Lie algebroids, quantum Poisson algebroids, and Lie algebroid connections

TL;DR

This paper explores the theory of Lie algebroids over ringed spaces, establishing a correspondence with quantum Poisson algebras, and introduces new constructions for Lie algebroid connections and deformations.

Contribution

It generalizes Lie algebroids to ringed spaces, constructs a sheaf of twisted universal enveloping algebras, and links Lie algebroids with quantum Poisson algebras through a bijective correspondence.

Findings

Universal enveloping algebroid has a natural filtration.

Established a bijective correspondence between sheaves of quantum Poisson algebras and Lie algebroids.

Constructed a sheaf of twisted universal enveloping algebras for non-flat connections.

Abstract

In this paper, we consider Lie algebroids over commutative ringed spaces. Lie algebroids over ringed spaces unify the existing notion of Lie algebroids over smooth manifolds, complex manifolds, analytic spaces, algebraic varieties, and schemes. We show that the universal enveloping algebroid of a Lie algebroid possesses a natural filtration that yields a structure of a sheaf of quantum Poisson algebras. We establish a bijective correspondence between sheaves of quantum Poisson algebras and Lie algebroids. We show that this correspondence leads to an adjunction between the two categories. We discuss this bijective correspondence in particular cases of Lie algebroids over ringed spaces and highlight the subsequent results. To characterize non-flat Lie algebroid connections, we construct a sheaf of twisted universal enveloping algebras for a Lie algebroid using Lie algebroid (hyper) cohomology. We show that our construction yields some of the existing constructions for Lie-Rinehart algebras and holomorphic Lie algebroids. As another application, we study the deformation groupoid of a Lie algebroid using the second hypercohomology of the Lie algebroid.