Poisson Manifold - Example (Lie-Poisson Manifold)

Example (Lie-Poisson Manifold)

If g is a finite-dimensional Lie algebra and g* is its dual vector space, then the Lie bracket induces a Poisson structure on g*.

More precisely, we identify the cotangent bundle of the manifold g*, i.e., the dual of g* with the given Lie algebra g. Then for two functions f₁ and f₂ on g*, and a point x ∈ g*, we may define

,

where the Lie bracket is computed in g through the isomorphism:

.

If e_k are local coordinates on g, then the Poisson bivector field is given by

,

where the are the structure constants of g.

The symplectic leaves of this Lie-Poisson manifold are the co-adjoint orbits of the Lie algebra used for the orbit method.