Atiyah classes and dg-Lie algebroids for matched pairs

Understand

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.