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

Abstract

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 homological perturbation lemma, we provide a new construction of Sti\'{e}non--Vitagliano--Xu's contraction relating the cochain complex $\big(\Gamma(\pi^! L),\mathit{Q}\big)$ of sections of $\pi^! L$ to the Chevalley--Eilenberg complex $(\Gamma(\Lambda^\bullet A^\vee\otimes(L/A)),d^{\mathrm{Bott}})$ of the Bott representation. Using this contraction, we construct two isomorphisms: the first identifies the cohomology of 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 from the 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 of isomorphisms identifies the Atiyah class and the Todd class of the dg Lie algebroid $\pi^! L$ with the Atiyah class and the Todd class of the Lie pair $(L,A)$, respectively.