sl2 highest-weight lowering formula
= sl2 highest-weight lowering formula
{title2=$XY^{k+1}v=(k+1)(m-k)Y^kv$}
In the <sl2 Lie algebra>, if $Xv=0$ and $Hv=mv$, then the <commutator> relations give $HY^jv=(m-2j)Y^jv$ and $XY^{k+1}v=(k+1)(m-k)Y^kv$. Induct using $XY=YX+H$. This controls the <raising operator> on a string formed by the <lowering operator> and yields the <classification of finite-dimensional sl2 representations>.