Infinitesimal ideal systems and the Atiyah class

TL;DR

This paper provides a geometric interpretation of the Atiyah class for Lie pairs, showing it vanishes under certain ideal system conditions, and introduces a new characterization of infinitesimal ideal systems.

Contribution

It offers a new geometric perspective on the Atiyah class of Lie pairs and characterizes ideal conditions for infinitesimal ideal systems.

Findings

Atiyah class vanishes when the subalgebroid is the fiber of an ideal system

New characterization of the ideal condition for infinitesimal ideal systems

Provides geometric interpretation of Atiyah class in Lie algebroid context

Abstract

This short note gives a geometric interpretation of the Atiyah class of a Lie pair. It proves that it vanishes if the subalgebroid is the kernel of a fibration of Lie algebroids. In other words, the Atiyah class of a Lie pair vanishes if the subalgebroid is the fiber of an ideal system in the Lie algebroid. In order to prove this, a new characterisation of the ideal condition for an infinitesimal ideal system is found.