Download 5.1 The Lie algebra of a Lie group Recall that a Lie group is a group

5.1 The Lie algebra of a Lie group
Recall that a Lie group is a group with a manifold structure such that the multiplication G × G → G
and the inverse I : G → G are smooth maps. Fixing one of the arguments in the multiplication we obtain
diffeomorphisms Lg : G → G and Rg : G → G, defined by Lg (g ′ ) = gg ′ and Rg (g ′ ) = g ′ g. They satisfy
the composition properties Lg1 ◦ Lg2 = Lg1 g2 and Rg1 ◦ Rg2 = Rg2 g1 . A vectorfield v ∈ X(G) is said to be
left-invariant (resp. right–invariant) if T Lg (v) = v (resp T Rg (v) = v) for all g ∈ G. In the following we will
concentrate mostly on the left-invariant vectorfields. Because of (3.3.7), the Lie bracket of two left-invariant
vectorfields is a left-invariant vectorfield. Therefore, the left-invariant vectorfields form a subalgebra of the
infinite-dimensional algebra X(G), called the Lie algebra of G and denoted L(G).
The identity element of the group will be denoted e. If v̄ ∈ Te G is a vector tangent to G at the identity,
we can define a unique left-invariant vectorfield v that coincides with v̄ in the identity, by:
v(g) = T Lg (v̄) .
(5.1.1)
There is therefore a one-to-one correspondence between elements of Te G and left-invariant vectorfields. Thus,
the dimension of the Lie algebra L(G) is equal to dimG.
The inverse map I : g 7→ g −1 is a diffeomorphism of G; since Rg ◦ I = I ◦ L−1
g , the tangent map
T I maps left-invariant vectorfields to right-invariant vectorfields and vice-versa. The map J : v 7→ −v is
an anti-isomorphism of the algebra of vectorfields X(G). It maps left-invariant vectorfields to left-invariant
vectorfields and right-invariant vectorfields to right-invariant vectorfields. Both J and T I act as the inversion
v 7→ −v in Te G, the tangent space at the identity. The composite map J ◦ T I is an anti-isomorphism on the
algebra of vectorfields X(G) that acts as the identity in Te G and maps left-invariant vectorfields to rightinvariant vectorfields and vice-versa. It maps a left-invariant vectorfield v to the right-invariant vectorfield
that coincides with v at the identity. Thus, the algebra of right-invariant vectorfields is antiisomorphic to
L(G).
Let {ea }, where a = 1, . . . , dim G, be a linear basis in Te G. We call La and Ra the left-invariant and
right-invariant vectorfields that coincide with ea at the identity. Since T Lg is an isomorphism, the sets
of vectorfields {La }, where a = 1, . . . , dim G define linear frames in Tg G for any g. Thus, a Lie group is
parallelizable.
The Lie brackets of the left-invariant frame fields {La } are left-invariant vectorfield and therefore the
structure functions of this field of bases (defined in (3.2.6)) are constant:
[La , Lb ] = fab c Lc ;
(5.1.2)
The coefficients fab c are called the structure constants of the Lie algebra L(G). Applying the anti-isomorphism
J ◦ T I, we find that the right-invariant frame fields satisfy
[Ra , Rb ] = −fab c Rc ;
(5.1.3)
Example 5.1.1. The Lie algebra of the linear group. Following up example 2.1.2, the matrix
elements Mab can be used as coordinates on the general linear group GL(n). Consider a curve
M (t) in GL(n). It is represented by a one-parameter family of matrices Mab (t). Given a function
f ∈ C ∞ (GL(n)), we have
∂f dMab ∂f
d
mab
f (c(t))
=
=
dt
∂Mab dt t=0
∂Mab
t=0
where the coordinate components of the tangent vector m are
d
mab = Mab (t)
.
dt
t=0
One can also write this equation as
m=
d
M (t)
∈ TM(0) GL(n) ,
dt
t=0
1
where M and m are matrices. In this way the space tangent to GL(n) at a point can be identified
with the space of all n × n matrices. The left and right actions of the group on tangent vectors are
then represented
by left and right matrix multiplications. For example if m is defined as above,
d
= mg ∈ TM(0)g (GL(n)). In particular, if M (0) = 1 the Lie algebra of the general
dt M (t)g t=0
linear group GL(n) consists of all n × n matrices.
