Marsden-Weinstein symplectic reduction theorem

Statement

Let be a compact, connected Lie group with Lie algebra , and let be a Hamiltonian G-space.

Generalizations on hypotheses

There are some small generalizations that could be made the hypotheses of the theorem, but this is the version I need for my current research, so I’m sticking with compactness.

Suppose for , acts freely on the manifold . Then there is a unique symplectic form on such that

where is the projection onto the quotient manifold and is the inclusion.

In this way, is a symplectic manifold and is called the symplectic or Marsden-Weinstein quotient.

Proof

We need to understand both the smooth manifold structure and the symplectic structure of the quotient.

Smooth manifold structure

First, we need to check that all the setup makes sense. In other words, we must verify is a smooth manifold.

Note that is a closed subgroup of and so is a Lie group itself.

Claim: is a submanifold of .

We can prove this using the regular value theorem, so we turn our attention to

However, we need a little help from some lemmas first.

Lemma 1: tangent space of orbits

Proof

Let , and be the 1-parameter subgroup such that and .

Then (see exponential map). Therefore, for the smooth curve

and , so

Next, let . Then there exists a curve for an interval such that and .

Since the action map is smooth, that means that there is some curve such that . Note that and for some . This means that for the orbit map

we have

Therefore, .

Lemma 2: kernel and image of

and Where is the Lie algebra of for . (See also annihilator of vector space.)

Proof

First, by the definition of the moment map for any and . Thus,

Next, let , so for some . For any , we can think of as a linear function on . Thus,

Let . That means that , so

Thus,

Hence, and .

Then, note that , so .

Thus, since , this means that .

By the property of annihilators, , so .

This means that and therefore .

Back to construction of smooth structure

Note that for . This is because for ,

Since by the hypotheses, acts freely on ,

Thus by lemma 2, This means that for every , is surjective, so is a regular value. Therefore, by the regular value theorem is a submanifold of .

Thus, we can use this to look at . Since is -equivariant, for and ,

Thus, is invariant, so it is the union of orbits of . This means that the orbit space makes sense. Note that acts smoothly and freely by assumption, and since is compact, the action is certainly proper. Hence, by the quotient manifold theorem, there is a unique smooth structure such that

is a smooth submersion.

Symplectic structure

In order to talk about anything with the symplectic structure on , we need to understand the tangent spaces of the orbit space.

By construction, any quotient manifold is built such that

is a smooth submersion. Thus,

is surjective. Therefore, by the first isomorphism theorem,

Since is a local defining map for the subspace of the orbit containing , then we know

Hence,

Now applying this idea to the submanifold we are studying, we see

Change in notation

Note the change in notation for the orbit below. It is an intentional decision to show that the orbit is of the stabilizer, not the whole group.

Now that we have a better way of understanding the tangent space at points on the reduced manifold, we need to understand the symplectic form as it relates to the tangent spaces.

Building the symplectic form

Since there is a canonical symplectic form when taking the quotient with an isotropic subspace, it suffices to show that is isotropic and

Since is a locally defining map for by construction then by lemma 2,

Same as above, since is a locally defining map for then we know that for ,

is a subspace.

Hence,

and is isotropic. Note when restricting to the submanifold then

Thus, let

Is this a symplectic form?

Antisymmetric: inherits this property from .

Non-degenerate: By this lemma, the canonical form is non-degenerate.

Closed: Since is surjective,

Well-defined: Checked pointwise by the lemma.