Contents
Chapter 1. Lie groups 1
1. Lie groups 1
2. Lie algebras 3
3. Subgroups and subalgebras 7
4. Homomorphisms of Lie groups and algebras 11
5. The exponential map 13
6. Homogeneous spaces 16
7. The adjoint representation 20
8. Fundamental vector fields 22
9. Locally isomorphic Lie groups 24
10. Problems 26
0
CHAPTER 1
Lie groups
1. Lie groups
Definition 1.1. A Lie group G is a group, which is at the same time a smooth
manifold in such a way that
• the multiplication µ : G × G → G is smooth,
• the inverse ν : G → G is smooth.
By a homomorphism of Lie groups we understand a smooth group homomorphisms.
Notation. We denote by e the unit and write a−1
instead of ν(a). We will be
using the left and right translations λa, ρa : G → G defined by
λa(b) = ab ρa(b) = ba
Theorem 1.2. The smoothness of the inverse follows from the smoothness of
the multiplication.
Proof. The defining equation for the inverse is µ(a, ν(a)) = e. By the implicit
function theorem it is enough to verify that the derivative of µ(a, −) at a−1
is
invertible. This follows from the fact that µ(a, −) = λa has an inverse λa−1 .
Remark. Every Lie group is a topological group, i.e. a group and a topological
group such that the multiplication and the inverse are continuous. The fifth
Hilbert problem states that every topological group G that is at the same time a
(topological) manifold admits a smooth structure for which G becomes a Lie group.
This was proved in 1952 (in fact the structure is even analytic). If time permits we
will get to the implication C2
⇒ C∞
.
Let M, N be smooth manifolds. Then the projections p : M × N → M and
q : M × N → N provide the canonical isomorphism
(p∗, q∗) : T(x,y)(M × N)
∼=
−−→ TxM × TyN.
The inverse isomorphism is obtained from the inclusions
iy : M → M × N jx : N → M × N
a → (a, y) b → (x, b)
Under the above identification the pair (X, Y ) ∈ TxM × TyN corresponds to
(iy)∗X + (jx)∗Y ∈ T(x,y)(M × N).
Lemma 1.3. The following formulae hold for A, B ∈ TeG:
µ∗(A, B) = A + B, ν∗A = −A.
1
1. LIE GROUPS 2
Proof. These are just simple calculations
µ∗(A, B) = µ((ie)∗A + (je)∗B) = (µie)∗A + (µje)∗B = A + B
and by differentiating e = µ(a, ν(a)) in the direction A ∈ TeG we get
0 = µ∗(A, ν∗A) = A + ν∗A
Examples 1.4. The classical gropups:
• The general linear group GL(n, R) - the group of invertible matrices (aij).
Since GL(n, R) ⊆ Rn×n
can be described as GL(n, R) = det−1
(R−{0}) it
is an open subset and hence a manifold. Multiplication is clearly smooth
(even algebraic).
• The general linear group GL(n, C) with coefficients in C. We think of
GL(n, C) as a subgroup of GL(2n, R) via the identification Cn
= Rn
⊕iRn
.
The embedding becomes
A + iB →
A −B
B A
On the other hand GL(n, C) ⊆ Cn×n
is again open and hence a manifold.
• The special linear groups
SL(n, R) = {A ∈ GL(n, R) | det A = 1}
SL(n, C) = {A ∈ GL(n, C) | det A = 1}
are certainly closed submanifolds and also subgroups. Later we will prove
Theorem. Every closed subgroup of a Lie group is a submanifold and
with the submanifold smooth structure a Lie group (i.e. a Lie subgroup).
• Let β : Rn
×Rn
→ R be a bilinear form represented by a matrix B = (bij).
A linear map α : Rn
→ Rn
is said to preserve β if
β(αx, αy) = β(x, y) ⇐⇒ AT
BA = B
Such linear automorphisms clearly form a closed subgroup of GL(n, R).
– Specifically for β = , , the scalar product, we have B = E, the
identity matrix and we obtain the orthogonal group
O(n, R) = {A ∈ GL(n, R) | AT
A = E}
and also the special orthogonal group
SO(n, R) = O(n, R) ∩ SL(n, R)
– Consider on R2n
the (nondegenerate antisymmetric) bilinear form
n
i=1
(xiyn+i − yixn+i)
with its matrix J =
0 E
−E 0
. The group of linear automorphisms
preserving this form is called the symplectic group Sp(2n, R). Analogously
we obtain Sp(2n, C).
2. LIE ALGEBRAS 3
• The unitary group U(n) = {A ∈ GL(n, C) | ¯AT
A = E} and the special
unitary group SU(n) = U(n) ∩ SL(n, C). There is also a complex
orthogonal group which is different from the unitary group. One of the
main qualitative differences is that O(n, C) is a complex manifold and a
complex Lie group (reason being that the defining equation AT
A = E
is holomorphic unlike that for the unitary group - it contains complex
conjugation).
• The spin group Spin(n). We will say more about it later. It is related to
SO(n, R) by a short exact sequence of groups
1 → Z/2 → Spin(n) → SO(n, R) → 1.
• Sp(n) = {A ∈ GL(n, H) | ¯AT
A = E}, the group of linear automorphisms
of the quaternionic space Hn
preserving the scalar product. Also Sp(n) =
Sp(2n, C) ∩ U(2n).
2. Lie algebras
Definition 2.1. A vector space L over R is called a Lie algebra if there is
given a bilinear map [ , ] : L × L → L satisfying
• the antisymmetry: [X, X] = 0,
• the Jacobi identity: [[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0.
From bilinearity we obtain
0 = [X + Y, X + Y ] = [X, X] + [X, Y ] + [Y, X] + [Y, Y ] = [X, Y ] + [Y, X]
implying [Y, X] = −[X, Y ].
Example 2.2. The vector fields on a smooth manifold M with the bracket
[X, Y ]:
X =
i
Xi
∂
∂xi , Y =
i
Yi
∂
∂xi =⇒ [X, Y ] =
i,j
Xj
∂Yi
∂xj − Yj
∂Xi
∂xj
∂
∂xi
Let L be a finite dimensional Lie algebra and e1, . . . , en its basis. Then [ei, ej] =
k ck
ijek. The numbers ck
ij are called the structure constants of L with respect to
the basis. They satisfy the following identities:
• ck
ji = −ck
ij,
• k(ck
ijcm
kl + ck
jlcm
ki + ck
licm
kj) = 0.
Conversely, by giving the basis e1, . . . , en and the structure constants ck
ij satisfying
the above equalities we obtain a Lie algebra L. The complete classification of Lie
algebras is not yet known.
Example 2.3. Let V be a vector space and denote L = hom(V, V ). On L we
define a bracket
[f, g] = f ◦ g − g ◦ f.
In this way we obtatin a Lie algebra gl(V ).
For a Lie group G we define g = Lie(G) = TeG as a vector space. Now we
proceed to introduce a bracket on g.
2. LIE ALGEBRAS 4
Definition 2.4. A vector field X : G → TG is called left-invariant if (λa)∗ ◦
X = X ◦ λa for any a ∈ G.
TG
(λa)∗
GG TG
G
X
yy
λa
GG G
X
yy
In other words X is λa-related with itself which we denote by X ∼λa
X.
Remark. The f-relatedness of vector fields X and Y has the following characterization
via the flow lines, easily verified by differentiating both sides.
f(FlX
t (x)) = FlY
t (f(x))
In other words f transfers the flow lines of X into the flow lines of Y . We will use
this property quite often.
Remark. Let A ∈ TeG be an arbitrary vector. It defines a vector field λA :
G → TG by the formula λA(a) = (λa)∗A. This vector field is clearly left-invariant
as
λA(ab) = (λab)∗A = (λaλb)∗A = (λa)∗((λb)∗A) = (λa)∗(λA(b))
It remains to verify its smoothness. Since (λa)∗A = µ∗(0a, A) this is achieved by
the following diagram
TG × TG
T µ
GG TG
G
(0,A)
yy
λA
ee
with (0, A) being the map with components the zero section 0 and the constant
map sending everything onto A.
Theorem 2.5. Let X, Y be left invariant vector fields. Then X + Y , kX and
