Extensions of Lie algebras of differential operators

TL;DR

This paper introduces the concept of D-Lie algebras, explores their properties and categories, and provides explicit constructions for non-abelian extensions of these algebras, extending classical Lie algebra extension theory.

Contribution

It defines D-Lie algebras, studies their properties and categories, and constructs all non-abelian extensions explicitly, generalizing previous work on Lie algebra extensions.

Findings

Defined D-Lie algebras with a canonical central element D

Developed the category of D-Lie algebras and modules

Provided explicit constructions for non-abelian extensions

Abstract

The aim of this note is to introduce the notion of a $\operatorname{D}$-Lie algebra and to prove some elementary properties of $\operatorname{D}$-Lie algebras, the category of $\operatorname{D}$-Lie algebras, the category of modules on a $\operatorname{D}$-Lie algebra and extensions of $\operatorname{D}$-Lie algebras. A $\operatorname{D}$-Lie algebra is an $A/k$-Lie-Rinehart algebra equipped with an $A\otimes_k A$-module structure and a canonical central element $D$ and a compatibility property between the $k$-Lie algebra structure and the $A\otimes_k A$-module structure. Several authors have studied non-abelian extensions of Lie algebras, super Lie algebras, Lie algebroids and holomorphic Lie algebroids and we give in this note an explicit constructions of all non-abelian extensions a $\operatorname{D}$-Lie algebra $\tilde{L}$ by an $A$-Lie algebra $(W,[,])$ where $\tilde{L}$ is projective as left $A$-module and $W$ is an $A\otimes_k A$-module with $IW=0$ for $I$ the kernel of the multiplication map. As a corollary we get an explicit construction of all non-abelian extensions of an $A/k$-Lie-Rinehart algebra $(L,\alpha)$ by an $A$-Lie algebra $(W,[,])$ where $L$ is projective as left $A$-module.