Raising and lowering on binary homogeneous polynomials
= Raising and lowering on binary homogeneous polynomials
{title2=$P=x\partial_y,\quad Q=y\partial_x,\quad[P,Q]e_k=(n-2k)e_k$}
On the degree-$n$ <homogeneous polynomials> use $e_k=x^{n-k}y^k$. Then $Pe_k=ke_{k-1}$ and $Qe_k=(n-k)e_{k+1}$; both maps are nilpotent of index $n+1$. Their <commutator> is diagonal in this basis with distinct <eigenvalues> $n,n-2,\ldots,-n$. Thus the raising and lowering operators are not <diagonalizable> for $n\geq1$, while their <commutator> is.