As is known to all, the Lie algebra $\frak{sl}\_2$ admits a very nice representation on
$$
\mathbb{K}[X,Y]
$$
the polynomials in two variables, given by
$$
E \mapsto X\frac{\partial }{\partial Y}, ~~ F \mapsto Y\frac{\partial }{\partial X}, ~~ H \mapsto X\frac{\partial }{\partial X} - Y\frac{\partial }{\partial Y}.
$$
Does an analogous representation exist for general $\frak{sl}\_n$?



A representation of $\frak{sl}_n$ as partial derivatives on polynomials
The group $GL(V)$ acts on a vector space $V$ by linear automorphisms, and this induces the action of its Lie algebra $\mathfrak{gl}(V)$, i.e. a homomorphism to the Lie algebra of vector fields (differentiations). Explicitly
$$e\_{ij} \mapsto X\_i \partial\_j.$$
The composition of two differentiations is a second order differential operator, and the commutator is another differentiation. All these polynomial differential operators naturally act on the ring $\mathbb{K}[V]$ of polynomial functions.


For $\mathfrak{sl}(V) \subset \mathfrak{gl}(V)$ just restrict to a subalgebra. For explicit operators like $E,F,H$ choose a basis of $\mathfrak{sl}(V)$, e.g. $e\_{ij}$ for $i\neq j$ and $e\_{ii} - e\_{i+1,i+1}$.


