Formality theorems for Hochschild chains in the Lie algebroid setting

Details Formality theorems for Hochschild chains in the Lie algebroid setting Formality theorems for Hochschild chains in the Lie algebroid setting

2005-04-19

arxiv.org/abs/math/0504372v3

Journal fur die reine und angewandte Mathematik (Crelles Journal) 612 (2007), 81-127

Mathematics

K-Theory and Homology

40 pages, no figures

Scientific paper

10.1515/CRELLE.2007.085

In this paper we prove Lie algebroid versions of Tsygan's formality conjecture for Hochschild chains both in the smooth and holomorphic settings. In the holomorphic setting our result implies a version of Tsygan's formality conjecture for Hochschild chains of the structure sheaf of any complex manifold and in the smooth setting this result allows us to describe quantum traces for an arbitrary Poisson Lie algebroid. The proofs are based on the use of Kontsevich's quasi-isomorphism for Hochschild cochains of R[[y_1,...,y_d]], Shoikhet's quasi-isomorphism for Hochschild chains of R[[y_1,...,y_d]], and Fedosov's resolutions of the natural analogues of Hochschild (co)chain complexes associated with a Lie algebroid.

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