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 $\mathcal{U}({\mathcal{O}}_X,\mathcal{L})$ 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 Poincaré-Birkhoff-Witt theorem and Cartier-Milnor-Moore theorem for the Lie algebroid.

Published in Communications in Algebra (2022)

References in corpus (6)

• Infinite-dimensional vector bundles in algebraic geometry (an introduction)
• Deformations of algebras in noncommutative geometry
• Lie algebroid cohomology as a derived functor
• Geometric structures and Lie algebroids
• An index theorem for Lie algebroids
• Eight lectures on Oka manifolds