Lie subalgebras of Differential Operators in one Variable

Abstract

Let $\mathrm{Witt}$ be the Lie algebra generated by the set $\{L_i|i\in \mathbb{Z}\}$ and $\mathrm{Vir}$ its universal central extension. Let $\mathrm{Diff}(V)$ be the Lie algebra of differential operators on $V=\mathbb{C}[[z]]$, $\mathbb{C}((z))$ 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.

To appear in the Mediterranean Journal of Mathematics, Vol. 16 (2019), issue no. 6