Structure of the coadjoint orbits of Lie groups

TL;DR

This paper investigates the geometric structure of coadjoint orbits of Lie algebras, revealing they are fiber bundles with affine subspaces as fibers, and applies symplectic geometry to derive new insights and results.

Contribution

It introduces a new geometric perspective on coadjoint orbits, generalizing previous results to arbitrary Lie algebras with ideals, and provides new proofs and conditions related to Lie algebra indices and orbit integrality.

Findings

Coadjoint orbits are fiber bundles with affine subspaces as fibers.

The fibers are isotropic submanifolds defined by coadjoint representations.

New proof of the formula for the index of Lie algebra and a necessary condition for orbit integrality.

Abstract

We study the geometrical structure of the coadjoint orbits of an arbitrary complex or real Lie algebra ${\mathfrak g}$ containing some ideal ${\mathfrak n}$. It is shown that any coadjoint orbit in ${\mathfrak g}^*$ is a bundle with the affine subspace of ${\mathfrak g}^*$ as its fibre. This fibre is an isotropic submanifold of the orbit and is defined only by the coadjoint representations of the Lie algebras ${\mathfrak g}$ and ${\mathfrak n}$ on the dual space ${\mathfrak n}^*$. The use of this fact and an application of methods of symplectic geometry give a new insight into the structure of coadjoint orbits and allow us to generalize results derived earlier in the case when ${\mathfrak g}$ is a split extension using the Abelian ideal ${\mathfrak n}$ (a semidirect product). As applications, a new proof of the formula for the index of Lie algebra and a necessary condition of integrality of a coadjoint orbit are obtained.