Example 5.1.2: The Lie algebra of the groups SO(3) and SU (2). Consider the space R3 with
cartesian coordinates y1 , y2 , y3 . Let Ri (α) denote a counterclockwise rotation by an angle α around
the axis i. Every rotation R ∈ SO(3) can be written uniquely in the form R = R3 (Φ)R2 (Θ)R3 (Ψ),
where 0 < Φ ≤ 2π, 0 ≤ Θ ≤ π, 0 < Ψ ≤ 2π. Thus the Euler angles Φ, Θ, Ψ can be taken
as coordinates on SO(3). This definition is in agreement with the usual definition of spherical
coordinates
y1 = r sin Θ cos Φ ,
(5.1.2.1a)
y2 = r sin Θ sin Φ ,
y3 = r cos Φ .
(5.1.2.1b)
(5.1.2.1c)
In fact, if we consider a sphere of radius r, the rotation with Euler angles Φ, Θ, Ψ maps the north
pole y1 = 0, y2 = 0, y3 = r to the point on the sphere whose spherical coordinates are exactly Θ,
Φ. The infinitesimal generators of the algebra of SO(3) are the matrices Ti with matrix elements
(Ti )jk = −ǫijk . They satisfy the algebra [Ti , Tj ] = ǫijk Tk . We have Ri (α) = exp(αTi ). Therefore
R(Θ,Φ, Ψ) = exp(ΦT3 ) exp(ΘT2 ) exp(ΨT3 )


cos Θ cos Φ cos Ψ − sin Φ sin Ψ − cos Θ cos Φ sin Ψ − sin Φ cos Ψ sin Θ cos Φ .
=  cos Θ sin Φ cos Ψ + cos Φ sin Ψ − cos Θ sin Φ sin Ψ + cos Φ cos Ψ sin Θ sin Φ 
− sin Θ cos Ψ
sin θ sin Ψ
cos Θ
Now consider the group SU (2). An element U ∈ SU (2) is a complex matrix
x4 + ix3 x2 + ix1
U=
−x2 + ix1 x4 − ix3
(5.1.2.2)
(5.1.2.3)
with det U = x21 + x22 + x23 + x24 = 1. So SU (2) is the unit sphere in R4 . The unit element of SU (2) is
the point with x1 = x2 = x3 = 0, x4 = 1. The Lie algebra of SU (2) consists of the skew-hermitian
matrices. It can be identified geometrically with the plane x4 = 1 in R4 . We take as basis in the
Lie algebra the matrices ti = − 2i σi where σi are the Pauli matrices. These generators satisfy the
algebra [ti , tj ] = ǫijk tk . The Lie algebras of SU (2) and SO(3) are thus isomorphic. A concrete
isomorphism is given by mapping ti to Ti . This also defines an isomorphism of a neighbourhood
of the identity in SU (2) to a neighbourhood of the identity in SO(3), and we can use this fact to
introduce the Euler angles as coordinates on SU (2). Define:
U (Θ, Φ, Ψ) = exp(Φt3 ) exp(Θt2 ) exp(Ψt3 )
i
i
− sin Θ
cos Θ
2 exp − 2 (Φ + Ψ)
2 exp − 2 (Φ − Ψ)
.
=
i
i
sin Θ
cos Θ
2 exp 2 (Φ − Ψ)
2 exp 2 (Φ + Ψ)
(5.1.2.4)
Every matrix U ∈ SU (2) can be written in this way, provided that 0 < Φ ≤ 2π, 0 ≤ Θ ≤ π,
0 < Ψ ≤ 4π. There is a homomorphism from SU (2) to SO(3) that maps U (Θ, Φ, Ψ) to R(Θ, Φ, Ψ).
It is a double covering because the range of Ψ in U is twice the range of Ψ in R.
