
Journal of Lie Theory 22 (2012), No. 1, 251268 Copyright Heldermann Verlag 2012 Structure of the Coadjoint Orbits of Lie Algebras Ihor V. Mykytyuk Institute of Mathematics, Pedagogical University, Podchorazych Str. 2, 30084 Cracow, Poland and: Inst. of Applied Problems of Mathematics and Mechanics, Naukova Str. 3b, 79601 Lviv, Ukraine mykytyuk_i@yahoo.com We study the geometrical structure of the coadjoint orbits of an arbitrary complex or real Lie algebra g containing some ideal n. It is shown that any coadjoint orbit in g* is a bundle with the affine subspace of g* as its fibre. This fibre is an isotropic submanifold of the orbit and is defined only by the coadjoint representations of the Lie algebras g and n on the dual space n*. The use of this fact gives a new insight into the structure of coadjoint orbits and allows us to generalize results derived earlier in the case when g is a semidirect product with an Abelian ideal n. As an application, a necessary condition of integrality of a coadjoint orbit is obtained. Keywords: Coadjoint orbit, integral coadjoint orbit. MSC: 57S25, 17B45, 22E45, 53D20 [ Fulltextpdf (347 KB)] for subscribers only. 