Atiyah classes and dg-Lie algebroids for matched pairs
Panagiotis Batakidis, Yannick Voglaire

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 of algebroids we construct a dg-manifold structure on the -graded manifold such that the inclusion and the projection are morphisms of dg-manifolds. The vertical tangent bundle then inherits a structure of dg-Lie algebroid over . When the Lie pair comes from a matched pair of Lie algebroids, we show that the inclusion 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.
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