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Journal of Lie Theory 22 (2012), No. 1, 251--268
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

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

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