weight lattice

The weight lattice ${\mathrm{\Lambda }}_W$ of a root system $R\subset E$ is the lattice

$${\mathrm{\Lambda }}_W=\left\{e\in E\right|\frac{(e,\alpha )}{(\alpha ,\alpha )}\in \mathbb{Z}\text{ for all }r\in R\}.$$

Weights which lie in the weight lattice are called . If $R\subset 𝔥$ is the root system of a semi-simple Lie algebra $𝔤$ with Cartan subalgebra $𝔥$, then ${\mathrm{\Lambda }}_W$ is exactly the set of weights appearing in finite dimensional representations of $𝔤$.