- 2.5k Downloads
In this chapter we prove, among other things, that the coadjoint orbits of a Lie group are symplectic manifolds. These symplectic manifolds are, in fact, the symplectic leaves for the Lie-Poisson bracket. This result was developed and used by Kirillov, Arnold, Kostant, and Souriau in the early to mid 1960s, although it had important roots going back to the work of Lie, Borel, and Weil. (See Kirillov [1962, 1976b], Arnold [1966a], Kostant , and Souriau .) Here we give a direct proof. Alternatively, one can give a proof using general reduction theory, as in Marsden and Weinstein  and Abraham and Marsden .
KeywordsSymplectic Form Symplectic Manifold Poisson Structure Symplectic Structure Coadjoint Orbit
Unable to display preview. Download preview PDF.