Classification of Lie algebras of differential operators

Abstract

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. We classify $\tilde{L}$-connections in the case when the canonical quotient $L$ of $\tilde{L}$ is projective as $A$-module. Any $\tilde{L}$-connection is determined by a 2-cocycle $f\in \operatorname{Z}^2(\operatorname{Der}_k(A),A)$ and an $L$-connection $(E,\nabla)$. We introduce the correspondence and Chow-operator of an $\tilde{L}$-connection. The aim of this construction is to relate connections on D-Lie algebras to algebraic cycles an the category of correspondences. The Chow-operator cannot be defined for an ordinary connection on an $A/k$-Lie-Rinehart algebra. It depends in a non-trivial way on the right $A$-module structure on $\tilde{L}$ and the canonical quotient $A/k$-Lie-Rinehart algebra $L$ has no such structure.