λ\lambda is a .

Interaction with roots (semisimple context)

When g\mathfrak g is semisimple and h\mathfrak h is a , g\mathfrak g decomposes into gα\mathfrak g_\alpha. For X∈gαX\in\mathfrak g_\alpha and v∈Vλv\in V_\lambda,

X⋅v∈Vλ+α,X\cdot v \in V_{\lambda+\alpha},

so root vectors “shift” weights by roots.

Context

For a finite-dimensional representation of a complex semisimple Lie algebra, the Cartan subalgebra acts semisimply, giving

V=⨁λVλ.V=\bigoplus_\lambda V_\lambda.

This weight-space decomposition is one of the main inputs to highest-weight methods and depends crucially on complete reducibility phenomena (compare ).