Abstract
In this survey chapter we discuss various approaches and known results, concerning the following question: when is it possible to find a commutative extension of a Poisson-commutative subalgebra in \(C^\infty (X)\) (where X is a Poisson manifold) to a commutative subalgebra in the deformation quantization of X, the algebra \(\mathscr {A}(X)\). A case of particular interest, which we consider with certain detail is the situation, when \(X=\mathfrak g^*\) and the commutative subalgebra is constructed by the argument shift method.
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Sharygin, G., Konyaev, A. (2018). Survey of the Deformation Quantization of Commutative Families. In: Buchstaber, V., Konstantinou-Rizos, S., Mikhailov, A. (eds) Recent Developments in Integrable Systems and Related Topics of Mathematical Physics. MP 2016. Springer Proceedings in Mathematics & Statistics, vol 273. Springer, Cham. https://doi.org/10.1007/978-3-030-04807-5_8
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