[X, Y ] are again left-invariant.
Proof. Since X and Y are λa related with X and Y respectively, the same is
true for their sum, multiples and bracket.
Definition 2.6. The vector space g = Lie(G) = TeG together with the bracket
[A, B] = [λA, λB]e is called the Lie algebra of the Lie group G.
Remark. For every finite dimensional Lie algebra g there exists a Lie group
G for which Lie(G) = g.
We would like to explain now why this is a reasonable object of study. We have
seen that the first derivative at e does not see anything from the structure of the
Lie group. The second derivative does but in order to make sense of the second
derivative one has to fix the coordinate charts (which we will do later and for them
the second derivative will be described exactly by the Lie bracket). Without a fixed
choice of the charts the second derivative only makes sense when the first derivative
vanishes at that point which is not the case for the product. The way out is to
“subtract from µ the sum of the two coordinates” by considering
[ , ] : G × G −→ G
(a, b) → aba−1
b−1
2. LIE ALGEBRAS 5
We will see shortly that the first derivative of the commutator vanishes at e and
the essential part of the second derivative is exactly the Lie bracket.
Notation. Let X, Y be two vector fields on a manifold M. Then we denote
(FlX
t )∗
Y (x) = (FlX
−t)∗Y (FlX
t (x)) ∈ TxM
the pullback of Y along the flow FlX
t of X. For each x ∈ M it is defined for t small.
Lemma 2.7. d
dt t=t0
(FlX
t )∗
Y (x) = (FlX
t0
)∗
[X, Y ](x).
Proof. First assume that t0 = 0 and let f : M → R be a smooth function.
We differentiate f in the direction of the left hand side:
d
dt t=0
(FlX
t )∗
Y (x) f = d
dt t=0
(FlX
t )∗
Y (x)f
= d
dt t=0
(FlX
−t)∗Y (FlX
t (x))f
= d
dt t=0
Y (FlX
t (x))(f ◦ FlX
−t)
= Y (x)(−Xf) + d
dt t=0
(Y f)(FlX
t (x))
= −(Y Xf)(x) + (XY f)(x) = [X, Y ](x) f
For a general t0 we have (FlX
t )∗
Y (x) = (FlX
t0
)∗
(FlX
t−t0
)∗
Y (x). Since (FlX
t0
)∗
is a
linear map we can interchange with d
dt .
Corollary 2.8. The following conditions are equivalent:
• [X, Y ] = 0,
• (FlX
t )∗
Y = Y , i.e. Y is FlX
t -related with itself for all t,
• FlX
t FlY
s (x) = FlY
s FlX
t (x), i.e. the flow lines commute.
In general we have FlY
−s FlX
−t FlY
s FlX
t (x) = x + st[X, Y ](x) + o(s, t)2
.
Proof. Differentiating twice we get
∂
∂t t=0
∂
∂s s=0
FlY
−s FlX
−t FlY
s FlX
t (x) = ∂
∂t t=0
−Y (x) + (FlX
t )∗
Y (x) = [X, Y ](x)
The remaining derivatives of order at most two are clearly zero.
Example 2.9. Let M = G, a Lie group. What does [A, B] for A, B ∈ g
express? Let us consider the following integral curves
• ϕ(t) the flow line of λA with ϕ(0) = e,
• ψ(t) the flow line of λB with ψ(0) = e.
A flow line of λA through a general a ∈ G is easily a · ϕ : t → aϕ(t). This follows
from the λa-relatedness of λA with itself: d
dt (aϕ(t)) = (λa)∗
d
dt ϕ(t). In other words
FlλA
t = ρϕ(t)
This implies that ϕ(t1 + t2) = ϕ(t1)ϕ(t2) and it is a homomorphism of groups. We
now compute
FlλB
−s FlλA
−t FlλB
s FlλA
t (x) = ϕ(t)ψ(s)ϕ(−t)ψ(−s) = ϕ(t)ψ(s)ϕ(t)−1
ψ(s)−1
.
In other words the group theoretic commutator [ϕ(t), ψ(s)] has a Taylor polynomial
[ϕ(t), ψ(s)] = [A, B]st + o(s, t)2
2. LIE ALGEBRAS 6
This can also be rewritten as d2
[ , ](e,e)((A, 0), (0, B)) = [A, B]. The Lie bracket
thus measures the non-commutativity of the Lie group. More precisely [A, B] = 0
if and only if all the elements ϕ(t) commute with all ψ(s). We will see later that
the connection between commutativity of G and vanishing of the bracket works
perfectly for connected Lie groups.
Definition 2.10. Let L, L be two Lie algebras. A linear map ϕ : L → L is
called a homomorphism of Lie algebras if ϕ[A, B]L = [ϕA, ϕB]L .
Theorem 2.11. Let f : G → H be a (smooth) homomorphism of Lie groups.
Then its derivative f∗ : g → h at e is a homomorhpism of Lie algebras.
Proof. Let us rewrite f(ab) = f(a)f(b) using the left translations as
f ◦ λa = λf(a) ◦ f
Differentiating in the direction A ∈ g we obtain f∗(λa)∗A = (λf(a))∗f∗A or
f∗λA(a) = λf∗A(f(a))
which means that λA is f-related to λf∗A. Since the bracket respects relatedness,
[λA, λB] must be f-related to [λf∗A, λf∗B]. Evaluating at e yields the result.
Definition 2.12. A smooth map f : G → H between Lie groups is a local
isomorphism if it is both a homomorphism and a local diffeomorphism at e (i.e. the
derivative f∗e : g → h is an isomorphism).
Two Lie groups G, H are called locally isomorphic if there exist neighbourhoods
U e and V e, in G and H respectively, together with a diffeomorphism f : U →
V which satisfies:
• f(ab) = f(a)f(b) whenever a, b, ab ∈ U,
• f−1
(ab) = f−1
(a)f−1
(b) whenever a, b, ab ∈ V .
Clearly if there exists a local isomorphism f : G → H then G and H are locally
isomorphic.
Theorem 2.13. Locally isomorphic groups have isomorphic Lie algebras.
Example 2.14. The additive groups R and T = SU(1) (the group of complex
units in C) are locally isomorphic. We think of the first as the group of translations
of the line while the second is the group of rotations of the circle (or C for that
matter). This is because there exists a local isomorphism R → T sending t → e2πit
.
Definition 2.15. Let L, L be Lie algebras. On their product L × L we
consider the bracket
[(X1, Y1), (X2, Y2)] = ([X1, X2]L, [Y1, Y2]L ).
We call L × L together with this bracket the product of Lie algebras L and L .
Theorem 2.16. Lie(G × H) ∼= Lie(G) × Lie(H).
Proof. The projections p : G × H → G and q : G × H → H are homomorphisms
and hence they induce homomorphisms of the Lie algebras in question. This
means
p∗[(X1, Y1), (X2, Y2)] = [p∗(X1, Y1), p∗(X2, Y2)] = [X1, X2]
and similarly for q. The canonical isomorphism (p∗, q∗) : Lie(G × H) → g × h is
then an isomorphism of Lie algebras.
3. SUBGROUPS AND SUBALGEBRAS 7
Remark. With the above Lie algebra structure L × L forms a product in the
category of Lie algebras. The previous proof is then just a demonstration of the
fact that Lie is a functor and preserve products (which is obvious from the fact that
this happens already at the level of tangent vector spaces at e).
What happens if we change sides? Denoting ρA the right-invariant vector field
with value A at e the next theorem asserts that the Lie bracket defined via the
right-invariant vector fields agrees with the usual one up to the minus sign.
Theorem 2.17. For A, B ∈ g the following holds: [ρA, ρB]e = −[λA, λB]e.
Proof. Consider the opposite group G∗
with multiplication a ∗ b = ba. The
inverse ν : G∗
→ G is a group homomorphism and
[A, B]∗
= [λ∗
A, λ∗
B]e = [ρA, ρB]e
Thus −[ρA, ρB]e = ν∗[A, B]∗
= [ν∗A, ν∗B] = [−A, −B] = [λA, λB]e.
Corollary 2.18. For a commutative group G the bracket on its Lie algebra is
identically zero.
3. Subgroups and subalgebras
Definition 3.1. A Lie subalgebra L ⊆ L is a vector subspace closed under
[ , ].
Theorem 3.2. If H ⊆ G is both a submanifold and a subgroup then h ⊆ g is a
Lie subalgebra.
Proof. In the diagram
H × H
µ
GG
 •

