Central Extensions of Lie Algebras, Dynamical Systems, and Symplectic Nilmanifolds

We describe the relations between Euler’s equations on central extensions of Lie algebras and Euler’s equations on the original algebras that we extend. We consider a special infinite sequence of central extensions of nilpotent Lie algebras constructed from the Lie algebra of formal vector fields on the line, and describe the orbits of coadjoint representations for these algebras. By using the compact nilmanifolds constructed from these algebras by I. K. Babenko and the author, we show that the covering Lie groups for symplectic nilmanifolds can have any rank as solvable Lie groups.