Atiyah classes and dg-Lie algebroids for matched pairs

For every Lie pair (L,A) of algebroids we construct a dg-manifold structure on the Z-graded manifold M=L[1]⊕L∕A such that the inclusion ι:A[1]→M and the projection p:M→L[1] are morphisms of dg-manifolds. The vertical tangent bundle TpM then inherits a structure of dg-Lie algebroid over M. When the Lie pair comes from a matched pair of Lie algebroids, we show that the inclusion ι 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.