H •
ι

G × G µ
GG G
the map µ (which exists since H is a subgroup) is smooth since H is a submanifold.
Hence H is a Lie group and the inclusion ι : H → G is a homomorphism. Thus its
derivative ι∗ : h → g is a homomorphism of Lie algebras (saying that the bracket of
h is a restriction of the bracket on g) and its image is therefore a subalgebra.
Example 3.3. Consider R2
. Then every line {(x, kx) | x ∈ R} (for k ∈ R) is
a subgroup (and a submanifold). Now consider the torus T2
= R2
/Z2
. Again we
get subgroups for any k ∈ R. For k ∈ Q this subgroup is a submanifold but not for
irrational k when this subgroup is dense.
Definition 3.4. A subset A ⊆ M of a smooth manifold M is called an initial
submanifold (of dimension k) if for each x ∈ A there exists a chart
ϕ : U
∼=
−−→ Rm
= Rk
× Rm−k
such that ϕ−1
(Rk
× {0}) is exactly the path component of U ∩ A containing x.
3. SUBGROUPS AND SUBALGEBRAS 8
Theorem 3.5. Every initial submanifold is the image of an (essentially unique)
injective immersion i satisfying the following universal property:
A •
i

N
g
bb
f
GG M
For every smooth map f : N → M with the property f(N) ⊆ i(A) the unique map
g : N → A satisfying ig = f is also smooth.
Proof. Let ϕ : U −→ Rm
be a chart on N from the definition of an initial
submanifold. Declare its restriction
Cx(U ∩ A)
∼=
−−→ Rk
× {0}
to the path component of U ∩A containing x to be a chart for A. This does endow A
with a smooth structure. It differs from the subspace topology (which is inevitable)
but the inclusion is clearly an injective immersion.
We verify the universal property for inclusions of initial submanifolds. Let
y ∈ N with f(y) = x and V a path connected neighbourhood of y which maps into
U. Since its image is also path connected it must be contained in U ∩ A. Thus g
in the chart provided by ψ is just a restriction of f and hence smooth.
Suppose now that i : A → M is another injective immersion with the same
image as i. Then there exists a factorization
A
h GG p
i 33
Axn
i~~
M
with h an immerison and a bijection at the same time. Since its inverse is also an
immersion by the same argument h must be in fact a diffeomorphism.
Remark. It is also true that any injective immersion i satisfying the above
universal property is in fact an inclusion of an initial submanifold but we will not
need this fact.
Remark. We have not proved that A has a countable basis for its topology.
In fact A might well have an uncountable number of components. However each of
the components of A is second countable.
Definition 3.6. A Lie subgroup H ⊆ G is an initial submanifold which is at
the same time a subgroup.
Theorem 3.7. A Lie subgroup H ⊆ G with its canonical smooth structure (and
multiplication) is a Lie group. Moreover h ⊆ g is a Lie subalgebra.
Proof. The whole proof is contained in the diagram
H × H
µ
GG
 •

H •
ι

G × G µ
GG G
3. SUBGROUPS AND SUBALGEBRAS 9
Our new definition includes the wild subgroups of the torus T2
. In fact we
are able to construct a Lie subgroup for any Lie subalgebra of g. To motivate
our construction observe that for a Lie subgroup H ⊆ G and a ∈ H we have
TaH = (λa)∗h and H is an integral submanifold of the left invariant distribution
determined by h.
More generally for a linear subspace P ⊆ g of dimension k the left translations
(λa)∗P =: λP (a) ⊆ TaG form a k-dimensional distribution λP on G. This distribution
is smooth: if A1, . . . , Ak is a basis of P then λA1 (a), . . . , λAk
(a) is a basis of
λP (a).
A distribution S on M is called involutive if for every two vector fields X, Y ∈ S
their bracket [X, Y ] also lies in S.
Theorem 3.8 (Frobenius theorem). If S is involutive then for every x ∈ M
there exists a local coordinate system y1
, . . . , ym
in a neighbourhood U of x such that
the vector fields ∂
∂y1 , . . . , ∂
∂yk form a basis of the distribution S on U. In particular
S is integrable.
Proof. Let X1, . . . , Xk be local vector fields which, near x, span the distribution
S and let us choose a coordinate system around x in which Xi(x) = ∂
∂xi . We
then define a map
ϕ : Rm
⊇ U −→ M
(t1
, . . . , tm
) −→ FlX1
t1 · · · FlXk
tk (0, . . . , 0, tk+1
, . . . , tm
)
The partial derivatives at the origin clearly consist of the vectors ∂
∂xi and thus ϕ is
a local diffeomorphism.
Let us compute the partial derivative with respect to ti
for i ≤ k at a general
point.
∂ϕ
∂ti
= FlX1
t1
∗
· · · Fl
Xi−1
ti−1
∗
Xi Fl
Xi+1
ti+1 · · · FlXm
tm (x)
To conclude the proof it is therefore enough to show that for any Y belonging to S
the pullbacks (FlY
t )∗
Xi also belong to S. Denote this pullback by
Yi(t) = (FlY
t )∗
Xi(x)
and write [Y, Xi] = aijXj. By Lemma 2.7 the paths Yi(t) satisfy the following
system of differential equations
d
dt Yi(t) = (FlY
t )∗
[Y, Xi] = aij(FlY
t (x))Yj(t)
We have Yi(0) = Xi(x) ∈ S(x) and since the system is linear we must have Yi(t) ∈
S(x) for all t. Namely applying any linear form α to this system we see that α(Yi(t))
satisfy the very same linear system of differential equations. Using the uniqueness
and the existence of the zero solution we see that α(Yi(0)) = 0 for all i implies
α(Yi(t)) = 0 for all i and t.
By an integral submanifold we will now understand a connected initial submanifold
A ⊆ M for which TxA = Sx for all x ∈ A. A maximal integral submanifold is
one that is not contained in any bigger.
Theorem 3.9. If S is involutive then to every point x ∈ M there exists a
unique maximal integral submanifold going through that point.
3. SUBGROUPS AND SUBALGEBRAS 10
Proof. We will obtain this initial submanifold as the set A of all points y ∈ M
which can be joined with x by a path γ : I → M tangent to the distribution S,
i.e. with the properties
• γ(0) = x, γ(1) = y,
• ˙γ = d
dt γ ∈ S.
We need to verify that A is indeed an initial submanifold, maximality should be
obvious. In a coordinate chart ϕj : Uj → Rm
from the Frobenius theorem Uj ∩ A
is clearly the disjoint union
(ck+1,...,cm)∈Cj
Rk
× {(ck+1, . . . , cm)}
It is enough to show that each Cj is at most countable since every countable subset
of Rm−k
is totally disconnected (in between any two distinct x, y in a countable set
X ⊆ R there lies some z ∈ X). First we prove an auxiliary fact:
Let B be an integral submanifold which is second countable. Then B intersects
each Uj in at most a countable number of leaves Rk
× {(ck+1, . . . , cm)}: if, by
contradiction, the number was uncountable then choosing a point from B in each
leaf we would find an uncountable discrete subset of B.
In particular every leaf of ϕj intersects at most countable number of leaves
of ϕk. Now start with A0 = {x} and at each step “leaf complete” Ai to obtain
Ai+1. Then A = Ai and it is second countable, hence intersects only a countable
number of leaves of each ϕj.
Let us return to a linear subspace P ⊆ g and the distribution λP on G.
Lemma 3.10. λP is involutive if and only if P is a Lie subalgebra.
Proof. Since [X, fY +gZ] = f[X, Y ]+(Xf)Y +g[X, Z]+(Xg)Z it is enough
to check the brackets of vector fields of the form λA with A ∈ P. But [λA, λB] =
λ[A,B].
Theorem 3.11. Let h ⊆ g be a Lie subalgebra. Then the maximal integral
submanifold H passing through e is a Lie subgroup.
Proof. Let a ∈ H. Since (λa−1 )∗λh = λh, the map λa−1 preserves integral
submanifolds. As λa−1 (a) = e and both a, e ∈ H we must have λa−1 (H) = H and
thus a−1
b ∈ H for all a, b ∈ H.
Now we tackle the uniqueness issue. First a lemma.
Lemma 3.12. Let f : G → H be a homomorphism of Lie groups whose derivative
at identity is surjective. Then the image of f is a union of components of
H.
Proof. The image is certainly a subgroup which is open. Since any open
subgroup is necessarily also closed (its complement being a union of cosets which
are open) the assertion follows.
Remark. Later we will use a simple variation of this lemma: Let U be a
connected neighbourhood of e in a Lie group G. Then the subgroup generated by
U is exactly the connected component Ge of G containing e.
Theorem 3.13. Let H1, H2 ⊆ G be two connected Lie subgroups. Then H1 =
H2 if and only if Lie(H1) = Lie(H2).
4. HOMOMORPHISMS OF LIE GROUPS AND ALGEBRAS 11
Proof. Let h denote the common Lie algebra of H1 and H2 and let H be
the maximal integral submanifold of λh passing through e. Since both Hi are also
integral submanifolds they must be contained in H and the inclusions Hi → H are
both injective and surjective by the previous lemma and thus H = Hi.
4. Homomorphisms of Lie groups and algebras
Lemma 4.1. A group homomorphism f : G → H which is smooth near e is
smooth everywhere.
Proof. This is a classical homogeneity argument. Denoting by U the neighbourhood
of e where f is smooth pick any a ∈ G and consider the diagram
U
f
GG
λa

