Matrix representation of universal enveloping $U(sl(2,\mathbb{C}))$ generators $ e^k,f^k$ Wiki describes the Lie algebra of universal enveloping algebra $U(sl(2,\mathbb{C}))$ with generators $$e^k,f^k,h$$
but Wiki says $e^k,f^k$ are not the matrix power of $sl(2,\mathbb{C}) $   generators $$e ,f ,h.$$

Question; What are the matrix representation of $U(sl(2,\mathbb{C}))$   generators $ e^k,f^k$?


 A: This is a legitimate confusion and I think the downvote is not justified.
You should not think of $e^k$ as $e \cdot \dots \cdot e \in M_2(\Bbb C)$, but more as formal objects that acts like $e$ composed $k$ times on any representation of $sl_2$.  The matrix product $e \cdot e = 0$. It represents how $e^2 \in U(sl_2)$ acts on $\Bbb C^2$, but it doesn't mean it that $e^2$ will act by zero on other representation !
This is because for a representation $V$, $e^k$ will acts as $\rho(e)^k$ (matrix product) when $\rho : sl_2 \to End(V)$ is a representation.
For example, the matrix form of $e^2$ acting on $Sym^2(\Bbb C^2)$ is $\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 2 \end{pmatrix}$.
More generally, to answer your question "what are the matrix form of $e^k$ ? " you need to pick a representation of $\rho : sl_2 \to End(V)$ and compute $\rho(e)^k$ . You can work out quite easily the general pattern, remembering that finite-dimensional irreducible representations of $\mathfrak {sl}_2$ are given by $Sym^n(\Bbb C^2)$.
