Tangent cohomology and cup-product for the Kontsevich quantization. (Cohomologie tangente et cup-produit pour la quantification de Kontsevich.) (French) Zbl 1051.53072

Manchon, Dominique; Torossian, Charles

Ann. Math. Blaise Pascal 10, No. 1, 75-106 (2003).

This paper gives a detailed proof of the fact that Kontsevich’s formality quasi-isomorphism is compatible with cup-products on tangent cohomology spaces. Let \(M\) be a smooth manifold. The Kontsevich formality theorem states the existence of a quasi-isomorphism \(L\infty\) between the Lie algebra \(g_1\) of polyvector fields on \(M\) and the Lie algebra \(g_2\) of polydifferential operators on \(M\).
In the flat case: \(M=\mathbb R^d\), the quasi-isomorphism between \(g_1\) and \(g_2\), noted \(U\), is such that the derivative of \(U\), at any formal 2-tensor \(h\gamma\), induces an isomorphism of graded commutative algebras from Poisson cohomology space to Hochschild cohomology relative to the deformed multiplication built from \(h\gamma\) via \(U\).
In the present work, the authors clarify the proof of this important result, in a pedagogical and illuminating way, and with all signs and orientations precised.

Reviewer: Angela Gammella (Nogent sur Oise)

MSC:

53D55

Deformation quantization, star products

Keywords:

Kontsevich quantization; formality quasi-isomorphism; Poisson cohomology; Hochschild cohomology

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