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

Abstract

HSUAN-YI LIAO A . 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 Lie algebroid π ! L of L along the projection π : A[1] → M . We give a new construction of Stiénon-Vitagliano-Xu's contraction for Γ(π ! L), Q by the homological perturbation lemma. Using this contraction, we prove that the Atiyah class and the Todd class of the dg Lie algebroid π ! L can be identified with the Atiyah class and the Todd class of the Lie pair (L, A), respectively.