Hamiltonian group actions and moment maps

Definition 6.3

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 .)


Exercise 6.5

If and are smooth functions on , define the Poisson bracket of and by

Show that 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.

 


Definition 6.4

We denote by the unique (connected) subgroup of with

 


Definition 6.5

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 given by
is Hamiltonian.

 


Example 6.5

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

and hence 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.


Definition 6.6

A submanifold is called isotropic if , and Lagrangian if in addition .

 


Remark 6.5

Lagrangian submanifolds appear in quantum mechanics as the smallest submanifolds on which a which a wavefunction can be localised. (cf. Heisenberg uncertainty principle)

 


Example 6.6

Consider acting on via

The corresponding Hamiltonians are generating the rotation.

 


Example 6.7

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).

 


Definition 6.7

Let be a symplectic manifold with a Hamiltonian action of and associated Hamiltonians . The moment map of the action is given by

 


Remark 6.6

The moment map for a torus action is well-defined up to translation in .

 


Example 6.8

For acting on as in Example 6.6, have

The moment image is
For example, for , the moment image is the quadrant in below.

For every point on the interior, we have

is a free -orbit.

 


In general, we have the following cool fact.


Theorem 6.1

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.


Definition 6.8

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


Exercise 6.6

In the setup of Definition of the moment map, assume that satisfies

Show that the following are equivalent.

  1. for all and .
  2. The map

is a homomorphism of (left) -modules, where acts on via the adjoint action, and on by .

  1. The map is a homomorphism of -modules.
  2. The map is a homomorphism of Lie algebras.
 


Lemma 6.1

Let be a compact Lie group with a Hamiltonian action on a symplectic manifold . Then there exists an equivariant moment map .

 
Proof

We have a commutative diagram of -modules (MISSING) with exact rows, where

Since the bottom row is an exact sequence of finite-dimensional -modules with compact, it has a -equivariant splitting. In particular, we can construct a map of -modules covering as shown, which induces an equivariant moment map .

 
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