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Tamarkin’s proof of Kontsevich formality theorem. (English) Zbl 1081.16014

The paper is devoted to an extended version of lectures given at a Luminy colloquium on “Quantification par déformation”. The main goal is to explain in details the proof by Tamarkin of an algebraic version of Kontsevich’s formality theorem: Let \(A\) be a polynomial algebra over a field \(k\) of characteristic zero and let \({\mathcal C}=C^*(A;A)\) be the cohomological complex of \(A\) with coefficients at \(A\). The dg Lie algebra \({\mathcal C}[1]\) is formal, that is, \({\mathcal C}[1]\) is isomorphic in the homotopy category of dg Lie algebras to its cohomology. The needed background (about operads, cooperads, formality, etc., until Etingof-Kazhdan theory) for understanding the proof is given in details.

MSC:

16E45 Differential graded algebras and applications (associative algebraic aspects)
17B55 Homological methods in Lie (super)algebras
16S80 Deformations of associative rings
18D50 Operads (MSC2010)
58B34 Noncommutative geometry (à la Connes)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
53D55 Deformation quantization, star products
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References:

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