Lie subalgebras of Differential Operators in one Variable

Abstract: Let $\operatorname{Witt}$ be the Lie algebra generated by the set ${L_i\,\vert\, i \in {\mathbb Z}}$ and $\operatorname{Vir}$ its universal central extension. Let $\operatorname{Diff}(V)$ be the Lie algebra of differential operators on $V=\mathbb{C}[[z]]$, $\mathbb{C}((z))$ or $V=\mathbb{C}(z)$. We explicitly describe all Lie algebra homomorphisms from $\mathfrak{sl}(2)$, $\operatorname{Witt}$ and $\operatorname{Vir}$ to $\operatorname{Diff}(V)$ such that $L_0$ acts on $V$ as a first order differential operator.