Consider the $3$ dimensional Lie algebra $L$ of differential operators $$x=u\frac{\partial{ }}{\partial{v}},\text{ } y=v\frac{\partial{ }}{\partial{u}}, \text{ }h=u\frac{\partial{ }}{\partial{u}}-v\frac{\partial{ }}{\partial{v}}$$ on the space of smooth complex vaued functions in $u$ and $v$.
(1) Prove that they satisfy satisfy $$[xy]=h, \text{ }[hx]=2x, \text{ }[hy]=-2y$$
(2) Prove using the killing form that $L$ is semisimple. Since $L$ is $3$ dimensional, this forces $L$ to be simple.
(3) Prove that the set of homogeneous polynomials of degree $n$ in $u$ and $v$ is a irreducible representation space of $L$.
I have proved the first two parts. But I am unable to prove the last part. Any ideas?
The adjoint operators for $L$ are given by $$ ad (x)=\begin{pmatrix} 0&0 &-2\\0&0&0\\0&1&0\end{pmatrix}, ad (y)=\begin{pmatrix} 0&0&0\\ 0&0&2\\-1&0&0\end{pmatrix}, ad(h)=\begin{pmatrix} 2&0&0\\0&-2&0\\0&0&0\end{pmatrix}. $$ Hence the Killing form $\kappa(x,y)={\rm tr}(ad(x)ad(y))$ has matrix $$ \kappa=\begin{pmatrix} 0&4&0\\4&0&0\\0&0&8\end{pmatrix}, $$ which clearly is non-degenerate for characteristic different from $2$. Hence for characteristic zero $L$ is simple, and isomorphic to $\mathfrak{sl}_2(K)$.
Now the action by homogeneous polynomials is an irreducible $L$-module $V_n$. A proof is given in almost every text on Lie algebras. Here is an online-link:
The Story of $\mathfrak{sl}(2, \mathbb{C})$ and its Representations or Watch Charlotte Multiply $2 × 2$ Matrices with Her Left Hand. The proof is given in Proposition $4$ on page $9$, and is prepared in section $5$.
Also related are the following questions:
Formulas give irreducible representation, $SL(2, \mathbb{C})$.
Representation of $\text{sl}(2,\mathbb{C})$