3Lie groups
III Differential Geometry
3 Lie groups
We now have a short digression to Lie groups. Lie groups are manifolds with
a group structure. They have an extraordinary amount of symmetry, since
multiplication with any element of the group induces a diffeomorphism of the Lie
group, and this action of the Lie group on itself is free and transitive. Effectively,
this means that any two points on the Lie group, as a manifold, are “the same”.
As a consequence, a lot of the study of a Lie group reduces to studying
an infinitesimal neighbourhood of the identity, which in turn tells us about
infinitesimal neighbourhoods of all points on the manifold. This is known as the
Lie algebra.
We are not going to go deep into the theory of Lie groups, as our main focus
is on differential geometry. However, we will state a few key results about Lie
groups.
Definition
(Lie group)
.
A Lie group is a manifold
G
with a group structure
such that multiplication
m
:
G × G → G
and inverse
i
:
G → G
are smooth
maps.
Example. GL
n
(R) and GL
n
(C) are Lie groups.
Example. M
n
(R) under addition is also a Lie group.
Example. O(n) is a Lie group.
Notation.
Let
G
be a Lie group and
g ∈ G
. We write
L
g
:
G → G
for the
diffeomorphism
L
g
(h) = gh.
This innocent-seeming translation map is what makes Lie groups nice. Given
any local information near an element
g
, we can transfer it to local information
near
h
by applying the diffeomorphism
L
hg
−1
. In particular, the diffeomorphism
L
g
:
G → G
induces a linear isomorphism D
L
g
|
e
:
T
e
G → T
g
G
, so we have a
canonical identification of all tangent spaces.
Definition
(Left invariant vector field)
.
Let
X ∈ Vect
(
G
) be a vector field.
This is left invariant if
DL
g
|
h
(X
h
) = X
gh
for all g, h ∈ G.
We write Vect
L
(G) for the collection of all left invariant vector fields.
Using the fact that for a diffeomorphism F , we have
F
∗
[X, Y ] = [F
∗
X, F
∗
Y ],
it follows that Vect
L
(G) is a Lie subalgebra of Vect(G).
If we have a left invariant vector field, then we obtain a tangent vector at
the identity. On the other hand, if we have a tangent vector at the identity, the
definition of a left invariant vector field tells us how we can extend this to a left
invariant vector field. One would expect this to give us an isomorphism between
T
e
G
and
Vect
L
(
G
), but we have to be slightly more careful and check that the
induced vector field is indeed a vector field.
Lemma. Given ξ ∈ T
e
G, we let
X
ξ
|
g
= DL
g
|
e
(ξ) ∈ T
g
(G).
Then the map
T
e
G → Vect
L
(
G
) by
ξ 7→ X
ξ
is an isomorphism of vector spaces.
Proof.
The inverse is given by
X 7→ X|
e
. The only thing to check is that
X
ξ
actually is a left invariant vector field. The left invariant part follows from
DL
h
|
g
(X
ξ
|
g
) = DL
h
|
g
(DL
g
|
e
(ξ)) = DL
hg
|
e
(ξ) = X
ξ
|
hg
.
To check that
X
ξ
is smooth, suppose
f ∈ C
∞
(
U, R
), where
U
is open and
contains e. We let γ : (−ε, ε) → U be smooth with ˙γ(0) = ξ. So
X
ξ
f|
g
= DL
g
(ξ)(f) = ξ(f ◦ L
g
) =
d
dt
t=0
(f ◦ L
g
◦ γ)
But as (
t, g
)
7→ f ◦ L
g
◦ γ
(
t
) is smooth, it follows that
X
ξ
f
is smooth. So
X
ξ
∈ Vect
L
(G).
Thus, instead of talking about
Vect
L
(
G
), we talk about
T
e
G
, because it
seems less scary. This isomorphism gives T
e
G the structure of a Lie algebra.
Definition
(Lie algebra of a Lie group)
.
Let
G
be a Lie group. The Lie algebra
g
of
G
is the Lie algebra
T
e
G
whose Lie bracket is induced by that of the
isomorphism with Vect
L
(G). So
[ξ, η] = [X
ξ
, X
η
]|
e
.
We also write Lie(G) for g.
In general, if a Lie group is written in some capital letter, say
G
, then the
Lie algebra is written in the same letter but in lower case fraktur.
Note that dim g = dim G is finite.
Lemma. Let G be an abelian Lie group. Then the bracket of g vanishes.
Example.
For any vector space
V
and
v ∈ V
, we have
T
v
V
∼
=
V
. So
V
as a
Lie group has Lie algebra
V
itself. The commutator vanishes because the group
is commutative.
Example.
Note that
G
=
GL
n
(
R
) is an open subset of
M
n
, so it is a manifold.
It is then a Lie group under multiplication. Then we have
gl
n
(R) = Lie(GL
n
(R)) = T
I
GL
n
(R) = T
I
M
n
∼
=
M
n
.
If A, B ∈ GL
n
(R), then
L
A
(B) = AB.
So
DL
A
|
B
(H) = AH
as L
A
is linear.
We claim that under the identification, if ξ, η ∈ gl
n
(R) = M
n
, then
[ξ, η] = ξη − ηξ.
Indeed, on G, we have global coordinates U
j
i
: GL
n
(R) → R where
U
j
i
(A) = A
j
i
,
where A = (A
j
i
) ∈ GL
n
(R).