H
λf(a)

aU
f
GG H
in which aU is a neighbourhood of a and thus f is smooth everywhere.
The essential idea of this section is to construct homomorphisms through their
graphs. Let us consider ϕ : g → h, a linear map between Lie algebras. The graph
of ϕ is the subset Graph(ϕ) = {(A, ϕ(A)) | A ∈ g}.
Lemma 4.2. Graph(ϕ) is a Lie algebra if and only if φ is a homomorphism of
Lie algebras.
Proof. By the definition of the bracket in the product
[(A, ϕ(A)), (B, ϕ(B))] = ([A, B], [ϕ(A), ϕ(B)])
which lies in Graph(ϕ) if and only if [ϕ(A), ϕ(B)] = ϕ[A, B].
Let ϕ : g → h be now a homomorphism of Lie algebras, Graph(ϕ) ⊆ g × h its
graph, a Lie subalgebra. There exists a unique connected Lie subgroup F ⊆ G × H
with Lie(F) = Graph(ϕ). Assuming that the composition F → G × H → G is a
diffeomorphism F will be a graph of a homomorphism f : G → H with f∗ = ϕ. In
general however this projection is only a local diffeomorphism: its derivative at e is
the isomorphism Graph(ϕ) → g and at other points this follows from the diagram
TeF
f∗
∼=
GG
(λa)∗
∼=

TeG
(λf(a))∗∼=

TaF
f∗
GG Tf(a)G
Definition 4.3. A continuous map f : X → Y is a covering if for each y ∈ Y
there exists its neighbourhood U such that
f−1
(U)
∼= GG
f
33
c∈C
U
c∈C
id
~~
U
Lemma 4.4. Every local isomorphism of Lie groups is a covering.
4. HOMOMORPHISMS OF LIE GROUPS AND ALGEBRAS 12
Proof. Let f : G → H be the local isomorphism, U a, V b open neighbourhoods
for which f|U : U
∼=
−−→ V with inverse g. Then we will show that
f−1
(V ) =
k∈ker f
k · U
Therefore let x ∈ f−1
(V ). Then x = (x · g(f(x))−1
) · g(f(x)) is the decomposition.
Also kx = k x implies that x(x )−1
= k−1
k ∈ ker f and thus f(x) = f(x ). Since
f in injective on U, x = x and necessarily k = k .
The proof is finished by recalling that the image of f is a union of components
(so that for any b the a above exists).
Theorem 4.5. Let X be a path connected and locally simply connected topological
space. Then X is simply connected if and only if every connected covering
of X is a global homeomorphism.
Before going into the proof we draw a corollary:
Theorem 4.6. Let G be a simply connected Lie group, H any Lie group. Then
for every homomorphism ϕ : g → h of Lie algebras there exists a unique homomorphism
of Lie groups f : G → H with the property f∗ = ϕ. For connected G the
uniqueness part is still valid.
Proof. The above constructed homomorphism F → G is a covering and according
to the previous theorem a diffeomorphism. Thus F is the graph of f.
Corollary 4.7. Two simply connected Lie groups G and H are isomorphic if
and only if their Lie algebras are isomorphic.
The assumption of simple connectivity is essential: the canonical projection
map R → R/Z = T is a homomorphism but there is no non-trivial homomorphism
in the opposite direction despite the fact Lie R = Lie T.
Proof of Theorem 4.5. Let us construct the universal covering of X. Set
˜X = {[γ] | γ : (I, 0) → (X, x)}
where [γ] denotes the class with respect to homotopies preserving both endpoints.
The projection p : ˜X → X sends [γ] → γ(1). Then clearly
• p−1
(x) ∼= π1(X, x).
• p is a covering: Let U be a simply connected neighbourhood of x . Then
p−1
(U) ∼=
[γ]
γ(0)=x
γ(1)=x
[γ] ∗ {[δ] | δ : (I, 0) → (U, y)}
in bijection with U by
simple connectivity
This bijection defines a topology on ˜X for which p is a covering. Therefore
˜X is a smooth manifold if X was to start with (again we leave out the
proof that ˜X is second countable).
Remark. We have shown that π1(X, x) is at most countable since
p−1
(x) is discrete and X second countable.
5. THE EXPONENTIAL MAP 13
• p is universal: let q : Y → X be a covering with connected Y and let
y ∈ q−1
(x). Then there exists a unique f : ˜X → Y satisfying qf = p and
f(˜x) = y where ˜x = [x] ∈ ˜X is the class of the constant path
( ˜X, ˜x)
f
∃!
GG
p
55
(Y, y)
q
{{
(X, x)
This is about the path lifting property: the path γ : (I, 0) → (X, x)
has a unique continuous lift to ( ˜X, ˜x), namely t → [γ|[0,t]]. Denote the
unique lift to (Y, y) by ˜γ. Since the lifts must be preserved f must send
[γ] → ˜γ(1).
• If π1(X, x) = {e} then ˜X → X is a homeomorphism: it is a local homeomorphism
from the definition of a covering and surjective from the path
connectedness of X. We will prove injectivity. Let p[γ] = p[δ], i.e.
γ, δ : (I, 0, 1) → (X, x, x )
The concatenation γ ∗ δ−1
is a loop in X, hence contractible to a point
which gives [γ] = [δ].
5. The exponential map
Definition 5.1. By a one-parameter subgroup in G we understand a homomorphism
γ : R → G.
Theorem 5.2. For every A ∈ g there exists a unique one-parameter subgroup
γA : R → G such that ˙γA(0) = A.
Proof. R is simply connected and Lie R = R with the trivial bracket and thus
a homomorphism R → g of Lie algebras is the same thing as a linear map.
The one-parameter subgroup γA is an integral curve of λA and more generally
for every a ∈ G the curve t → a · γA(t) is:
d
dt t=t0
aγA(t) = d
dt t=t0
aγA(t0)γA(t − t0) = (λaγA(t0))∗A = λA(a · γA(t0))
Theorem 5.3. The flow of the left-invariant vector field λA is
FlλA
t (a) = aγA(t) = ργA(t)(a)
Moreover λA is complete (the integral curves are defined for all t ∈ R).
Definition 5.4. The map exp : g → G sending A → γA(1) is called the
exponential map of the Lie group G.
Example 5.5. For G = (R+
, ·) the associated Lie algebra is Lie G = R, the
left-invariant vector field λA(a) = (λa)∗A = aA. The equation for the flow is
d
dt γA = γAA
and its solution is clearly γA(t) = etA
. Hence exp(A) = eA
.
5. THE EXPONENTIAL MAP 14
Example 5.6. More generally for G = GL(n, R) the exponential map is
exp : gl(n, R) −→ GL(n, R)
A −→ eA
=
∞
k=0
1
k!
Ak
Theorem 5.7. It holds exp(tA) = γA(t).
Proof. γA(t) = FlλA
t·1 (e) = Flt·λA
1 (e) = FlλtA
1 (e) = exp(tA).
Theorem 5.8. The map exp : g → G is smooth and a diffeomorphism on a
neighbourhood of 0.
Proof. The vector field λA depends smoothly on A and thus also exp. We
compute the derivative of exp by considering a curve t → tA in g. Its image under
exp is t → exp(tA) = γA(t) whose derivative at 0 is ˙γA(0) = A. We conclude that
exp∗ = id : g → g.
Theorem 5.9. For every homomorphism of Lie groups the following diagram
commutes.
G
f
GG H
g
exp
yy
f∗
GG h
exp
yy
Proof. f(γA(t)) is a one-parameter subgroup with initial speed f∗A and thus
equal to γf∗A(t). Evaluating at t = 1 yields the result.
Lemma 5.10. Let f : G → H be a homomorphism of Lie groups with G connected
and let K ⊆ H be a Lie subgroup. Then f(G) ⊆ K if and only if f∗(g) ⊆ k.
Proof. Suppose that f∗(g) ⊆ k. Then f(exp(g)) = exp(f∗(g)) ⊆ exp(k) ⊆ K.
Since exp(g) is a neighbourhood of e in G, f−1
(K) is an open subgroup of G. As
G is connected f−1
(K) must equal G.
Theorem 5.11. Let ϕ : R → G be a continuous group homomorphism. Then
ϕ is smooth.
Proof. In a neighbourhood of 0 ∈ R we can write uniquely ϕ(t) = exp(A(t))
with X(t) a continuous path in g starting at 0. We would like to show that X(t)
is linear. Let ϕ[−t0, t0] ⊆ exp U where U is a ball centered at 0 and such that exp
is a diffeomorphism on 2U. Let n ∈ N. We will show that kX t0
n = X k t0
n for
0 ≤ k ≤ n by induction on k. For k = 0 or k = 1 this is clear. Assuming the
statement true for k write
(k + 1)X t0
n = kX t0
n + X t0
n ∈ 2U
Since
exp (k + 1)X t0
n = exp X t0
n
k+1
= ϕ t0
n
k+1
= ϕ (k + 1)t0
n = exp X (k + 1)t0
n
and exp is injective on 2U this finishes the induction step. As a particular case
nX t0
n = X(t0) and thus X k
n t0 = k
n X(t0) which easily holds also for all integers
k with |k| ≤ n. From continuity X(rt0) = rX(t0) for all r ∈ [−1, 1]. Since ϕ|[−t0,t0]
5. THE EXPONENTIAL MAP 15
is now linear and hence smooth, it is smooth everywhere by the usual argument
(homogeneity).
Theorem 5.12. Let G, H be Lie groups and f : G → H a continuous group
homomorphism between them. Then f is smooth.
Proof. Pick a basis A1, . . . , Am in g and define a map ϕ : Rm
→ G by
(t1, . . . , tm) → exp(t1A1) · · · exp(tmAm)
Clearly ϕ is a diffeomorphism near 0. It is called a coordinate chart of a second
kind (the first kind is exp itself). The composition fϕ is the map
(t1, . . . , tm) → f(exp(t1A1)) · · · f(exp(tmAm))
which is smooth: each continuous one-parameter subgroup f(exp(tiAi)) is smooth
by the previous theorem and so is their product. Again we can globalize by homo-
geneity.
Theorem 5.13 (The closed subgroup theorem). Let H ⊆ G be a subgroup (in
the algebraic sense) which is also closed as a subspace of a Lie group G. Then H
is a submanifold and thus a Lie subgroup.
Proof. We divide the proof into a few steps:
• Define
h = {˙γ(0) | γ : (R, 0) → (G, e) a smooth curve}
Then h is a linear subspace since ˙γ1(0) + ˙γ2(0) = d
dt t=0
(γ1(t) · γ2(t)) and
k ˙γ(0) = d
dt t=0
γ(kt).
• Let An ∈ g be a sequence converging to A and let tn > 0 converge to
0 ∈ R. We claim that if exp(tnAn) ∈ H then exp(tA) ∈ H for all t ∈ R.
We may suppose that t > 0. Choose mn ∈ N in such a way that |t−mntn|
is minimal. Then |t − mntn| → 0 and consequently mntnAn → tA. But
exp(mntnAn) = exp(tnAn)mn
∈ H and since H is closed it follows that
exp(tA) ∈ H too.
• We show that h = {A ∈ g | exp(tA) ∈ H ∀t ∈ R}. The inclusion ⊇ follows
from the definition of h. For the reverse inclusion let A ∈ g be ˙γ(0) for
some curve γ : R → H. For t small we write γ(r) = exp(A(t)). Then
A = ˙γ(0) = exp∗( ˙A(0)) = ˙A(0) = lim
n→∞
A( 1
n )
1
n
Setting An = nA 1
n → A and tn = 1
n we have
exp(tnAn) = exp A 1
n = γ 1
n ∈ H.
and by the previous point exp(tA) ∈ H for all t ∈ R.
• Let k ⊆ g be a linear subspace complementary to h. We claim that there
exists a neighbourhood 0 ∈ W ⊆ k such that exp(W) ∩ H = {e}. By
contradiction let Bn → 0 be a sequence in k such that exp(Bn) ∈ H.
With respect to some norm on k consider An = Bn
|Bn| . By passing to a
subsequence we may assume that An converges to some A ∈ k. Putting
tn = |Bn| we have exp(tnAn) = exp(Bn) ∈ H and thus exp(tA) ∈ H for
all t ∈ R. By the previous point A ∈ h, a contradiction to A ∈ k.
6. HOMOGENEOUS SPACES 16
• Define ϕ : h × k → G by (A, B) → exp A · exp B. We will show that there
exists a neighbourhood 0 ∈ V ⊆ h for which the restriction
ϕ : V × W
∼=
−−→ U ⊆ G
is a diffeomorphism onto its image U (which is easy) and such that
U ∩ H = ϕ(V × {0}).
Therefore let x ∈ U ∩ H be in the image, x = exp A · exp B. As both
x, exp A ∈ H, also exp B ∈ H. By the previous point B = 0.
Thus we found a chart at e which flattens out H. Charts at other points are
obtained by translation.
6. Homogeneous spaces
Definition 6.1. By a left action of a Lie group G on a smooth manifold M we
understand a smooth map : G×M → M satisfying e = id and a ◦ b = ab where
we write a = (a, −). The algebraic content is a homomorphism G → Diff(M).
The right action r : M × G → M has to satisfy re = id and ra ◦ rb = rba.
We will write a(x) = a · x and ra(x) = xa.
Remark. A right action of G is the same as a left action of the opposite group
G∗
.
Definition 6.2. The orbit of a point x ∈ M is the subset Gx = {ax | a ∈ G}.
We call the action transitive if there is only one orbit in M or equivalently if
Gx = M for every x ∈ M.
The stabilizer subgroup of a point x ∈ M is the (closed) subgroup
Sx = {a ∈ G | ax = x}.
The action is called free if the stabilizer subgroup of each point is trivial, Sx = {e}
for every x ∈ M. The action is called effective if a = b implies a = b, i.e. if the
homomorphism G → Diff(M) is injective.
Set theoretically the action yields a diagram
G
(−,x)
GG
p
44
M
G/Sx
aSx → ax an injective map
``
and if the action is transitive then G/Sx → M is even a bijection. Naturally G/Sx
is a topological space, a quotient of G:
U ⊆ G/Sx is open ⇐⇒ p−1
(U) ⊆ G is open.
Theorem 6.3. Let H ⊆ G be a closed subgroup of a Lie group G. Then there
exists a unique smooth structure on G/H for which p : G → G/H is a submersion.
6. HOMOGENEOUS SPACES 17
Proof. First we will demonstrate uniqueness in a more general context. The
idea here is that surjective submersions are quotient objects:
M
g
GG
f

