Helge Maakestad: Generalized enveloping algebras, connections and characteristic classes
Helge Maakestad gives the Seminar in Algebra and Algebraic Geometry:
Generalized enveloping algebras, connections and characteristic classes
Given a Lie-Rinehart algebra L, it follows the category of L-connections is a small abelian category. Hence a well known theorem from category theory says the category conn(L) of L-connections is equivalent to a subcategory of the category mod(R) of modules on an associative ring R. This correspondence does not preserve injective and projective objects. I will give a functorial construction of a universal algebra U(L) of the Lie-Rinehart algebra L with the property that there is an equivalence of categories between the category conn(L) and the category mod(U(L)) preserving injective and projective objects. I will use U(L) to give an explicit construction of cohomology and homology groups of arbitrary connections. Such an explicit construction was previously known to exist for flat L-connections. Because of the functorial nature of the construction it generalize to the case of an arbitrary sheaf of Lie-Rinehart algebras L on a scheme X relative to a base scheme S. I will relate U(L) to the generalized universal enveloping algebra U(B,L,f) introduced in the previous lecture on the subject. I will use U(L) to study characteristic classes and the Chern character for Lie-Rinehart algebras. I will construct for any Lie-Rinehart algebra L which is projective as B-module, a characteristic ring Char(L) which is a sub-ring of the image Im(Ch) of the Chern character in H^*(L,B). I will discuss how one may study Im(Ch) using Char(L).