The enveloping algebra of a Lie algebra of differential operators

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

Differential operators on projective modules

In this paper we give explicit formulas for differential operators on a finitely generated projective module E on an arbitrary commutative unital ring A. We use the differential operators constructed to give a simple formula for the curvature of a connection on a Lie-Rinehart algebra in terms of the fundamental matrix of E. This gives an explicit formula for the curvature of a connection on E defined in terms of an idempotent for E. We also consider the notion of a stratification on the module E induced by a projective basis. It turns out few stratifications are induced by a projective basis.Comment: Corollaries 3.3, 3.4 and 3.5 removed because of an erro

Extensions of Lie algebras of differential operators

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.Comment: 12.03.2019: Some corrections. 15.04.2019: Theorem 2.14 added. 28.06.2019: Example 3.4 added. 11.07.2019: References added. 10.11.2020: Minor revisio