Let $\alpha$ be an arbitrary scalar in $\mathbb{C}$ and let $V(\alpha)$ be an infinite dimensional $\mathbb{C}$-vector space (with a countable basis). The formulas $h.v_i=(\alpha -2i).v_i$, $f.v_i=(i+1).v_{i+1}$ and $e.vi=(\alpha-i+1).v_{i-1}$ define an $sl(2,\mathbb{C})$-module structure.
If $\alpha+1=i$ is positive then $v_i$ is a maximal vector (it is easy to see this). Now I need to see that $v_0 \mapsto v_i$ induces a homomorphism $\phi: V(\alpha-2i) \rightarrow V(\alpha)$, that $\phi$ is injective, $Im(\phi)$ and $V(\alpha)/Im(\phi)$ is irreducible. I proved that $\phi$ is injective (I found $\phi(v_i)=(i+1)...(i+j)v_{i+j}$ )and I have an idea how to prove irreducibility but I don't exactly know what the image look like. Thx.