moment map

Definition

A smooth function is called a moment map if

• (i.e. ) for all
• is -equivariant

Where

• is the fundamental vector field of of the Lie algebra of the Lie group .
• which evaluates in

Motivation

The goal is to form a systematic way to choose the Hamiltonian functions needed to make a Hamiltonian action.

So, we are looking for some map (this is sometimes called the comoment map)

such that

We know that we need this map to be linear on , so instead of defining the map itself, we can map into which guarantees linearity.

Consider a map

then using the pairing for all and get a map

So the goal is to define such that

is a Lie algebra homomorphism (using the Poisson bracket on ).

Equivariance

Equivariance in context

What does equivariance mean here? Usually, equivariance is taken to mean So in this context, we consider the coadjoint action of on :

Why require equivariance? At first glance, this would seem like a somewhat arbitrary condition.

Theorem statement

A map is equivariant if and only if is a Lie algebra homomorphism (using the definitions of given above.)

Proof

Assume is equivariant, then we must show that

Using the definition of the Poisson bracket for symplectic manifolds,

Where is the flow at time of the vector field .

Note since , (using the exponential map) then

Next assume . It remains to show that

Since is connected, for every , so it suffices to prove that this works just under the image of , i.e.

Note that this is the same condition as

where (respectively ) represents the flow of the fundamental vector field of on () at time .

Recall that given a smooth function between smooth manifolds, if two vector fields and are -related then

Hence, it suffices to show that the vector fields are -related. That is,

Note that , so and .

Let , which can be canonically identified as . Then for

because is a linear function on . Then we see that

Thus, it is clear that

So is equivariant.