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 Stiénon--Vitagliano--Xu's contraction relating the cochain complex $(\Gamma ({\pi }^{!}L),Q)$ 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 $(\Gamma (({\pi }^{!}L{)}^{\vee }\otimes \mathrm{End}({\pi }^{!}L)),Q)$ with the Chevalley--Eilenberg cohomology $H_{\mathrm{CE}}^{\bullet }(A,(L/A{)}^{\vee }\otimes \mathrm{End}(L/A))$ arising from the Bott representation, while the second identifies the cohomologies $H^{\bullet }(\Gamma (\Lambda ({\pi }^{!}L{)}^{\vee }),Q)$ and $H_{\mathrm{CE}}^{\bullet }(A,\Lambda (L/A{)}^{\vee })$. 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.

28 pages. Some examples and references added. Some typos corrected. Application to g-manifolds added. To appear in Communications in Mathematical Physics

References in corpus (6)

• The Atiyah class of a dg-vector bundle
• $Q$-Manifolds and Mackenzie Theory
• Exponential map and $L_{\infty }$ algebra associated to a Lie pair
• Dg manifolds, formal exponential maps and homotopy Lie algebras
• Hochschild cohomology of dg manifolds associated to integrable distributions
• Keller admissible triples and Duflo theorem

Cited by in corpus (1)

• $A_{\infty }$-Algebras from Lie Pairs