If is a symplectic manifold, the group of symplectomorphisms of is the infinite-dimensional Lie group
By Remark 6.3, the Lie algebra of is
the space of symplectic vector fields on
. (This sits inside the space of all smooth vector fields, which is the Lie algebra of the group of diffeomorphisms of
If and are smooth functions on , define the Poisson bracket of and by
is a Lie bracket on
, and that the map
is a Lie algebra homomorphism with respect to the usual Lie bracket of vector fields. Deduce that the space of Hamiltonian vector fields
is a Lie subalgebra.
We denote by the unique (connected) subgroup of with
Let be a connected Lie group acting on a symplectic manifold . We say that the action is Hamiltonian if the action of factors through
Equivalently, the action is Hamiltonian if for every
, associated vector field
Let be a torus. Then with Lie bracket . If we have a Hamiltonian action of on , then if is a basis for , we get a collection of Hamiltonians on . Since , passing to the corresponding vector fields , we have
is constant for all
. With a little more work, we can actually show that
As a slight aside, note that since the span the tangent space to each orbit, this implies that when restricted to a -orbit. Submanifolds of this type appear sufficiently often in symplectic topology that they have a special name.
A submanifold is called isotropic if , and Lagrangian if in addition .
Lagrangian submanifolds appear in quantum mechanics as the smallest submanifolds on which a which a wavefunction can be localised. (cf. Heisenberg uncertainty principle)
Consider acting on via
The corresponding Hamiltonians are
Consider with symplectic form given by the area form. (If we identify with , then this is also the natural symplectic structure coming from the Fubini-Study form.) Then the action on given by rotation about the -axis is Hamiltonian, and the Hamiltonian is just the height function (projection to -coordinate).
Let be a symplectic manifold with a Hamiltonian action of and associated Hamiltonians . The moment map of the action is given by
The moment map for a torus action is well-defined up to translation in .
For acting on as in Example 6.6, have
The moment image is
For example, for
, the moment image is the quadrant in
For every point on the interior, we have
is a free
In general, we have the following cool fact.
If is the moment map for a Hamiltonian torus action, then is a convex polytope.
We can define moment maps for more general group actions as follows.
Let be a connected compact Lie group with Lie algebra and a Hamiltonian action on a symplectic manifold . A map is called an equivariant moment map if
- for all and , and
- for .
Here is (the inverse of) the dual to the adjoint action , which is the map induced on the tangent space at the identity of by
In the setup of Definition of the moment map, assume that satisfies
Show that the following are equivalent.
- for all and .
- The map
is a homomorphism of (left)
via the adjoint action, and on
- The map is a homomorphism of -modules.
- The map is a homomorphism of Lie algebras.
Let be a compact Lie group with a Hamiltonian action on a symplectic manifold . Then there exists an equivariant moment map .
We have a commutative diagram of -modules
with exact rows, where
Since the bottom row is an exact sequence of finite-dimensional
compact, it has a
-equivariant splitting. In particular, we can construct a map of
as shown, which induces an equivariant moment map