Atiyah and Todd classes of regular Lie algebroids

TL;DR

This paper investigates how Atiyah and Todd classes associated with regular Lie algebroids behave with respect to the Atiyah sequence, revealing their compatibility and restrictions within the sequence structure.

Contribution

It proves that Atiyah and Todd classes of dg manifolds from regular Lie algebroids respect the Atiyah sequence, extending understanding of their functorial properties.

Findings

Atiyah and Todd classes restrict to the Lie algebra bundle K.

They project onto classes of the integrable distribution F.

Classes respect the Atiyah sequence structure.

Abstract

For any regular Lie algebroid $A$, the kernel $K$ and the image $F$ of its anchor map $\rho_A$, together with $A$ itself fit into a short exact sequence, called Atiyah sequence, of Lie algebroids. We prove that Atiyah and Todd classes of dg manifolds arising from regular Lie algebroids respect the Atiyah sequence. That is, the Atiyah and Todd classes of $A$ restrict to the Atiyah and Todd classes of the bundle $K$ of Lie algebras on the one hand, and project onto the Atiyah and Todd classes of the integrable distribution $F \subseteq T_M$ on the other hand.