Highest-weight vectors and modules

Assume the Axiom of Choice. Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$, a chosen positive system with nilpotent subalgebra ${\mathfrak{n}}^{+}$ (Positive and negative nilpotent subalgebras and the Borel), and let $V$ be a representation of $\mathfrak{g}$. A highest weight vector of $V$ is a nonzero vector $v\in V_{\lambda }$, for some $\lambda \in {\mathfrak{h}}^{\ast }$, with ${\mathfrak{n}}^{+}\cdot v=0$ (Weight and weight space); the functional $\lambda$ is then called the weight of $v$. A highest weight module of highest weight $\lambda$ is a representation $V$ generated, as a $\mathfrak{g}$-module, by a highest weight vector of weight $\lambda$: the smallest subrepresentation of $V$ containing $v$ is $V$ itself.

By Lie representations are U(g)-modules the action of $\mathfrak{g}$ extends uniquely to a unital action of $U(\mathfrak{g})$, so the subrepresentation generated by $v$ is exactly $U(\mathfrak{g})v$; this is the content of the word "generated" above and makes the notion independent of any choice of generators. A highest weight vector $v$ satisfies $H\cdot v=\lambda (H)v$ for all $H\in \mathfrak{h}$ by the definition of $V_{\lambda }$. The zero representation is not a highest weight module, since a highest weight vector is required to be nonzero.