Formality theorems for Hochschild chains in the Lie algebroid setting

Abstract

In this paper we prove Lie algebroid versions of Tsygan's formality conjecture for Hochschild chains both in the smooth and holomorphic settings. Our result in the holomorphic setting implies a version of Tsygan's formality conjecture for Hochschild chains of the structure sheaf of any complex manifold. The proofs are based on the use of Kontsevich's quasi-isomorphism for Hochschild cochains of [[y1, , yd]], Shoikhet's quasi-isomorphism for Hochschild chains of [[y1, , yd]], and Fedosov's resolutions of the natural analogues of Hochschild (co)chain complexes associated with a Lie algebroid. In the smooth setting we discuss an application of our result to the description of quantum traces for a Poisson Lie algebroid. © Walter de Gruyter 2007.