# How to find Darboux coordinates?

I would like to find local Darboux coordinates for symplectic structures on coadjoint orbits of some nilpotent Lie group. At first, I thought that this would be not very hard, and that it would be possible to find a change variables by simply playing with the Poisson bracket relations. But eventually I've got stuck.

So, is there a more or less general way to find Darboux coordinates on coadjoint orbits?

There is a large literature on this. I would recommend the papers

• MR0379752 Vergne, Michèle La structure de Poisson sur l'algèbre symétrique d'une algèbre de Lie nilpotente. Bull. Soc. Math. France 100 (1972), 301–335.

• MR0787118 Kamalin, S. A.; Perelomov, A. M. Construction of canonical coordinates on polarized coadjoint orbits of Lie groups. Comm. Math. Phys. 97 (1985), no. 4, 553–568.

• MR0976688 Bonnet, Pierre Paramétrisation du dual d'une algèbre de Lie nilpotente. Ann. Inst. Fourier (Grenoble) 38 (1988), no. 3, 169–197.

• MR0958263 Pedersen, Niels Vigand On the symplectic structure of coadjoint orbits of (solvable) Lie groups and applications. I. Math. Ann. 281 (1988), no. 4, 633–669.

• MR2538595 Arnal, Didier; Currey, Bradley; Dali, Bechir Construction of canonical coordinates for exponential Lie groups. Trans. Amer. Math. Soc. 361 (2009), no. 12, 6283–6348.

which have among their main results, the construction of (global) Darboux coordinates on coadjoint orbits of nilpotent (and more generally exponential solvable) Lie groups.

• Found them =) Unfortunately I was not able to fully understand the algorithm and mainly reverse-engineered the process from Pedersen's examples. But I'll try to get through these papers in future. Thanks again. – Andrey Jul 28 '14 at 14:20
• @Andrey Yeah, the subject tends to produce rather terse papers. If memory serves, I found Pedersen the hardest-going and Bonnet the easiest to follow. As you say, it helps to just do a lot of examples. – Francois Ziegler Jul 28 '14 at 15:18