摘要
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 π ! L of L along the bundle projection π: A[1] → M of the shifted vector bundle A[1]. Applying the homological perturbation lemma, we provide a new construction of Stiénon–Vitagliano–Xu’s contraction relating the cochain complex (Γ (π ! L) , Q) of sections of π ! L to the Chevalley–Eilenberg complex (Γ(Λ∙A∨⊗(L/A)),dBott) of the Bott representation. Using this contraction, we construct two isomorphisms: the first identifies the cohomology of the cochain complex (Γ((π!L)∨⊗End(π!L)),Q) with the Chevalley–Eilenberg cohomology HCE∙(A,(L/A)∨⊗End(L/A)) arising from the Bott representation, while the second identifies the cohomologies H∙(Γ(Λ(π!L)∨),Q) and HCE∙(A,Λ(L/A)∨) . We prove that this pair of isomorphisms identifies the Atiyah class and the Todd class of the dg Lie algebroid π ! L with the Atiyah class and the Todd class of the Lie pair (L, A), respectively.