Classification of finite-dimensional sl2 representations

For every $n\ge 0$ there is one irreducible ${\mathfrak{sl}}_2(\mathbb{C})$-module $V(n)$ of dimension $n+1$, with weights $n,n-2,\dots ,-n$, each of multiplicity one. Every finite-dimensional representation is a direct sum of these modules.

On any finite-dimensional ${\mathfrak{sl}}_2$-module, the lowering operator $f$ is injective from the $m$-weight space to the $(m-2)$-weight space whenever $m>0$. This follows on each irreducible summand from its standard weight string.