On Lie algebroid over algebraic spaces

TL;DR

This paper explores the properties of Lie algebroids over algebraic spaces, focusing on their universal enveloping algebroids, and establishes foundational theorems like PBW and Cartier-Milnor-Moore in this context.

Contribution

It introduces a sheaf-theoretic approach to universal enveloping algebroids over algebraic spaces and extends classical theorems to this setting.

Findings

Sheafification of universal enveloping algebras for Lie-Rinehart algebras.

Resemblance of universal enveloping algebroids to sheaves of bialgebras.

Proof of PBW and Cartier-Milnor-Moore theorems for Lie algebroids.

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 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 Poincar\'e-Birkhoff-Witt theorem and Cartier-Milnor-Moore theorem for the Lie algebroid.