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A formality theorem for Poisson manifolds. (English) Zbl 1066.53145

The standard Formality Theorem, which is a source of a proof of the Kontsevich’ result on deformation quantization of Poisson structures, relates differentiable Hochschild cochains for the associative algebra \(C^\infty(M)\) of smooth functions on a manifold \(M\) with its Hochschild cohomology, i.e. multivector fields on \(M\). This relation is a homomorphism ‘up to homotopy’ of graded Lie algebras: the Hochschild cochains \(\mathfrak g_2\) with the Gerstenhaber bracket on one hand and the multivector fields \(\mathfrak g_1\) with the Schouten bracket on the other.
The authors extend this result assuming that \((M,\pi)\) is a Poisson manifold. This puts a deformed associative algebra structure on \(C^\infty(M)[[\hbar]]\) given by a star-product \(*\) associated with \(\pi\) and defines deformed complexes: \((\mathfrak g_2[[\hbar]],b_*)\) with the deformed Hochschild differential associated with the deformed associative structure \(*\), and \((\mathfrak g_1[[\hbar]],\hbar\partial_\pi)\) with the Poisson cohomology differential \(\partial_\pi\). A version of the Hochschild-Kostant-Rosenberg result states that these complexes are quasi-isomorphic.
The main result in the paper is a formality theorem for these two differential graded Lie algebras. In fact, as the Gerstenhaber bracket with the cup-product can be extended to a homotopy Gerstenhaber algebra, the required homomorphism can be chosen even Gerstenhaber ‘up to homotopy’. The proof follows the ideas of D. Tamarkin [Another proof of M. Kontsevich formality theorem, math.QA/9803025] and makes use of the Etingof-Kazhdan’s results on dequantization and quantization of Lie bialgebras [P. Etingof and D. Kazhdan, Sel. Math., New Ser. 4, No. 2, 213–231, 233–269 (1998; Zbl 0915.17009)].

MSC:

53D55 Deformation quantization, star products
17B55 Homological methods in Lie (super)algebras
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
18D50 Operads (MSC2010)
16S80 Deformations of associative rings

Citations:

Zbl 0915.17009
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