P
N
h
bb
If f is a surjective submersion and g any smooth map which factors through f settheoretically,
i.e. such that ker f ⊆ ker g (or more concretely f(x) = f(x ) implies
g(x) = g(x )), then the unique map h satisfying g = hf is smooth. This follows
easily from the fact that f admits smooth local sections (and h is thus a composition
of g with such a section).
The uniqueness now follows formally since in the diagram
G
p
}}
p
33
G/H
id
DD
G/H
id
ll ←− possibly different smooth structures
the unique factorization maps are the identity maps and the fact that they are both
smooth means precisely that the two smooth structures coincide.
It remains to prove the existence. Let k ⊆ g be a linear subspace complementary
to h. There are neighbourhoods 0 ∈ V ⊆ k, 0 ∈ W ⊆ h and e ∈ U ⊆ G such that
ϕ : V × W −→ U
(A, B) −→ exp A · exp B
is a diffeomorphism and U ∩ H = ϕ({0} × W). Let 0 ∈ V ⊆ V be such that
(exp V )−1
· (exp V ) ⊆ U
which is possible by continuity of the operations. Suppose now that A1, A2 ∈ V
are such that (exp A1) · H = (exp A2) · H. Then (exp A1)−1
· exp A2 ∈ U ∩ H and
is equal to exp B for a unique B ∈ W. Multiplying back
ϕ(A2, 0) = ϕ(A1, B)
which implies A1 = A2 and B = 0. This says that the map
f : V × H −→ G
(A, b) −→ (exp A) · b
is injective. Since it is also a local diffeomorphism on V × (exp W) by translation
it is so everywhere and f is in fact a diffeomorphism onto its image.
We have now identified a neighbourhood of H ⊆ G with a product V × H and
in such a way that the cosets a · H lying in this “chart” are of the form {A} × H.
Thus the map
ψ : V ∼= V × {e} → V × H → G
p
−−→ G/H
embeds V as a neighbourhood of the coset eH ∈ G/H. We therefore declare it a
chart on G/H. In this way the map p becomes the projection V × H → V and
6. HOMOGENEOUS SPACES 18
thus a submersion. To get a chart near arbitrary aH redefine f as
fa : V × H −→ G
(A, b) −→ (exp A) · a · b
The transition map between the resulting charts ψa and ψa is the composition
V
exp
−−−→ U
ρa a−1
−−−−−→ U
f−1
−−−−→ V × H −→ V
with all arrows smooth and ρa a−1 only locally defined.
Definition 6.4. The manifold G/H is called a homogeneous space.
Theorem 6.5. The orbit of each point is an immersed submanifold (i.e. image
of an injective immersion).
Proof. Consider the diagram
G
(−,x)
GG
p
44
M
G/Sx
f
``
with the map f smooth by the previous theorem. We need to show that it is
an immersion (on the other hand it is injective almost by the definition of Sx).
Suppose first that for A ∈ g its image p∗A is sent to 0 ∈ TxM by f∗. Then
d
dt t=0
exp(tA)x = 0. On the other hand
d
dt t=t0
exp(tA)x = d
dt t=t0
exp(t0A) exp((t − t0)A)x
= ( exp(t0A))∗
d
dt t=t0
exp((t − t0)A)x
0
= 0
Thus exp(tA)x = x for all t ∈ R and exp(tA) ∈ Sx implying that A ∈ ker p∗ and
p∗A = 0. This finishes the proof that f is an immersion at eSx. At other points
this is guaranteed by the homogeneity:
eSx_

