Lie algebras of differential operators III: Classification

Helge Øystein Maakestad

Published 2019-05-23Version 1

In a previous paper we introduced the notion of a D-Lie algebra $\tilde{L}$. A D-Lie algebra $\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions. We used this notion to define the universal enveloping algebra of the category of $\tilde{L}$-connections and to define the cohomology and homology of an arbitrary connection. In this note we introduce the canonical quotient $L$ of a D-Lie algebra $\tilde{L}$ and use this to classify D-Lie algebras where $L$ is projective as $A$-module. We define for any 2-cocycle $f\in \operatorname{Z}^2(\operatorname{Der}_k(A),A)$ a functor $F_{f}(-)$ from the category of $A/k$-Lie-Rinehart algebras to the category of D-Lie algebras and classify D-Lie algebras with projective canoncial quotient using the functor $F_{f}(-)$. We prove a similar classification for non-abelian extensions of D-Lie algebras.