weight (Lie algebras)

Let 𝔥 be an abelian Lie algebra, V a vector spaceMathworldPlanetmath and ρ:𝔥EndV a representation. Then the representation is said to be diagonalisable, if V can be written as a direct sum


where 𝔥* is the dual spaceMathworldPlanetmathPlanetmath of 𝔥 and

Vλ={vVρ(h)v=λ(h)v for all h𝔥}.

Now let 𝔤 be a semi-simple Lie algebra. Fix a Cartan subalgebraMathworldPlanetmath 𝔥, then 𝔥 is abelian. Let ρ:𝔤EndV be a representation whose restriction to 𝔥 is diagonalisable. Then for any λ𝔥*, the space Vλ is the weight space of λ with respect to ρ. The multiplicity of λ with respect to ρ is the dimensionPlanetmathPlanetmathPlanetmath of Vλ:


If the multiplicity of λ is greater than zero, then λ is called a weight of the representation ρ.

A representation of a semi-simple Lie algebra is determined by the multiplicities of its weights.

Title weight (Lie algebras)
Canonical name WeightLieAlgebras
Date of creation 2013-03-22 13:11:42
Last modified on 2013-03-22 13:11:42
Owner GrafZahl (9234)
Last modified by GrafZahl (9234)
Numerical id 7
Author GrafZahl (9234)
Entry type Definition
Classification msc 17B20
Synonym weight
Defines diagonalisable
Defines diagonalizable
Defines multiplicity
Defines weight space