Algebraic connections on projective modules with prescribed curvature

Abstract

In this paper we generalize classical results on Lie algebras and universal enveloping algebras of Lie algebras to Lie-Rinehart algebras. We define for any Lie-Rinehart algebra $L$ and any cocycle $f$ in $Z^2(L,B)$, a universal enveloping algebra $U(B,L,f)$ with the property that the category of left modules on $U(B,L,f)$ is equivalent to the category of modules with an $L$-connection where the curvature has "type $f$". A connection of curvature "type $f$" is a special case of a non-flat connection. We also study deformations of filtered algebras and prove a relationship between the deformation groupoid $A(\operatorname{Sym}_B^*(L))$ of the Lie-Rinehart algebra $L$ and $H^2(L,B)$. We also give an explicit realization of the category of $L$-connections as a category of left modules on an associative ring $U(L)$. We use the associative ring $U(L)$ to give a definition of cohomology and homology of arbitrary $L$-Connections. In the construction of the ring $U(L)$ we implicitly introduce the notion "D-Lie algebra" for the first time.