The left–invariant and right–invariant vectorfields which coincide at the identity with the basis
α
vectors Ta are denoted La = Lα
a ∂α and Ra = Ra ∂α respectively. In the coordinate basis of the
Euler angles they are
L1 = sin Ψ
∂
1
∂
∂
−
cos Ψ
+ cot Θ cos Ψ
,
∂Θ sin Θ
∂Φ
∂Ψ
2
(5.1.2.5a)
L2 = cos Ψ
L3 =
∂
1
∂
∂
+
sin Ψ
− cot Θ sin Ψ
,
∂Θ sin Θ
∂Φ
∂Ψ
(5.1.2.5b)
∂
,
∂Ψ
(5.1.2.5c)
∂
1
∂
∂
− cot Θ cos Φ
+
cos Φ
,
∂Θ
∂Φ sin Θ
∂Ψ
∂
1
∂
∂
− cot Θ sin Φ
+
sin Φ
,
R2 = cos Φ
∂Θ
∂Φ sin Θ
∂Ψ
∂
R3 =
.
∂Φ
(5.1.2.5d)
R1 = − sin Φ
(5.1.2.5e)
(5.1.2.5f )
A direct calculation gives the Lie brackets
[La , Lb ] = ǫabc Lc ,
(5.1.2.6a)
[Ra , Rb ] = − ǫabc Rc .
(5.1.2.6b)
It follows immediately from the semigroup property of flows that the integral curves of the left-invariant
vectorfields through the identity are one-parameters subgroups of G. The same is true for the right-invariant
vectorfields. Conversely, the vectors tangent to the one-parameter subgroups are
both left- and right
invariant: denoting c(t) a one-parameter group (with c(0) = e) and v(t) = dc(t+w)
the vector tangent
dw
w=0
to the curve at c(t),
T Lc(s) (v(t)) =
d
d
Lc(s) c(t + w)
=
c(t + s + w)
= v(t + s)
dw
dw
w=0
w=0
(5.1.4)
and similarly T Rc(s) (v(t)) = v(t + s).
Given g ∈ G and a one–parameter subgroup c(t) with tangent vector v at c(0), consider now the curve
gc(t) = Rc(t) (g). We have
d
d
Rc(t) (g)
= Lg (c(t))
= T Lg (v) = v a La (g)
dt
dt
t=0
t=0
(5.1.5)
One can read this equation by saying that the flow generated by a left-invariant vectorfield consists of right
translations (and similarly the flow generated by a right-invariant vectorfield consists of left multiplications).
From this one calculates
d
=0
(5.1.6)
[La , Rb ] = LLa Rb = T Rc(t) (Rb )
dt
t=0
where c(t) is the one-parameter group tangent to La in the origin. This reflects the fact that the right and
left multiplications commute.
Since left and right multiplications commute, if v is a left-invariant vectorfield, T Rg (v) is also a leftinvariant vectorfield. The map T Lg ◦ T Rg−1 is an isomorphism of the vectorspace Te G ≈ L(G) onto itself
called Ad(g). The map Ad : G → End(L(G)) is called the adjoint representation. In a given basis {La } it
has a matrix representation Ad(g)a b , such that
Ad(g)Lb = Ad(g)a b La .
(5.1.7)
The components of the left– and right–invariant vectorfields relative to a coordinate basis form n×n matrices
α
Lα
a and Ra :
∂
∂
, Ra = Raα α .
(5.1.8)
La = Lα
a
∂y α
∂y
The adjoint representation matrices can be written in terms of the components of the invariant vectorfields:
Ad(g)a b = R(g)aα L(g)α
b .
3
(5.1.9)
where R(g)aα is the inverse matrix of R(g)α
a.
5.2 Invariant forms
A field of p-forms α is said to be left- (resp. right-) invariant if L∗g α = α (resp Rg∗ α = α). As with
vectorfields, the space of invariant p-forms is in one-to-one correspondence with the space Λpe G. One can
a
choose bases of one-forms La = Laα dy α and Ra = Rα
dy α , dual to the invariant vectorfields (5.1.2):
a
La (Lb ) = Laα Lα
b = δb ;
a α
Ra (Rb ) = Rα
Rb = δba .