G/Sx
f
GG
a
∼=

M
a
∼=

aSx G/Sx
f
GG M
Example 6.6. Fix v ∈ R2
and consider the following action of R on R2
R × R2
−→ R2
(t, u) −→ u + tv
Clearly the orbit of u is the line u + Rv. Passing to the torus T2
= R2
/Z2
we see
that orbits need not be embedded submanifolds.
Remark. In general every orbit is an initial submanifold.
Corollary 6.7. For a transitive action the map f : G/Sx → M is a diffeo-
morphism.
6. HOMOGENEOUS SPACES 19
Proof. From Sard’s theorem it easily follows that smooth bijections exist only
between manifolds of the same dimension. Hence the immersion f is in fact a local
diffeomorphism. Being also bijective it is a diffeomorphism by the inverse function
theorem.
Examples 6.8. Examples of homogeneous spaces:
• Let V be a vector space. Then GL(V ) acts transitively on V − {0} and
thus V − {0} ∼= GL(V )/Sv where v ∈ V − {0}.
• The sphere Sn−1
with the action of O(n) is a homogeneous space, Sn−1 ∼=
O(n)/O(n − 1) where O(n − 1) is thought of as a subgroup of O(n) consisting
of block matrices
O(n − 1) ∼=
A 0
0 1
∈ O(n) A ∈ O(n − 1)
• The n-dimensional affine space is acted upon by the group
GA(n) =
A v
0 1
∈ GL(n + 1) A ∈ GL(n), v ∈ R
of affine transformations, namely we identify a point x ∈ Rn
with a vector
( x
1 ) in Rn+1
and then
A v
0 1
x
1
=
Ax + v
1
The origin is preserved exactly by the subgroup
GL(n) =
A 0
0 1
∈ GA(n) A ∈ GL(n)
describing Rn
as GA(n)/ GL(n). Similarly with GL(n) replaced by O(n)
we arrive at Rn ∼= Euc(n)/O(n) with Euc(n) denoting the group of (not
necessarily origin preserving) isometries of Rn
.
• The Stiefel manifold (of orthonormal k-frames in V )
Sk(V ) = {(v1, . . . , vk) | vi, vj = δij}
has as examples S1(V ), the unit sphere in V , Sn(Rn
) = O(n). For general
Sk(Rn
) there is a natural action of O(n) componentwise:
A(v1, . . . , vk) = (Av1, . . . , Avk)
The stabilizer of the k-tuple (e1, . . . , ek) of the first k vectors of the standard
basis is clearly
O(n − k) ∼=
E 0
0 C
∈ O(n) C ∈ O(n − k)
Thus Sk(Rn
) ∼= O(n)/O(n − k).
• The Grassmann manifold Gk(V ) of all k-dimensional subspaces of V is
naturally a quotient of Sk(V ), namely by the means of the map
Sk(V ) −→ Gk(V )
(v1, . . . , vk) −→ span(v1, . . . , vk)
7. THE ADJOINT REPRESENTATION 20
The O(n)-action on Sk(Rn
) passes to Gk(Rn
) with the stabilizer of Rk
being
O(k) × O(n − k) ∼=
B 0
0 C
∈ O(n) B ∈ O(k), C ∈ O(n − k)
and thus providing Gk(Rn
) ∼= O(n)/O(k) × O(n − k).
Theorem 6.9. Let N ⊆ G be a closed normal subgroup. Then G/N with its
canonical smooth structure is a Lie group.
Proof. The left vertical arrow in
G × G
µ
GG
p×p

