Lie Subalgebras of Differential Operators in One Variable

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