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تسجيل مستخدم جديدA proof of Tsygans formality conjecture for Hamiltonian actions
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In this short note we prove an equivariant version of the formality of multidiffirential operators for a proper Lie group action. More precisely, we show that the equivariant Hochschild-Kostant-Rosenberg quasi-isomorphism between the cohomology of the equivariant multidifferential operators and the complex of equivariant multivector fields extends to an $L_infty$-quasi-isomorphism. We construct this $L_infty$-quasi-isomorphism using the $G$-invariant formality constructed by Dolgushev. This result has immediate consequences in deformation quantization, since it allows to obtain a quantum moment map from a classical momentum map with respect to a $G$-invariant Poisson structure.
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Proofs of Tsygans formality conjectures for chains would unlock important algebraic tools which might lead to new generalizations of the Atiyah-Patodi-Singer index theorem and the Riemann-Roch-Hirzebruch theorem. Despite this pivotal role in the trad
itional investigations and the efforts of various people the most general version of Tsygans formality conjecture has not yet been proven. In my thesis I propose Fedosov resolutions for the Hochschild cohomological and homological complexes of the algebra of functions on an arbitrary smooth manifold. Using these resolutions together with Kontsevichs formality quasi-isomorphism for Hochschild cochains of R[[y_1, >..., y_d]] and Shoikhets formality quasi-isomorphism for Hochschild chains of R[[y_1,..., y_d]] I prove Tsygans formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold. The construction of the formality quasi-isomorphism for Hochschild chains is manifestly functorial for isomorphisms of the pairs (M, abla), where M is the manifold and abla is an affine connection on the tangent bundle. In my thesis I apply these results to equivariant quantization, computation of Hochschild homology of quantum algebras and description of traces in deformation quantization.
Boris Shoikhet noticed that the proof of lemma 1 in section 2.3 of math.QA/0504420 contains an error. In this note I give a correct proof of this lemma which was suggested to me by Dmitry Tamarkin. The correction does not change the results of math.QA/0504420.
In this paper we provide a quantization via formality of Poisson actions of a triangular Lie algebra $(mathfrak g,r)$ on a smooth manifold $M$. Using the formality of polydifferential operators on Lie algebroids we obtain a deformation quantization o
f $M$ together with a quantum group $mathscr{U}_hbar(mathfrak{g})$ and a map of associated DGLAs. This motivates a definition of quantum action in terms of $L_infty$-morphisms which generalizes the one given by Drinfeld.
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