G
p

G/N × G/N GG G/N
is a surjective submersion therefore the dotted arrow (the multiplication in G/N)
is smooth.
7. The adjoint representation
Definition 7.1. By a representation of G we understand a left action of G on
a vector space V by linear maps (automorphisms), i.e. for which each a : V → V
is linear. Equivalently ρ : G → GL(V ) is a (smooth) homomorphism of Lie groups.
Definition 7.2. A representation of a Lie algebra L on a vector space V is a
homomorphism π : L → gl(V ) of Lie algebras. More concretely π is a linear map
for which π[X, Y ](v) = πX ◦ πY (v) − πY ◦ πX(v).
Definition 7.3. A linear subspace W ⊆ V is called invariant with respect to
a representation ρ if ρ(a)(W) ⊆ W for all a ∈ G. Analogously it is called invariant
with respect to a representation π if π(X)(W) ⊆ W for all X ∈ L.
Theorem 7.4. Let G be a connected Lie group and ρ its representation on V ,
ρ∗ : g → V the induced representation of g. Then W ⊆ V is invariant with respect
to ρ if and only if it is invariant with respect to ρ∗.
Proof. Consider the following subgroup of GL(V )
GL(V, W) = {ϕ ∈ GL(V ) | ϕ(W) ⊆ W}.
It is easy to show that
gl(V, W) = Lie(GL(V, W)) = {ϕ ∈ gl(V ) | ϕ(W) ⊆ W}.
The statement then becomes a special case of Lemma 5.10.
Let : G × M → M be a left action and x ∈ M its fixed point (i.e. Sx = G).
Then ρ : G → GL(TxM) given by a → ( a)∗x is smooth by
TG × TM
∗
GG TM
G × TxM
c1
0×id
yy
ρ
WW
7. THE ADJOINT REPRESENTATION 21
and consequently a representation of G on TxM. We apply these general considerations
to the action of G on itself via conjugation (inner automorphisms):
(a, b) −→ inta b = aba−1
Now e ∈ G is a fixed point and we define Ad : G → GL(g) as above
Ad(a)B = (inta)∗B
Moreover Ad(a) ∈ AutLie(g) since inta is a homomorphism of Lie groups. We denote
the induced representation by ad : g → gl(g) (in fact Der(g)).
Theorem 7.5. For each A, B ∈ g it holds ad(A)(B) = [A, B].
Proof. We compute
ad(A)(B) = ∂
∂s s=0
Ad(exp(sA))(B)
= ∂
∂s s=0
∂
∂t t=0
intexp(sA) exp(tB)
= ∂
∂s s=0
∂
∂t t=0
exp(sA) exp(tB) exp(−sA)
= ∂
∂s s=0
∂
∂t t=0
FlλA
−s FlλB
t FlλA
s (e)
= ∂
∂s s=0
(FlλA
−s)∗λB(FlλA
s (e)) = [λA, λB]e = [A, B]
Theorem 7.6. If H ⊆ G is a normal subgroup then h ⊆ g is an ideal, i.e. a
linear subspace such that [g, h] ⊆ h (meaning [A, B] ∈ h for all A ∈ g and B ∈ h).
Proof. Since aHa−1
⊆ H or inta H ⊆ H we differentiate to get Ad(a)(h) ⊆ h
and finally ad(g)h ⊆ h.
Theorem 7.7. Let H be a connected Lie subgroup of a connected Lie group G
such that h ⊆ g is an ideal. Then H is a normal subgroup.
Proof. We have ad : g → gl(g, h). Since G is connected Ad : G → GL(g, h). It
is enough to show that inta(exp tB) ∈ H for all B ∈ h since the subgroup generated
by such elements is the whole group H. But exp tB is a one-parameter subgroup
and inta a homomorphism, thus inta exp tB is also a one-parameter subgroup in G
with initial speed d
dt t=0
inta exp tB = Ad(a)B ∈ h.
Theorem 7.8. Let ϕ : G → H be a homomorphism of Lie groups. Then its
kernel is a closed normal subgroup K ⊆ G and its Lie algebra k is the kernel of
ϕ∗ : g → h.
Proof. A ∈ k iff exp tA ∈ K iff exp(t · ϕ∗A) = ϕ(exp tA) = e iff ϕ∗A = 0.
Definition 7.9. The centre C of a Lie group G is the set
C = {a ∈ G | ab = ba ∀b ∈ G}
In other words, C is the kernel of int : G → Aut(G).
Theorem 7.10. The centre of a connected Lie group G is the kernel of the
adjoint representation Ad.
Proof. a ∈ C iff inta(G) = e iff Ad(a)g = 0 iff Ad(a) = 0.
8. FUNDAMENTAL VECTOR FIELDS 22
Definition 7.11. The centre of a Lie group L is the ideal
Z = {X ∈ L | [X, Y ] = 0 ∀Y ∈ L}
In other words, Z is the kernel of ad : L → gl(L).
Theorem 7.12. For a connected Lie group G, the centre Z of g the Lie algebra
of the centre C of G.
Proof. Since C = ker(Ad), its Lie algebra Lie(C) = ker(ad).
Remark. If the centre of L is zero then L can be embedded into gl(L) via the
representation ad.
Theorem 7.13 (Ado). Every Lie algebra can be embedded into gl(V ) for some
finite-dimensional vector space V .
Corollary 7.14. Every Lie algebra is induced from some Lie group.
Proof. By Ado’s theorem L ⊆ gl(n). Since gl(n) = Lie(GL(n)) one can find
a Lie subgroup of GL(n) corresponding to L.
8. Fundamental vector fields
Consider a left action : G × M → M. To every vector A ∈ g we associate a
vector field A on M by A(x) = ( (−, x))∗A. As usual A is smooth and is called
the fundamental vector field on M corresponding to A ∈ g. Analogously we define
fundamental vector fields for right actions.
Theorem 8.1. In the case of a left action of G on M it holds [ A, B] = −[A,B].
For the right action we obtain [rA, rB] = r[A,B].
Proof. On M × G consider the vector field (0, λA)(x, a) = (0x, λA(a)).
r∗(x,a)(0, λA) = (r(x, −))∗aλA = (r(xa, −))∗eA = rA(xa)
says that (0, λA) is r-related to rA. As the same is true for B we obtain for the
brackets that [(0, λA), (0, λB)] is r-related to [rA, rB]. But
[(0, λA), (0, λB)] = ([0, 0], [λA, λB]) = (0, λ[A,B])
which is r-related to r[A,B]. Thus [rA, rB] = r[A,B].
The last theorem can be expressed by saying that r : g → XM, A → rA is a
homomorphism of Lie algebras. The left action gives an antihomomorphism.
Definition 8.2. By a right infinitesimal action of a Lie group G on a manifold
M we understand a homomorphism R : g → XM of Lie groups. A right infinitesimal
action is called complete if RA is a complete vector field for each A ∈ g. Analogously
a left infinitesimal action is an antihomomorphism.
Example 8.3. The fundamental vector fields are complete: r(x, exp tA) =
x exp tA is an integral curve through x defined for all t ∈ R.
Remark. A left action is a homomorphism of Lie groups G → Diff(M) (with
infinite dimensional target). The induced Lie algebra homomorphism is g →
Lie(Diff(M)), the latter being XM but with the opposite bracket. As for finite
dimensional Lie groups we can “integrate” a homomorphism of Lie groups but here
under additional requirements - the completeness.
8. FUNDAMENTAL VECTOR FIELDS 23
Theorem 8.4. For a complete right infinitesimal action R : g → XM of a
simply connected Lie group G on M there exists a unique right action r : M ×G →
M of G on M such that RA is its fundamental vector field rA.
Remarks.
• The simple connectivity is necessary: for the action of G = R on itself
by translations the infinitesimal action rt = t passes to an infinitesimal
action of R on the quotient R/Z for which no action exists.
• The theorem holds locally without the assumptions of completeness and
simple connectivity.
• The usual translation between left and right yields an analogous statement
for left actions.
Proof. Let first r be an action of G on M. Let Sx denote the following
submanifold
Sx = {(xa, a) | a ∈ G} ⊆ M × G
The tangent space of Sx is
TSx = {(rA(xa), λA(a)) | a ∈ G, A ∈ g}
Thus Sx is an integral submanifold of the distribution (rA, λA) | A ∈ g .
Let us now start the actual proof of the theorem by considering the distribution
D = (RA, λA) | A ∈ g . Then D is involutive since
[(RA, λA), (RB, λB)] = ([RA, RB], [λA, λB]) = (R[A,B], λ[A,B]).
Let Sx be the maximal integral submanifold of D through (x, e) ∈ M × G. We
claim now that px : Sx → M × G → G is a diffeomorphism.
First we show that it is a covering. Fix a ∈ G and consider an arbitrary
(y, a) ∈ M × G. The computation
d
dt t=t0
(FlRA
t (y), a exp tA)
γ(t)
= (RA(FlRA
t0
(y)), λA(a exp t0A)) ∈ D
shows that γ(t) ∈ SA since it is tangent to the distribution D. Let U ⊆ g be
an open ball centered at 0 on which exp is a diffeomorphism. If (y, a) ∈ Sx then
also (FlRA
1 (y), a exp A) ∈ Sx for all A ∈ U and such points form an open neighbourhood
on which px is a diffeomorphism onto a exp U. If (z, b) ∈ Sx is arbitrary
with b ∈ a exp U then b = a exp A and thus the above subset considered for
(Fl
R−A
1 (z), b exp(−A)) contains (z, b). This finishes the proof that px is a covering
and in fact a diffeomorphism as G is simply connected.
We define for x ∈ M and a ∈ G the action by the requirement
(xa, a) ∈ Sx
By the previous part there is a unique choice for xa. We need to show that r
is smooth but first let us prove the axioms of an action. Clearly xe = x as
Sx is an integral manifold through (x, e). Consider now a left action of G on
M × G by a(y, b) = (y, ab). The distribution D is invariant under this action (as
(id, λa)∗(RA, λA) = (RA, λA)) and thus also its maximal integral submanifolds.
The requirement for our action r can be then rewritten as
Sx = aSxa = a(bS(xa)b)) = (ab)S(xa)b
9. LOCALLY ISOMORPHIC LIE GROUPS 24
As also Sx = (ab)Sx(ab) the maximal integral submanifolds S(xa)b and Sx(ab) must
also be equal proving (xa)b = x(ab).
A word about smoothness...
Definition 8.5. Consider two actions r and r of a Lie group G on manifolds
M and M . A map f : M → M is called equivariant if f(xa) = f(x)a.
Theorem 8.6. If f : M → M is equivariant then rA is f-related to rA.
Proof. The requirement from the definition is f ◦ r(x, −) = r (x, −) ◦ f. Applying
the derivatives of both sides to A we get f∗rA = rAf.
Theorem 8.7. Let f : M → M be a smooth map such that rA is f-related to
rA. If G is connected then f is equivariant.
Proof. Consider the set H ⊆ G of all a ∈ G for which f(xa) = f(x)a for
all x ∈ M. Then H is clearly a subgroup and thus we only need that it contains
a neighbourhood of e. But f(x exp tA) = f(FlrA
t (x)) = Fl
rA
t (f(x)) = f(x) exp tA,
hence exp g ⊆ H and H is open and therefore equal to G.
9. Locally isomorphic Lie groups
Let G be a connected Lie group. Recall that the universal covering of G is
˜G
p

{[γ] | γ : (I, 0) → (G, e)}
_

G γ(1)
with [γ] the homotopy class of γ relative to the boundary. ˜G is simply connected:
firstly π1
˜G → π1G (this works for any covering) since we can lift homotopies and
constant paths lift to constant paths. The image consists exactly of the classes of
loops that lift to loops. For ˜G if γ : I → G lifts to a loop its endpoints must be
equal ˜e = [γ] and the image is therefore trivial.
We give ˜G a structure of a Lie group: let γ, δ : (I, 0) → (G, e) be two paths.
Define their product to be the path
(γ · δ)(t) = γ(t)δ(t)
which easily passes to homotopy classes rel ∂I.
Theorem 9.1. The above multiplication on ˜G describes a structure of a Lie
group for which the projection p : ˜G → G is a local isomorphism (i.e. a homomorphism
and a local diffeomorphism).
Proof. The unit and inverses are also pointwise. The diagram
˜G × ˜G GG

˜G
local diffeomorphism

G × G
smooth
GG G
shows that the dotted arrow (the multiplication in ˜G) is smooth.
9. LOCALLY ISOMORPHIC LIE GROUPS 25
There is an action of π1G on ˜G, π1G × ˜G → ˜G given by
([α], [γ]) −→ [α] · [γ] = [α ∗ γ]
which respects the projection p : ˜G → G. Let Γ ⊆ π1G be a subgroup and consider
pΓ : ˜G/Γ → G
where ˜G/Γ is the space of orbits of the restriction of the action to Γ. Locally
U × π1G 1 
GG

˜G
p

U 1  GG G
and the action of Γ is by left multiplication in π1G. Thus the projection pΓ from
˜G/Γ to G is locally of the form
U × (π1G/Γ) → U
and in particular is a covering of G.
Theorem 9.2. Let G be a connected Lie group. Then the mapping
{subgroups Γ ⊆ π1G} −→ local isomorphisms ρ : G → G
with G any connected Lie group
/iso
Γ −→ (pΓ : ˜G/Γ → G)
is a bijection with inverse ρ −→ im(π1ρ : π1G → π1G).
Proof. The image of π1pΓ consists of those loops that lift to loops in ˜G/Γ.
These are precisely those in Γ. In the opposite direction any ρ fits into the diagram
˜G GG
88
p
&&
G
ρ
ÔÔ
˜G/Γ
∼=
VV
pΓ

