For a semisimple Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$, fix a set of positive roots $\Phi^+$. The Borel subalgebra is: $\mathfrak{b} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi^+} \mathfrak{g}_\alpha$

This is a maximal solvable subalgebra. The nilradical is $\mathfrak{n} = \bigoplus_{\alpha \in \Phi^+} \mathfrak{g}_\alpha$.

Let $V$ be a representation of $\mathfrak{g}$ and $\lambda \in \mathfrak{h}^*$ a weight. A non-zero vector $v_\lambda \in V$ is a highest weight vector of weight $\lambda$ if:

1. $H \cdot v_\lambda = \lambda(H) v_\lambda$ for all $H \in \mathfrak{h}$ (weight vector condition)
2. $E_\alpha \cdot v_\lambda = 0$ for all $\alpha \in \Phi^+$ (killed by positive root spaces)

The representation $V$ is a highest weight module if it is generated by a highest weight vector.

A weight $\lambda \in \mathfrak{h}^*$ is integral if $\langle \lambda, \alpha^\vee \rangle \in \mathbb{Z}$ for all roots $\alpha \in \Phi$, where $\alpha^\vee = 2\alpha / \langle \alpha, \alpha \rangle$ is the coroot.

A weight is dominant if $\langle \lambda, \alpha^\vee \rangle \geq 0$ for all positive roots $\alpha \in \Phi^+$.

The set of dominant integral weights is denoted $P^+$ or $\Lambda^+$.