Let $\mf{h}$ be an abelian Lie algebra, $V$ a vector space and $\rho\colon\mf{h}\to\End V$ a representation. Then the representation is said to be diagonalisable, if $V$ can be written as a direct sum \begin{equation*} V=\DirectSum_{\lambda\in\mf{h}^*}V_\lambda \end{equation*}where $\mf{h}^*$ is the dual space of $\mf{h}$ and \begin{equation*} V_\lambda=\{v\in V\mid\rho(h)v=\lambda(h)v\text{ for all }h\in\mf{h}\}. \end{equation*} Now let $\mf{g}$ be a semi-simple Lie algebra. Fix a Cartan subalgebra $\mf{h}$ , then $\mf{h}$ is abelian. Let $\rho\colon\mf{g}\to\End V$ be a representation whose restriction to $\mf{h}$ is diagonalisable. Then for any $\lambda\in\mf{h}^*$ , 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 $V_\lambda$ : \begin{equation*} \mult_\rho(\lambda):=\dim V_\lambda. \end{equation*}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.