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For a Lie algebroid $L$ and a Lie subalgebroid $A$, i.e. a Lie pair $(L,A)$,we study the Atiyah class and the Todd class of the pullback dg (i.e.differential graded) Lie algebroid $\pi^! L$ of $L$ along the bundle projection$\pi:A[1] \to M$ of the shifted vector bundle $A[1]$. Applying the homologicalperturbation lemma, we provide a new construction ofSti\'{e}non--Vitagliano--Xu's contraction relating the cochain complex$\big(\Gamma(\pi^! L),\mathit{Q}\big)$ of sections of $\pi^! L$ to theChevalley--Eilenberg complex $(\Gamma(\Lambda^\bulletA^\vee\otimes(L/A)),d^{\mathrm{Bott}})$ of the Bott representation. Using thiscontraction, we construct two isomorphisms: the first identifies the cohomologyof the cochain complex $(\Gamma((\pi^! L)^\vee\otimes\mathrm{End}(\pi^!L)),\mathit{Q})$ with the Chevalley--Eilenberg cohomology$H^\bullet_{\mathrm{CE}}(A,(L/A)^\vee\otimes\mathrm{End}(L/A))$ arising fromthe Bott representation, while the second identifies the cohomologies$H^\bullet(\Gamma(\Lambda(\pi^! L)^\vee),\mathit{Q})$ and$H^\bullet_{\mathrm{CE}}(A,\Lambda(L/A)^\vee)$. We prove that this pair ofisomorphisms identifies the Atiyah class and the Todd class of the dg Liealgebroid $\pi^! L$ with the Atiyah class and the Todd class of the Lie pair$(L,A)$, respectively.

Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs