On Lie algebroid over algebraic spaces

Abstract

We consider Lie algebroids over algebraic spaces (in short we call it as a-spaces) by considering the sheaf of Lie-Rinehart algebras. We discuss about properties of universal enveloping algebroid of a Lie algebroid over an a-space . This is done by sheafification of the presheaf of universal enveloping algebras for Lie-Rinehart algebras. We review the extent to which structure of the universal enveloping algebroid of Lie algebroids (over special a-spaces) resembles a sheaf of bialgebras. In the sequel we present a version of Poincare-Birkhoff-Witt theorem and Cartier-Milnor-Moore theorem for the Lie algebraoid.