I am trying to solve following exercise from Humphrey's Lie algebra:
Ler $R=F[X,Y]$ be polynomial ring in two variables over field $F$. Then $L=\mathfrak{sl}(2,F)$ acts naturally on subspace $\{mX+nY: m,n\in F\}$ it:
$$\begin{bmatrix} X \\ Y\end{bmatrix} \mapsto \begin{bmatrix} a & b\\ c & -a\end{bmatrix} \begin{bmatrix} X \\ Y\end{bmatrix}.$$ Extend this action to the ring $R=F[X,Y]$ by the derivation rule: $z.fg=(z.f)g+f(z.g)$ for $z\in L$ and $f,g\in F[X,Y]$.
Show that this extension is well defined and becomes an $L$-module.
I didn't get any intuition to prove wee-definedness; can you help me?
This looks something simple exercise, but, I was confused in understanding- prove that it is well-defined. What should I do actually to prove well-definedness?
What I did: consider $x=\begin{bmatrix} 0 & 1\\ 0 & 0\end{bmatrix}$, $y=\begin{bmatrix} 0 & 0\\ 1 & 0\end{bmatrix}$ and $h=\begin{bmatrix} 1 & 0\\ 0 & -1\end{bmatrix}$. Then
$$x.X^n=0, \,\,\,\, x.Y^n=nY^{n-1}X, \,\,\,\, y.X^n=nX^{n-1}Y, \,\,\,\, y.Y^n=0.$$ After this, what to do?
Consider $x=\pmatrix{a &b\cr c&d}$, you have $x.\pmatrix{X\cr Y}=\pmatrix{aX+bY\cr cX+dY}$. This means that $x.X=aX+bY$ and $x.Y=cX+dY$.
You can define recursively $x.X^n=x.(XX^{n-1})=(x.X)X^{n-1}+X(x.X^{n-1})=(aX+bY)X^{n-1}+X(x.X^{n-1})$.
Similar with $x.Y^n$
show $[x,y].P=x.(y.P)-y.(x.P)$.