weight (Lie algebras)

Let $𝔥$ be an abelian Lie algebra, $V$ a vector space and$\rho :𝔥\to \mathrm{End}V$ a representation. Then the representationis said to be diagonalisable, if $V$ can be written as a directsum

$$V=\underset{\lambda \in {𝔥}^{*}}{\oplus }{V}_{\lambda }$$

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

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

Now let $𝔤$ be a semi-simple Lie algebra. Fix a Cartan subalgebra$𝔥$, then $𝔥$ is abelian. Let $\rho :𝔤\to \mathrm{End}V$ be a representation whose restriction to $𝔥$ isdiagonalisable. Then for any $\lambda \in {𝔥}^{*}$, 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 $V_{\lambda }$:

$${\mathrm{mult}}_{\rho }(\lambda ):=dim{V}_{\lambda }.$$

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 themultiplicities of its weights.