G
with Γ = im(π1ρ). The top arrow exists by universality of ˜G. The dotted arrow
exists since loops in Γ lift to loops in G . It is an isomorphism of Lie groups.
Remark. We will show in the tutorial that π1G → ˜G is a homomorphism and
the action of π1G on ˜G is by left translations, i.e. ˜G/Γ is a quotient of ˜G by (a
central subgroup) Γ.
Example 9.3 (The universal covering of a commutative connected Lie group
G). Since Lie G = Rn
with zero bracket it is also the Lie algebra of the simply
connected Lie group Rn
(with vector addition) and thus ˜G = Rn
. Therefore G ∼=
Rn
/Γ where Γ is some discrete subgroup of Rn
. We will show now that Γ = Zk
⊆ Rn
in some coordinates on Rn
.
First reduction is to the case n = k, namely we have span Γ = Rk
⊆ Rn
and Γ
is still discrete in Rk
. We must show that Γ = Zk
in some coordinates on Rk
.
We start an induction by k = 1 which we proved in the tutorial. For the
induction step we may assume that Γ ⊆ R×Rk
= Rk+1
is such that the intersection
Γ ∩ R = 0 with the first coordinate axis is nonzero. Since it is also discrete it is
10. PROBLEMS 26
generated by some a0. In Rk
= Rk+1
/R consider its subgroup Γ/ a0 . We show
by contradiction that it is discrete. Namely assume the existence of a sequence
αn = (βn, γn) ∈ Γ with γn → 0 in Rk
. By adding a suitable multiple of a0 to
each αn we may assume that βn ∈ [−a0/2, a0/2] and by extracting a subsequence
we may further assume that αn converges. But then αn+1 − αn ∈ Γ converges to
0, a contradiction with Γ being discrete. By the induction hypothesis Γ/ a0 =
˜a1, . . . , ˜ak . We choose for each ˜ai an element ai ∈ Γ representing it. Then the
suitable basis in which Γ = Zk+1
is formed by (a0, a1, . . . , ak).
0 GG Za0
GG
∼=

Z{a0, a1, . . . , ak} GG

Z{˜a1, . . . , ˜ak} GG
∼=

0
0 GG Γ ∩ R GG Γ GG Γ/(Γ ∩ R) GG 0
Corollary 9.4. The only compact connected commutative Lie group of dimension
k is the torus Tk
= (S1
)k
.
Example 9.5. For n ≥ 3 we have π1 SO(n) ∼= Z/2. Therefore SO(n) possesses
a two-sheeted universal covering which is denoted by Spin(n) = SO(n). We will
show geometrically that π1 SO(n) = Z/2 in the tutorial. For higher n we have a
fibration
SO(n) → SO(n + 1) → Sn
whose long exact sequence of homotopy groups contains the following portion
0 = π2(Sn
) → π1(SO(n))
∼=
−→ π1(SO(n + 1)) → π1Sn
= 0
10. Problems
Problem 10.1. An algebra is a vector space A together with a bilinear map
· : A × A → A. Let A be now an associative algebra and define [ , ] : A × A → A
by [a, b] = a · b − b · a. Show that with this operation A forms a Lie algebra.
A special case of the previous is the algebra End(V ) of endomorphisms of a
vector space V together with their compositions. The induced Lie algebra is denoted
by gl(V ). The bracket of two endomorphisms ϕ, ψ is
[ϕ, ψ] = ϕ ◦ ψ − ψ ◦ ϕ
Problem 10.2. Let A be an algebra. A linear map D : A → A is called a
derivative if for all a, b ∈ A
D(a · b) = D(a) · b + a · D(b)
Show that derivatives form a Lie subalgebra Der(A) ⊆ gl(A).
Problem 10.3. Let C∞
M = C∞
(M, R) denote the algebra of all smooth
functions on M. Then every vector field X on M determines a mapping
C∞
M −→ C∞
(M)
f −→ Xf = df(X)
Show that this mapping is a derivative (in the algebraic sense). Also show that all
derivatives of C∞
M are of this form.
10. PROBLEMS 27
Let us now describe the Lie bracket of vector fields from this point of view:
[X, Y ] is simply the vector field corresponding to the bracket of the two derivatives
X and Y of C∞
M. This means that [X, Y ]f = XY f − Y Xf and this formula
determines a unique vector field [X, Y ].
It also holds that algebra homomorphisms C∞
N → C∞
M are in bijection with
smooth maps M → N. One may then rewrite the f-relatedness of vector fields X
and Y as
C∞
N
f∗
GG
Y

C∞
M
X

C∞
N
f∗
GG C∞
M
It is then a simple matter to show that Xi ∼f Yi implies [X1, X2] ∼f [Y1, Y2].
Problem 10.4. Compute the Lie algebra of the additive Lie group Rn
.
Problem 10.5. Compute the Lie algebra of the Lie group GL(n, R) from the
definition.
Problem 10.6. Compute the Lie algebra of the Lie group GL(n, R) from the
formula [A, B] = ∂2
∂s∂t
(s,t)=(0,0)
ϕ(t)ψ(s)ϕ(t)−1
ψ(s)−1
.
Problem 10.7. Compute the Lie algebra of the Lie group S3
= Sp(1) of unit
quaternions and show that it is isomorphic to R3
with the vector product ×.
Problem 10.8. Let B : Rn
× Rn
→ R be a bilinear form and denote by
G(B) = {A ∈ GL(n, R) | AT
BA = B} ⊆ GL(n, R)
the closed subgroup of all automorphisms preserving the form B. Compute the Lie
algebra of G(B).
Problem 10.9. Compute the Lie algebra of SO(n, R).
Problem 10.10. Let A be an algebra and denote by Aut(A) the group of all
algebra automorphisms of A. Compute its Lie algebra.
Problem 10.11. Determine all Lie algebras of dimension 2 over R.
Problem 10.12. Prove that the element −2 0
0 −1 of GL(2, R) lies in the component
of the unit E but not in the image of exp.
Problem 10.13. Let
G =





1 a b
0 1 c
0 0 1

 ∈ GL(3, R) a, b, c ∈ R



denote the Heisenberg group. Show that the bracket on Lie(G) is non-trivial and
exp is a global diffeomorphism.
Problem 10.14. Show that for G = S3
= Sp(1) the map exp is not a local
diffeomorphim at all points of g.
Problem 10.15. Find all connected one-dimensional Lie groups.
Problem 10.16. Show that a discrete normal subgroup of a connected Lie
group must Lie in the centre.
10. PROBLEMS 28
Problem 10.17. Let f : M → G be a smooth map from a manifold M to a Lie
group G. Denote by δlf the g-valued 1-form called the left logarithmic derivative
of f given by
δlf(x, X) = (λf(x)−1 )∗f∗X
(with (x, X) denoting a tangent vector X ∈ TxM). For example
δlid(a, A) = (λa−1 )∗A = ω(A)
the Maurer-Cartan form. Compute δlλb, δlρb, δlµb, δlνb and δl(f · g−1
).
As a corollary, for a connected manifold M two maps f, g : M → G satisfy
δlf = δf g if and only if f = c · g for some c ∈ G. There exists also a criterion for
determining whether a g-valued one-form is a left logarithmic derivative of a map
into G. This generalizes the integral calculus of functions.
Problem 10.18. Let ˜G be the universal covering of G. Show that π1G ⊆ ˜G is
a discrete and normal subgroup thus lying in the centre of ˜G.
Problem 10.19. Show that the image of the adjoint representation Ad :
Sp(1) → GL(3, R) is SO(3, R) and that its kernel is the subgroup {±1}. Thus
Sp(1) is the 2-fold (universal) covering of SO(3, R).
Problem 10.20. Let ϕ : Sp(1) × Sp(1) → SO(4, R) be the map sending (a, b)
to the orthogonal transformation of the quaternions x → axb−
1. Show that this
map is a 2-fold (universal) covering.
Problem 10.21. Compute the centre of SO(n, R) or even better the centralizer
CSO(n,R) GL(n, R)+. Try to determine all connected Lie groups with Lie algebra
so(n, R).
Problem 10.22. Try to determine the first few terms in the Baker-CampbellHausdorff
formula for
log(exp X · exp Y ) : g × g → g
where log is the (locally defined) inverse to exp.
A semidirect product of groups is a split short exact sequence
1 GG K GG G
p
GG H GG
i
ii 1
The subgroup K ⊆ G is normal being a kernel of p. The map f : H
i
−→ G
int
−−→
Aut(K) given by f(x)(y) = xyx−1
is a group homomorphism. For a ∈ G there are
uniquely determined k ∈ K and h ∈ H such that a = k · i(h). Namely h = p(a)
and k = a · i(h)−1
. Therefore as sets G ∼= K × H and the multiplication is given by
(k1, h1)·(k2, h2) = k1·i(h1)·k2·i(h2) = k1·f(h1)(k2)·i(h1h2) = (k1·f(h1)(k2), h1·h2)
The resulting group is denoted by K H = K f H.
Problem 10.23. Show that GA(n, R) is a semidirect product GA(n, R) ∼=
Rn
GL(n, R) where the action of GL(n, R) on Rn
is the standard one.
Problem 10.24. Let G be a Lie group. Show that µ∗ : TG×TG → TG endows
TG with a structure of a Lie group.
Problem 10.25. Show that TG is a semidirect product TG ∼= g G and
identify the involved action of G on g.
10. PROBLEMS 29
Problem 10.26. Compute the Lie algebra of a semidirect product K f H.
Problem 10.27. Determine the Lie algebra of TG.