A PBW theorem for inclusions of (sheaves of) Lie algebroids

TL;DR

This paper establishes a simple criterion based on Atiyah classes for when a Poincaré-Birkhoff-Witt theorem applies to inclusions of Lie algebroids, extending previous algebraic results.

Contribution

It introduces a straightforward criterion using Atiyah classes for Poincaré-Birkhoff-Witt type theorems in Lie algebroid inclusions, generalizing earlier Lie algebra results.

Findings

Provides a criterion based on Atiyah classes for PBW theorems in Lie algebroids

Uses the first infinitesimal neighbourhood Lie algebroid as a key tool

Extends previous Lie algebra inclusion results to Lie algebroids

Abstract

Inspired by the recent work of Chen-Sti\'enon-Xu on Atiyah classes associated to inclusions of Lie algebroids, we give a very simple criterium (in terms of those classes) for relative Poincar\'e-Birkhoff-Witt type results to hold. The tools we use (e.g. the first infinitesimal neighbourhood Lie algebroid) are straightforward generalizations of the ones previously developped by Caldararu, Tu and the author for Lie algebra inclusions.