By D. L. Johnson

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Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Quantum mechanics as a deformation of classical mechanics, Lett. Math. Phys. 1 (1977) 521–530 and Deformation theory and quantization, part I, Ann. of Phys. 111 (1978) 61–110. 4. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization, part II, Ann. of Phys. 111 (1978) 111–151 5. S. 144 (2002) 38–53. 6. A. Cattaneo, G. Felder and L. Tomassini, From local to global deformation quantization of Poisson manifolds, Duke Math.

F k+1 ) φ ∀ f j ∈ C ∞ (M ). (34) k Similarly we associate to a differential operator O ∈ Dpoly (M ) a section Oφ ∈ Γ(Dpoly ) determined by the fact that Oφ (fφ1 , . . , fφk+1 ) = O(f 1 , . . , f k+1 ) φ ∀ f j ∈ C ∞ (M ). 4. e. iff it is of the form Fφ D k for some F ∈ Tpoly (M ); a section of Dpoly is DF poly −horizontal if and only k if is of the form Oφ for some O ∈ Dpoly (M ). Dolgushev constructs his L∞ -morphism in two steps from the fiberwize Kontsevich formality from Ω(M, Tpoly ) to Ω(M, Dpoly ) building first a twist which depends only on the curvature and its covariant derivatives, then building a contraction using the vanishing of the DG cohomology.

X. Defining dx = (−1)|x| Q1 x [x, y] := Q2 (x ∧ y) = (−1)|x|(|y|−1) Q2 (x, y), (16) November 4, 2009 13:57 WSPC - Proceedings Trim Size: 9in x 6in ewmproc Deformation Quantisation and Connections 21 the above relations show that d is a differential on V , and [ , ] is a graded skewsymmetric bilinear map from V × V → V satisfying (−1)|x||z| [[x, y], z] + (−1)|y||x| [[y, z], x](−1)|z||y| [[z, x], y] + terms in Q3 = 0 and d[x, y] = [dx, y] + (−1)|x| [x, dy]. 1. Any L∞ –algebra (V, Q) so that all the Taylor coefficients Qn of Q vanish for n > 2 yields a differential graded Lie algebra and vice versa.

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