Algebraic connections on projective modules with prescribed curvature

In this paper we generalize some results on universal enveloping algebras of Lie algebras to Lie–Rinehart algebras and twisted universal enveloping algebras of Lie–Rinehart algebras. We construct for any Lie–Rinehart algebra L and any 2-cocycle f   in Z2(L,A)Z2(L,A) the universal enveloping algebra U(f)U(f) of type f. When L is projective as left A  -module we prove a PBW-Theorem for U(f)U(f) generalizing classical PBW-Theorems. We then use this construction to give explicit constructions of a class of finitely generated projective A-modules with no flat algebraic connections. One application of this is that for any Lie–Rinehart algebra L which is projective as left A-module and any cohomology class c   in H2(L,A)H2(L,A) there is a finite rank projective A-module E   with c1(E)=cc1(E)=c. Another application is to construct for any Lie–Rinehart algebra L which is projective as left A  -module a subring Char(L)Char(L) of H⁎(L,A)H⁎(L,A) – the characteristic ring of L  . The ring Char(L)Char(L) ring is defined in terms of the cohomology group H2(L,A)H2(L,A) and has the property that it is a non-trivial subring of the image of the Chern character ChQ:K(L)Q→H⁎(L,A). We also give an explicit realization of the category of L  -connections as a category of modules on an associative algebra Uua(L)Uua(L).