Abstract. In this paper, we compute the Gerstenhaber bracket on the Hochschild cohomology of C ∞ (M ) G for a finite group G acting on a compact manifold M . Using this computation, we obtain geometric descriptions for all noncommutative Poisson structures on C ∞ (M ) G when M is a symplectic manifold. We also discuss examples of deformation quantizations of these noncommutative Poisson structures.

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.Comment: 40 pages, no figure

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We show that any coboundary Lie bialgebra can be quantized. For this, we prove that Etingof-Kazhdan quantization functors are compatible with Lie bialgebra twists, and if such a quantization functor corresponds to an even associator, then it is also compatible with the operation of taking coopposites. We also use the relation between the Etingof-Kazhdan construction of quantization functors and the alternative approach to this problem, which was established in a previous work.