Extensions of Lie algebras of differential operators

The aim of this note is to introduce the notion of a $\mathrm{D}$-Lie algebra and to prove some elementary properties of $\mathrm{D}$-Lie algebras, the category of $\mathrm{D}$-Lie algebras, the category of modules on a $\mathrm{D}$-Lie algebra and extensions of $\mathrm{D}$-Lie algebras. A $\mathrm{D}$-Lie algebra is an $A\mathrm{/}k$-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 $\mathrm{D}$-Lie algebra $\tilde{L}$ by an $A$-Lie algebra $(W,[,])$ where $\tilde{L}$ is projective as left $A$-module and $W$ is an $A\otimes_k A$-modulewith$IW=0$for$I$ the kernel of the multiplication map. As a corollary we get an explicit construction of all non-abelian extensions of an $A\mathrm{/}k$-Lie-Rinehart algebra $(L,\alpha )$ by an $A$-Lie algebra $(W,[,])$ where $L$ is projective as left $A$-module.