On sheaves of Lie-Rinehart algebras

TL;DR

This paper develops a theory of sheaves of Lie-Rinehart algebras over spaces, introducing morphisms and homotopy concepts, and applies it to smooth manifolds to analyze their geometric structure.

Contribution

It introduces morphisms and homotopy groups for sheaves of Lie-Rinehart algebras, generalizing Lie algebroid concepts and applying them to manifold foliations.

Findings

Sheaves induce a partition of the manifold into leaves.

Leaves correspond to orbits of the fundamental groupoid.

Generalization of homotopy groups for Lie-Rinehart sheaves.

Abstract

We study sheaves of Lie-Rinehart algebras over locally ringed spaces. We introduce morphisms and comorphisms of such sheaves and prove factorization theorems for each kind of morphism. Using this notion of morphism, we obtain (higher) homotopy groups and groupoids for such objects which directly generalize the homotopy groups and Weinstein groupoids of Lie algebroids. We consider, the special case of sheaves of Lie-Rinehart algebras over smooth manifolds. We show that, under some reasonable assumptions, such sheaves induce a partition of the underlying manifold into leaves and that these leaves are precisely the orbits of the fundamental groupoid.