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 $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$ of a Lie algebroid $\mathcal{L}$ over an $a$-space $(X, \mathcal{O}_X)$. This is done by sheafification of the presheaf of universal enveloping algebras of Lie-Rinehart algebras. We review the extent to which the structure of the universal enveloping algebroid of Lie algebroids (over special $a$-spaces) resembles a bialgebroid structure, and present a version of Poincare-Birkhoff-Witt theorem and Cartier-Milnor-Moore theorem for this type of structure.