(5.2.1)
The forms La and Ra are called the left and right Maurer–Cartan forms. Instead of thinking of them as
a set of real one-forms, one can also regard them as a single Lie-algebra-valued one-form L = La La and
R = Ra Ra . When acting on a vector v ∈ Tg G, L(v) = La (v)La = v a La , i.e. it gives the left-invariant
vectorfield that coincides with the vector v at the point g. In particular L and R act as the identity on left–
and right–invariant vectorfields. In the case of a matrix group, one often uses the notation
L(g) = g −1 dg ,
R(g) = dgg −1 .
(5.2.2)
Consider first the general linear group. A nondegenerate matrix M ∈ GL(n) can be parametrized by a
set of n2 coordinates y α . Equation (5.2.2) stands for L(M ) = M −1 ∂α M dy α , where M −1 ∂α M is a matrix,
hence an element of the Lie algebra of GL(n). For any group G, if ρ is a linear representation and if we set
M = ρ(g), we can think of M as a function of the coordinates y α of the group element g. Then M −1 ∂α M
is a matrix representing an element of the Lie algebra of G. See also example (5.2.1) below. This form is
left–invariant, because, for fixed g ′ , L(g ′ g) = (M ′ M )−1 d(M ′ M ) = L(g).
Since d commutes with the pullback, the exterior derivative of an invariant form is invariant. Therefore,
the differentials of the Maurer-Cartan forms can be decomposed on the basis of invariant two-forms. Using
the definition (4.2.1), (5.2.1) and (5.1.2) we obtain
dLa (Lb , Lc ) = Lb (La (Lc )) − Lc (La (Lb )) − La ([Lb , Lc ]) = −La ([Lb , Lc ]) = −fbc a .
(5.2.3)
From here we obtain the following Maurer-Cartan equations:
1
dLa + fbc a Lb ∧ Lc = 0 ;
2
1
dRa − fbc a Rb ∧ Rc = 0 .
2
(5.2.4)
Under the action of right translations we have
−1 a
(Rg∗ La )(Lb ) = La (T Rg (Lb )) = La (T Rg ◦ T L−1
) b
g (Lb )) = Ad(g
and therefore
Rg∗ La = Ad(g −1 )a b Lb .
(5.2.5)
Example 5.2.1. On the group SU (2) the left–invariant and right–invariant Maurer–Cartan
a
forms are given by L = U −1 dU = Laα dϕα ⊗ Ta and R = dU U −1 = Rα
dϕα ⊗ Ta respectively.
Introducing the matrix form (A.4) in (5.2.2) and differentiating, we can read off the components
L1 = sin ΨdΘ − sin Θ cos ΨdΦ ,
(5.2.1.1a)
L2 = cos ΨdΘ + sin Θ sin ΨdΦ ,
L3 = dΨ + cos ΘdΦ ,
(5.2.1.1b)
(5.2.1.1c)
R1 = − sin ΦdΘ + sin Θ cos ΦdΨ ,
(5.2.1.1d)
2
R = cos ΦdΘ + sin Θ sin ΦdΨ ,
R3 = dΦ + cos ΘdΨ .
(5.2.1.1e)
(5.2.1.1f )
One can then easily check equations (5.2.1) and the Maurer-Cartan equations
1
dLa = − ǫabc Lb ∧ Lc ,
2
1
dRa = ǫabc Rb ∧ Rc .
2
4
(5.2.1.3a)
(5.2.1.3b)
A left-invariant p-form can be written in the left-invariant basis
α=
1
αa ...a La1 ∧ . . . ∧ Lap ,
p! 1 p
(5.2.6)
where αa1 ...ap are constants. These components are related to the natural basis components by
αa1 ...ap = Laµ11 . . . Lµapp αµ1 ...µp .
(5.2.7)
Using the general formula (4.2.3), the exterior differential of an invariant form is
dα(v1 , . . . , vp+1 ) =
X
(−1)i+j+1 α([vi , vj ], . . . , v̂i , . . . , vj , . . . , vp+1 ) ,
(5.2.8)
i<j
or, in components,
(dα)a1 ...ap+1 = (p + 1) fa1 a2 b αba3 ...ap+1 + . . . .
