classification of finite-dimensional representations of semi-simple Lie algebras

If $𝔤$ is a semi-simple Lie algebra, then we say that an representation $V$has highest weight $\lambda$, if there is a vector $v\in V_{\lambda }$, the weight space of$\lambda$, such that $Xv=0$ for $X$ in any positive root space, and $v$ is called a highestvector, or vector of highest weight.

There is a unique (up to isomorphism) irreduciblefinite dimensional representation of $𝔤$ with highest weight $\lambda$ for any dominantweight $\lambda \in {\mathrm{\Lambda }}_W$, where ${\mathrm{\Lambda }}_W$ is the weight lattice of $𝔤$, andevery irreducible representation of $𝔤$ is of this type.