- journal article metadata
European Mathematical Society Publishing House
2017-07-02 23:45:01
Journal of Noncommutative Geometry
J. Noncommut. Geom.
JNCG
1661-6952
1661-6960
Global analysis, analysis on manifolds
General
10.4171/JNCG
http://www.ems-ph.org/doi/10.4171/JNCG
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
11
2017
2
Deformation quantization of integrable systems
Georgy
Sharygin
Lomonosov Moscow State University and ITEP, Moscow, Russia
Dmitry
Talalaev
Lomonosov Moscow State University and ITEP, Moscow, Russia
Quantization, integrable systems, Hochschild relative cohomology
In this paper we address the following question: is it always possible to choose a deformation quantization of a Poisson algebra $\mathcal{A}$ so that certain Poisson-commutative subalgebra $\mathcal{C}$ in it remains commutative? We define a series of cohomological obstructions to this, that take values in the Hochschild cohomology of $\mathcal{C}$ with coefficients in $\mathcal{A}$. In some particular case of the pair $(\mathcal{A},\mathcal{C})$ we reduce these classes to the classes of the Poisson relative cohomology of the Hochschild cohomology. We show, that in the case, when the algebra $\mathcal{C}$ is polynomial, these obstructions coincide with the previously known ones, those which were defined by Garay and van Straten.
Differential geometry
Functional analysis
741
756
10.4171/JNCG/11-2-9
http://www.ems-ph.org/doi/10.4171/JNCG/11-2-9