nLab coadjoint orbit

Definition

Given a Lie group $G$, it acts smoothly on the dual $g^*$ of its Lie algebra $g$ by the coadjoint action. The orbits of that action are called coadjoint orbits.

Coadjoint orbits are especially important in the orbit method of representation theory or, more generally, geometric quantization.

Sometimes coadjoint orbits are studied in the infinite-dimensional case (for example in study of Virasoro algebra).

Properties

As symplectic leaves of the Lie-Poisson structure

The dual $g^*$ of a (say finite-dimensional real) Lie algebra has a structure of a Poisson manifold with the Poisson structure due to A. Kirillov and Souriau, called the Lie-Poisson structure, namely for any $a\in g^*$,

$$\{ f, g\}(a) \;\coloneqq\; \langle [d f_a, d g_a],a\rangle$$

The coadjoint orbits are the symplectic leaves of that structure; hence each orbit is a symplectic manifold.

Examples

• Wilson loops and 1d Chern-Simons σ-models with target the coadjoint orbit

Related concepts

• orbit method

• flag variety, coset space

• Schubert calculus

References

• François Bayen, Moshé Flato, Christian Fronsdal, André Lichnerowicz, Daniel Sternheimer, pp. 66 in: Deformation theory and quantization. I. Deformations of symplectic structures., Annals of Physics 111 1 (1978) 61-110 [doi:10.1016/0003-4916(78)90224-5]

• Bradley N. Currey, The Structure of the Space of Coadjoint Orbits of an Exponential Solvable Lie Group, ransactions of the American Mathematical Society Vol. 332, No. 1 (Jul., 1992), pp. 241-269, (JSTOR)

In particle physics:

• Ismaël Ahlouche Lahlali, Josh A. O’Connor: Coset symmetries and coadjoint orbits [arXiv:2411.05918]