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2016

Deformation quantization and the action of Poisson vector fields

Sharygin, G.

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As one knows, for every Poisson manifold $M$ there exists a formal noncommutative deformation of the algebra of functions on it; it is determined in a unique way (up to an equivalence relation) by the given Poisson bivector.

Let a Lie algebra $\mathfrak g$ act by derivations on the functions on $M$.
The main question, which we shall address in this paper is whether it is possible to lift this action to the derivations on the deformed algebra.
It is easy to see, that when dimension of $\mathfrak g$ is $1$, the only necessary and sufficient condition for this is that the given action is by Poisson vector fields.

Deformation quantization and the action of Poisson vector fields · Around

Built on

I. M. Krichever Commutative rings of ordinary linear differential operators, Funktsional. Anal. i Prilozh. 12
1978
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M. Cahen, S. Gutt, M. De Wilde, Local cohomology of the algebra of smooth functions on a connected manifold, Lett. in Math. Phys. 4
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B.V.Fedosov Index theorems (in Russian), Russian Math Surveys, 1991, 65
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Earlier work this paper cites.

Similar

J.-L. Loday, Cyclic homology, Springer, 1992
1992
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M. Kontsevich, Deformation quantization of Poisson manifolds; Letters in Mathematical Physics, 66
2003
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B. Keller, Introduction to Kontsevich’s quantization theorem, Notes covering the material of a minicourse given at the EMALCA III, Morelia (México), August 2003
2003
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M. Kontsevich, Deformation quantization of Poisson manifolds, I; preprint: arXiv:q-alg/9709040
Cited in the paper.

Then

A. Belov-Kanel, M. Kontsevich The Jacobian conjecture is stably equivalent to the Dixmier conjecture; Moscow Mathematical Journal, 7
2007
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D. Calaque, M. van den Bergh, Hochschild cohomology and Atiyah classes; Advances in Mathematics 224
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