Hochschild cohomology of dg manifolds associated to integrable distributions

Abstract

For the field K = R or C, and an integrable distribution F ⊆ T K M = TM ⊗ R K on a smooth manifold M , we study the Hochschild cohomology of the dg manifold (F [1], dF ) and establish a canonical isomorphism with the Hochschild cohomology of the transversal polydifferential operators of F . In particular, for the dg manifold (T 0,1 X [1], ∂) associated with a complex manifold X, we prove that it is canonically isomorphic to the Hochschild cohomology HH • (X) of the complex manifold. As an application, we show that the Kontsevich-Duflo type theorem for the dg manifold (T 0,1 X [1], ∂) implies the Kontsevich-Duflo theorem for complex manifolds.

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

“…For example, one can find a contraction of (m, n)tensors in [7]. Also see [21,8]. Here, we formulate the contractions in terms of Hom spaces.…”

Internal symmetry of the $L_{\leqslant 3}$ algebra arising from a Lie pair

“…In the last decade much research on Lie pairs has been done following different strategies and the underlying mathematical structures: Atiyah classes arising from Lie pairs have been studied, using a variety of methods, see e.g. [2,8,9]; It is shown that geometric objects including Kapranov dg and Fedosov dg manifolds [19,30], algebraic objects such as Hopf algebras [7,11], Leibniz ∞ and L ∞ algebras can be derived from Lie pairs [1,6,20]; Also, in the context of Lie pairs, considerable attentions had been paid to Poincaré-Birkhoff-Witt isomorphisms [4,5], Kontsevich-Duflo isomorphisms [10,21], and Rozansky-Witten-type invariants [37], etc.…”