The linear dual vector space of a finite-dimensional Lie algebra has a natural Poisson structure of Kirillov-Kostant-Souriau which is called the linear Poisson structure. Its symplectic leaves are precisely the coadjoint orbits. The geometric quantization of linear Poisson structures is important in the representation theory; the corresponding star products are important basic examples in the study of deformation quantization (e.g. Gutt 11, section 2.2).

References

Kostant

S. Gutt, Lett. Math. Phys.

Mark Rieffel, Lie group convolution algebras as deformation quantizations of linear Poisson structures, Amer. J. of Mathematics 112, No. 4 (Aug., 1990), pp. 657-685, jstor

Simone Gutt, section 2.2. of Deformation quantization of Poisson manifolds, Geometry and Topology Monographs 17 (2011) 171-220 (pdf)

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