Atiyah classes and dg-Lie algebroids for matched pairs

TL;DR

This paper constructs dg-manifold structures from Lie pairs of algebroids, establishing links between Atiyah classes of dg-Lie algebroids, Lie pairs, and dDG-algebras, revealing new geometric and algebraic insights.

Contribution

It introduces a dg-manifold framework for Lie pairs and relates Atiyah classes across dg-Lie algebroids, Lie pairs, and dDG-algebras, providing a novel geometric perspective.

Findings

Constructed dg-manifold structures for Lie pairs.

Established quasi-isomorphisms relating Atiyah classes.

Linked Atiyah classes of dg-Lie algebroids to those of Lie pairs and dDG-algebras.

Abstract

For every Lie pair $(L,A)$ of algebroids we construct a dg-manifold structure on the $\mathbb{Z}$-graded manifold $\mathcal M=L[1]\oplus L/A$ such that the inclusion $\iota: A[1] \to \mathcal M$ and the projection $p:\mathcal M\to L[1]$ are morphisms of dg-manifolds. The vertical tangent bundle $T^p\mathcal M$ then inherits a structure of dg-Lie algebroid over $\mathcal M$. When the Lie pair comes from a matched pair of Lie algebroids, we show that the inclusion $\iota$ induces a quasi-isomorphism that sends the Atiyah class of this dg-Lie algebroid to the Atiyah class of the Lie pair. We also show how (Atiyah classes of) Lie pairs and dg-Lie algebroids give rise to (Atiyah classes of) dDG-algebras.