It is a general phenomenon that for any Lie algebrag, a Poisson bracket
on functions on the dual spaceg∗can be defined. This is because the Leibniz
property ensures that the Poisson bracket only depends on Ω, its restriction
to linear functions, and linear functions ong∗are elements ofg. So a Poisson
bracket on functions ong∗is given by first defining
Ω(X,X′) = [X,X′] (15.14)forX,X′∈g= (g∗)∗, and then extending this to all functions ong∗by the
Leibniz property.
Such a Poisson bracket on functions on the vector spaceg∗is said to provide a
“Poisson structure” ong∗. In general it will not provide a symplectic structure
ong∗, since it will not be non-degenerate. For example, in the case of the
Heisenberg Lie algebra
g∗=M⊕R
and Ω will be non-degenerate only on the subspaceM, the phase space, which
it will give a symplectic structure.
Digression.The Poisson structure ong∗can often be used to get a symplectic
structure on submanifolds ofg∗. As an example, takeg=so(3), in which case
g∗=R^3 , with antisymmetric bilinear formωgiven by the vector cross-product.
In this case it turns out that if one considers spheres of fixed radius inR^3 ,
ωprovides a symplectic form proportional to the area two-form, giving such
spheres the structure of a symplectic manifold.
This is a special case of a general construction. Taking the dual of the adjoint
representationAdong, there is an action ofg∈Gong∗by the representation
Ad∗, satisfying
(Ad∗(g)·l)(X) =l(Ad(g−^1 )X)
This is called the “co-adjoint” action ong∗. Picking an elementl∈g∗, the
orbitOlof the co-adjoint action turns out to be a symplectic manifold. It comes
with an action ofGpreserving the symplectic structure (the restriction of the
co-adjoint action ong∗to the orbit). In such a case the moment map
μ:Ol→g∗is just the inclusion map. Two simple examples are
- Forg=h 3 , phase spaceM =R^2 with the Heisenberg group action of
equation 15.13 is given by a co-adjoint orbit, takingl∈h∗ 3 to be the dual
basis vector to the basis vector ofh 3 given by the constant function 1 on
R^2. - Forg=so(3)the non-zero co-adjoint orbits are spheres, with radius the
length ofl, the symplectic form described above, and an action ofG=
SO(3)preserving the symplectic form.