A nondominant integral highest-weight module can be infinite-dimensional

Given: The Axiom of Choice, ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ with Cartan subalgebra $\mathfrak{h}=\mathbb{C}h$ and chosen positive system ${\mathrm{\Phi }}^{+}=\{\alpha \}$ with simple-root base $\mathrm{\Delta }=\{\alpha \}$, where $\alpha (h)=2$, with coroot $h_{\alpha }=h$ (Coroot of a Lie-algebra root, The special linear Lie algebra sl_2), the functional $\lambda \in {\mathfrak{h}}^{\ast }$ with $\lambda (h)=-1$, and the highest weight module $M(\lambda )$ of Verma modules for sl2.

The Axiom of Choice is assumed; it is inherited through the cited module and the general highest-weight and integral-weight definitions (The Axiom of Choice).

$\lambda$ is integral: $⟨\lambda ,{\alpha }^{\vee }⟩=\lambda (h_{\alpha })=\lambda (h)=-1\in \mathbb{Z}$, but it is not dominant, since $-1$ is not nonnegative (Integral, dominant, and strictly dominant weights, Coroot of a Lie-algebra root).

Every Verma module $M(\lambda )$ in Verma modules for sl2 has the infinite basis $v_k=f^k\cdot v_0$, $k\ge 0$, and is therefore infinite-dimensional. [example statement]