3Hamiltonian vector fields
III Symplectic Geometry
3.4 Hamiltonian actions
In the remainder of the course, we are largely interested in how Lie groups can
act on symplectic manifolds via Hamiltonian vector fields. These are known as
Hamiltonian actions. We first begin with the notion of symplectic actions.
Let (M, ω) be a symplectic manifold, and G a Lie group.
Definition
(Symplectic action)
.
A symplectic action is a smooth group action
ψ
:
G → Diff
(
M
) such that each
ψ
g
is a symplectomorphism. In other words, it
is a map G → Symp(M, ω).
Example.
Let
G
=
R
. Then a map
ψ
:
G → Diff
(
M
) is a one-parameter group
of transformations
{ψ
t
:
t ∈ R}
. Given such a group action, we obtain a complete
vector field
X
p
=
dψ
t
dt
(p)
t=0
.
Conversely, given a complete vector field X, we can define
ψ
t
= exp tX,
and this gives a group action by R.
Under this correspondence, symplectic actions correspond to complete sym-
plectic vector fields.
Example.
If
G
=
S
1
, then a symplectic action of
S
1
is a symplectic action of
R which is periodic.
In the case where
G
is
R
or
S
1
, it is easy to define what it means for an
action to be Hamiltonian:
Definition
(Hamiltonian action)
.
An action of
R
or
S
1
is Hamiltonian if the
corresponding vector field is Hamiltonian.
Example. Take (S
2
, ω = dθ ∧dh). Then we have a rotation action
ψ
t
(θ, h) = (θ + t, h)
generated by the vector field
∂
∂θ
. Since
ι
∂
∂θ
ω
= d
h
is exact, this is in fact a
Hamiltonian S
1
action.
Example. Take (T
2
, dθ
1
∧ dθ
2
). Consider the action
ψ
t
(θ
1
, θ
2
) = (θ
1
+ t, θ
2
).
This is generated by the vector field
∂
∂θ
1
. But
ι
∂
∂θ
1
ω
= d
θ
2
, which is closed but
not exact. So this is a symplectic action that is not Hamiltonian.
How should we define Hamiltonian group actions for groups that are not
R
or
S
1
? The simplest possible next case is the torus
G
=
T
n
=
S
1
× ··· × S
1
.
If we have a map
ψ
;
T
n
→ Symp
(
M, ω
), then for this to be Hamiltonian, it
should definitely be the case that the restriction to each
S
1
is Hamiltonian in
the previous sense. Moreover, for these to be compatible, we would expect each
Hamiltonian function to be preserved by the other factors as well.
For the general case, we need to talk about the Lie group of
G
. Let
G
be a
Lie group. For each g ∈ GG, there is a left multiplication map
L
g
: G → G
a 7→ ga.
Definition
(Left-invariant vector field)
.
A left-invariant vector field on a Lie
group G is a vector field X such that
(L
g
)
∗
X = X
for all g ∈ G.
We write
g
for the space of all left-invariant vector fields on
G
, which comes
with the Lie bracket on vector fields. This is called the Lie algebra of G.
If
X
is left-invariant, then knowing
X
e
tells us what
X
is everywhere, and
specifying
X
e
produces a left-invariant vector field. Thus, we have an isomor-
phism g
∼
=
T
e
G.
The Lie algebra admits a natural action of
G
, called the adjoint action. To
construct this, note that G acts on itself by conjugation,
ϕ
g
(a) = gag
−1
.
This fixes the identity, and taking the derivative gives
Ad
g
:
g → g
, or equivalently,
Ad
is a map
Ad
:
G → GL
(
g
). The dual
g
∗
admits the dual action of
G
, called
the coadjoint action. Explicitly, this is given by
hAd
∗
g
(ξ), xi = hξ, Ad
g
xi.
An important case is when
G
is abelian, i.e. a product of
S
1
’s and
R
’s, in which
case the conjugation action is trivial, hence the (co)adjoint action is trivial.
Returning to group actions, the correspondence between complete vector
fields and
R
/
S
1
actions can be described as follows: Given a smooth action
ψ : G → Diff(M) and a point p ∈ M, there is a map
G → M
g 7→ ψ
g
(p).
Differentiating this at e gives
g
∼
=
T
e
G → T
p
M
X 7→ X
#
p
.
We call
X
#
the vector field on
M
generated by
X ∈ g
. In the case where
G
=
S
1
or
R
, we have
g
∼
=
R
, and the complete vector field corresponding to the action
is the image of 1 under this map.
We are now ready to define
Definition
(Hamiltonian action)
.
We say
ψ
:
G → Symp
(
M, ω
) is a Hamiltonian
action if there exists a map µ : M → g
∗
such that
(i)
For all
X ∈ g
,
X
#
is the Hamiltonian vector field generated by
µ
X
, where
µ
X
: M → R is given by
µ
X
(p) = hµ(p), Xi.
(ii) µ
is
G
-equivariant, where
G
acts on
g
∗
by the coadjoint action. In other
words,
µ ◦ ψ
g
= Ad
∗
g
◦µ for all g ∈ G.
µ is then called a moment map for the action ψ.
In the case where G is abelian, condition (ii) just says µ is G-invariant.
Example. Let M = C
n
, and
ω =
1
2
X
j
dz
j
∧ d¯z
j
=
X
i
r
j
dr
j
∧ dθ
j
.
We let
T
n
= {(t
1
, . . . , t
n
) ∈ C
n
: |t
k
| = 1 for all k},
acting by
ψ
(t
1
,...,t
n
)
(z
1
, . . . , z
n
) = (t
k
1
1
z
1
, . . . , t
k
n
n
z
n
)
where k
1
, . . . , k
n
∈ Z.
We claim this action has moment map
µ : C
n
→ (t
n
)
∗
∼
=
R
n
(x
1
, . . . , z
n
) 7→ −
1
2
(k
1
|z
1
|
2
, . . . , k
n
|z
n
|
2
).
It is clear that this is invariant, and if X = (a
1
, . . . , a
n
) ∈ t
n
∈ R
n
, then
X
#
= k
1
a
1
∂
∂θ
1
+ ··· + k
n
a
n
∂
∂θ
n
.
Then we have
dµ
X
= d
−
1
2
X
k
j
a
j
r
2
j
= −
X
k
j
a
j
r
j
dr
j
= ι
X
#
ω.
Example.
A nice example of a non-abelian Hamiltonian action is coadjoint
orbits. Let
G
be a Lie group, and
g
the Lie algebra. If
X ∈ g
, then we get a
vector field
g
X
#
generated by
X
via the adjoint action, and also a vector field
g
X on g
∗
generated by the co-adjoint action.
If ξ ∈ g
∗
, then we can define the coadjoint orbit through ξ
O
ξ
= {Ad
∗
g
(ξ) : g ∈ G}.
What is interesting about this is that this coadjoint orbit is actually a symplectic
manifold, with symplectic form given by
ω
ξ
(X
#
ξ
, Y
#
ξ
) = hξ, [X, Y ]i.
Then the coadjoint action of
G
on
O
ξ
has moment map
O
ξ
→ g
∗
given by the
inclusion.