Under this chart, we have
X
ξ
|
A
= L
A
(ξ) =
X
i,j
(Aξ)
i
j
∂
∂U
i
j
A
=
X
i,j,k
A
i
k
ξ
k
j
∂
∂U
i
j
A
So we have
X
ξ
=
X
i,j,k
U
i
k
ξ
k
j
∂
∂U
i
j
.
So we have
[X
ξ
, X
η
] =
X
i,j,k
U
i
k
ξ
k
j
∂
∂U
i
j
,
X
p,r,q
U
p
q
η
q
r
∂
∂U
p
r
.
We now use the fact that
∂
∂U
i
j
U
p
q
= δ
ip
δ
jq
.
We then expand
[X
ξ
, X
η
] =
X
i,j,k,r
(U
i
j
ξ
j
k
η
k
r
− U
i
j
ξ
j
k
ξ
k
r
)
∂
∂U
i
r
.
So we have
[X
ξ
, X
η
] = X
ξη−ηξ
.
Definition
(Lie group homomorphisms)
.
Let
G, H
be Lie groups. A Lie group
homomorphism is a smooth map that is also a homomorphism.
Definition
(Lie algebra homomorphism)
.
Let
g, h
be Lie algebras. Then a Lie
algebra homomorphism is a linear map β : g → h such that
β[ξ, η] = [β(ξ), β(η)]
for all ξ, η ∈ g.
Proposition.
Let
G
be a Lie group and
ξ ∈ g
. Then the integral curve
γ
for
X
ξ
through
e ∈ G
exists for all time, and
γ
:
R → G
is a Lie group homomorphism.
The idea is that once we have a small integral curve, we can use the Lie
group structure to copy the curve to patch together a long integral curve.
Proof.
Let
γ
:
I → G
be a maximal integral curve of
X
ξ
, say (
−ε, ε
)
∈ I
. We fix
a t
0
with |t
0
| < ε. Consider g
0
= γ(t
0
).
We let
˜γ(t) = L
g
0
(γ(t))
for |t| < ε.
We claim that ˜γ is an integral curve of X
ξ
with ˜γ(0) = g
0
. Indeed, we have
˙
˜γ|
t
=
d
dt
L
g
0
γ(t) = DL
g
0
˙γ(t) = DL
g
0
X
ξ
|
γ(t)
= X
ξ
|
g
0
·γ(t)
= X
ξ
|
˜γ(t)
.
By patching these together, we know (
t
0
− ε, t
0
+
ε
)
⊆ I
. Since we have a fixed
ε that works for all t
0
, it follows that I = R.
The fact that this is a Lie group homomorphism follows from general proper-
ties of flow maps.
Example. Let G = GL
n
. If ξ ∈ gl
n
, we set
e
ξ
=
X
k≥0
1
k!
ξ
k
.
We set
F
(
t
) =
e
tξ
. We observe that this is in
GL
n
since
e
tξ
has an inverse
e
−tξ
(alternatively, det(e
tξ
) = e
tr(tξ)
6= 0). Then
F
0
(t) =
d
dt
X
k
1
k!
t
k
ξ
k
= e
tξ
ξ = L
e
tξ
ξ = L
F (t)
ξ.
Also, F (0) = I. So F (t) is an integral curve.
Definition
(Exponential map)
.
The exponential map of a Lie group
G
is
exp : g → G given by
exp(ξ) = γ
ξ
(1),
where γ
ξ
is the integral curve of X
ξ
through e ∈ G.
So in the case of G = GL
n
, the exponential map is the exponential map.
Proposition.
(i) exp is a smooth map.
(ii)
If
F
(
t
) =
exp
(
tξ
), then
F
:
R → G
is a Lie group homomorphism and
DF |
0
d
dt
= ξ.
(iii) The derivative
D exp : T
0
g
∼
=
g → T
e
G
∼
=
g
is the identity map.
(iv) exp
is a local diffeomorphism around 0
∈ g
, i.e. there exists an open
U ⊆ g
containing 0 such that exp : U → exp(U ) is a diffeomorphism.
(v) exp
is natural, i.e. if
f
:
G → H
is a Lie group homomorphism, then the
diagram
g G
h H
exp
Df|
e
f
exp
commutes.
Proof.
(i) This is the smoothness of ODEs with respect to parameters
(ii) Exercise.
(iii) If ξ ∈ g, we let σ(t) = tξ. So ˙σ(0) = ξ ∈ T
0
g
∼
=
g. So
D exp |
0
(ξ) = D exp |
0
( ˙σ(0)) =
d
dt
t=0
exp(σ(t)) =
d
dt
t=0
exp(tξ) = X
ξ
|
e
= ξ.
(iv) Follows from above by inverse function theorem.
(v) Exercise.
Definition
(Lie subgroup)
.
A Lie subgroup of
G
is a subgroup
H
with a smooth
structure on H making H an immersed submanifold.
Certainly, if H ⊆ G is a Lie subgroup, then h ⊆ g is a Lie subalgebra.
Theorem.
If
h ⊆ g
is a subalgebra, then there exists a unique connected Lie
subgroup H ⊆ G such that Lie(H) = h.
Theorem.
Let
g
be a finite-dimensional Lie algebra. Then there exists a (unique)
simply-connected Lie group G with Lie algebra g.
Theorem.
Let
G, H
be Lie groups with
G
simply connected. Then every Lie
algebra homomorphism g → h lifts to a Lie group homomorphism G → H.