Survey of the deformation quantization of commutative families (Q2275151)

It is known that every Poisson manifold \(X\) allows a deformation quantization, that is a non-commutative product on \(C^*(X)\). This paper adresses the problem to find a commutative (instead of a non-commutative algebra), quantizing a commutative Poisson subalgebra. Such commutative algebras have important applications. In this survey, the authors discuss various approaches and known results. The paper is self-contained. The case where \(X\) is the dual of a Lie algebra is studied with details. For the entire collection see [Zbl 1412.37003].