nLab linear Poisson structure

see at Lie-Poisson structure for more

Idea

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)