# Lie subalgebras of Differential Operators in one Variable

## TL;DR

This paper classifies all Lie algebra homomorphisms from key infinite-dimensional Lie algebras to differential operators on various function spaces, focusing on those where a specific element acts as a first-order operator.

## Contribution

It explicitly describes all homomorphisms from (2), Witt, and Vir to differential operators with a first-order action of L_0.

## Key findings

- Complete classification of homomorphisms for specified Lie algebras.
- Explicit descriptions of the images of these homomorphisms.
- Identification of conditions for the first-order action of L_0.

## 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.