weight (Lie algebras)

Let $\mathfrak{h}$ be an abelian Lie algebra, $V$ a vector space and $\rho\colon\mathfrak{h}\to\operatorname{End}V$ a representation. Then the representation is said to be diagonalisable, if $V$ can be written as a direct sum

$V=\bigoplus\limits_{\lambda\in\mathfrak{h}^{*}}V_{\lambda}$

where $\mathfrak{h}^{*}$ is the dual space of $\mathfrak{h}$ and

$V_{\lambda}=\{v\in V\mid\rho(h)v=\lambda(h)v\text{ for all }h\in\mathfrak{h}\}.$

Now let $\mathfrak{g}$ be a semi-simple Lie algebra. Fix a Cartan subalgebra $\mathfrak{h}$ , then $\mathfrak{h}$ is abelian. Let $\rho\colon\mathfrak{g}\to\operatorname{End}V$ be a representation whose restriction to $\mathfrak{h}$ is diagonalisable. Then for any $\lambda\in\mathfrak{h}^{*}$ , the space $V_{\lambda}$ is the weight space of $\lambda$ with respect to $\rho$ . The multiplicity of $\lambda$ with respect to $\rho$ is the dimension of $V_{\lambda}$ :

$\operatorname{mult}_{\rho}(\lambda):=\dim V_{\lambda}.$

If the multiplicity of $\lambda$ is greater than zero, then $\lambda$ is called a weight of the representation $\rho$ .

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