2005/04/30 by Damien Calaque, Vasiliy Dolgushev, Gilles Halbout
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math.KT #math.QA #msc:16E45 #msc:18G55 #msc:53C15
paper · pdf · doi:10.1515/crelle.2007.085
published as Journal fur die reine und angewandte Mathematik (Crelles Journal) 612 (2007), 81-127 · 40 pages, no figures
arxiv created 2006/09/26 · openalex publication_date 2007/01/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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 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. In the smooth setting we discuss an application of our result to the description of quantum traces for a Poisson Lie algebroid.