Algebraic connections on projective modules with prescribed curvature

摘要

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 Z(2) (L, A) the universal enveloping algebra U(f) of type f. When L is projective as left A-module we prove a PBW-Theorem for 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 H-2(L, A) there is a finite rank projective A-module E with c(1)(E) = c. Another application is to construct for any Lie-Rinehart algebra L which is projective as left A-module a subring Char(L) of H* (L, A) the characteristic ring of L. The ring Char(L) ring is defined in terms of the cohomology group H-2 (L, A) and has the property that it is a non-trivial subring of the image of the Chern character Ch(Q) : 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 U-ua (L). (C) 2015 Elsevier Inc. All rights reserved.