The enveloping algebra of a Lie algebra of differential operators

Abstract

The aim of this note is to prove various general properties of a generalization of the full module of first order differential operators on a commutative ring - a $\operatorname{D}$-Lie algebra. A $\operatorname{D}$-Lie algebra $\tilde{L}$ is a Lie-Rinehart algebra over $A/k$ equipped with an $A\otimes_k A$-module structure that is compatible with the Lie-structure. It may be viewed as a simultaneous generalization of a Lie-Rinehart algebra and an Atiyah algebra with additional structure. Given a $\operatorname{D}$-Lie algebra $\tilde{L}$ and an arbitrary connection $(\rho, E)$ we construct the universal ring $\tilde{U}^{\otimes}(\tilde{L},\rho)$ of the connection $(\rho, E)$. The associative unital ring $\tilde{U}^{\otimes}(\tilde{L},\rho)$ is in the case when $A$ is Noetherian and $\tilde{L}$ and $E$ finitely generated $A$-modules, an almost commutative Noetherian sub ring of $\operatorname{Diff}(E)$ - the ring of differential operators on $E$. It is constructed using non-abelian extensions of $\operatorname{D}$-Lie algebras. The non-flat connection $(\rho, E)$ is a finitely generated $\tilde{U}^{\otimes}(\tilde{L},\rho)$-module, hence we may speak of the characteristic variety $\operatorname{Char}(\rho,E)$ of $(\rho, E)$ in the sense of $D$-modules. We may define the notion of holonomicity for non-flat connections using the universal ring $\tilde{U}^{\otimes}(\tilde{L},\rho)$. This was previously done for flat connections. We also define cohomology and homology of arbitrary non-flat connections. The cohomology and homology of a non-flat connection $(\rho,E)$ is defined using $\operatorname{Ext}$ and $\operatorname{Tor}$-groups of a non-Noetherian ring $\operatorname{U}$. In the case when the $A$-module $E$ is finitely generated we may always calculate cohomology and homology using a Noetherian quotient of $\operatorname{U}$. This was previously done for flat connections.Comment: 24.3.2019: Corrections made on the definition of the universal ring and some new proofs added. 24.09.2019: Extended introduction and minor changes. 03.11.2019: A significant extension - 15 pages added. 21.07.2020: An example on finite dimensionality of cohomology and homology groups added (Ex. 3.21