On Lie algebroid over algebraic spaces
Ashis Mandal, Abhishek Sarkar

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 -spaces) by considering the sheaf of Lie-Rinehart algebras. We discuss about properties of universal enveloping algebroid of a Lie algebroid over an -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 -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.
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Taxonomy
TopicsAdvanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models
