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) WeightLieAlgebras 2013-03-22 13:11:42 2013-03-22 13:11:42 GrafZahl (9234) GrafZahl (9234) 7 GrafZahl (9234) Definition msc 17B20 weight diagonalisable diagonalizable multiplicity weight space