(5.2.9)
The formR ω = L1 ∧ . . . ∧ Ln (with n = dimG) is a left-invariant volume-form on G. If G is compact, the
volume V = G ω is finite. If α is any p-form on G, we define the G-averaged p-form ᾱ by
ᾱ =
1
V
Z
ω(g)L∗g α .
(5.2.10)
G
Clearly, ᾱ is a left-invariant form. Because of the property (4.2.8), if α is closed also ᾱ is closed. We shall
now prove that if G is connected, ᾱ − α is exact. We can write
Z
1
ᾱ − α =
ω(g)(L∗g α − α) .
V G
For each g ∈ G we can find a one-parameter group c(t) such that c(0) = 1 and c(1) = g. Then, denoting K
the right–invariant vectorfield tangent to c,
L∗g α
−α=
Z
0
1
d
dt L∗c(t) α =
dt
Z
1
dtLK α =
0
Z
0
1
dt diK α = d
Z
1
dt iK α .
0
This proves that in every cohomology class one can find a representative which is left-invariant. Using (5.2.9),
the exterior differential can be expressed as a purely algebraic operation. This Lie algebra cohomology, which
is due to Chevalley and Eilenberg, is isomorphic to the de Rham cohomology of G.
5.3 Group actions on manifolds
Let G be a Lie group and M a smooth manifold. One says that G acts on M from the left if there is a
map L : G × M → M such that:
L(e, x) = x ∀x ∈ M ,
L(g1 , L(g2 , x)) = L(g1 g2 , x) .
(We call e the identity in the group). Similarly, G acts on M from the right if there is a map R : M × G → M
such that:
R(x, e) = x ∀x ∈ M ,
R(R(x, g2 ), g1 ) = R(x, g2 g1 ) .
It is often convenient to define for each g ∈ G a map Lg : M → M by Lg (x) = L(g, x), and similarly in
the case of a right action Rg : M → M is defined by Rg (x) = R(x, g). The composition of these maps is
governed by the rule
Rg1 ◦ Rg2 = Rg2 g1 .
Lg1 ◦ Lg2 = Lg1 g2 ,
5
The maps Lg are diffeomorphisms of M to itself. The map g 7→ Lg is a homomorphism from G to the
diffeomorphism group of M ; it is called a realization of the (abstract) group G.
In the following we will discuss mainly left actions, but all definitions can be extended in an obvious
way to right actions. The action is said to be:
• effective if Lg (x) = x for all x implies g = e;
• free if Lg (x) = x for some x implies g = e (this is equivalent to saying that the maps Lg have no fixed
points);
• transitive if for any pair of points x1 , x2 in M there is a g in G such that Lg (x1 ) = x2 .
It is convenient to introduce some further definitions. The orbit through a point x0 ∈ M is the set Ox0
of all points x ∈ M such that x = Lg (x0 ) for some g ∈ G. By definition the group G acts transitively on each
orbit. The stabilizer (or isotropy group) of a point x0 ∈ M is the subgroup Hx0 of G such that Lg (x0 ) = x0
for g ∈ Hx0 . The stabilizers of two different points belonging to the same orbit are conjugate subgroups
of G. In fact if x = Lg (x0 ), the stabilizer Hx consists precisely of all elements of G which are of the form
ghg −1 , with h ∈ Hx0 . Thus the stabilizers of all points in the same orbit are isomorphic. By definition if the
action is free the stabilizer of each point is the trivial group consisting only of the identity. The relation of
belonging to the same orbit is an equivalence relation on M . Thus one can define the quotient space M/G.
This is also called the space of orbits, since every orbit of G in M corresponds to exactly one point of M/G.
In general, the space of orbits will not be a smooth manifold. This is due to the existence of orbits with
non-isomorphic stabilizers.
Let us consider some examples. If M is a vectorspace and the action of G on M is by linear transformations, then the action defines a linear representation of G (by contrast in the case when M is not a
linear space L is called a nonlinear realization of the group G). In this case the action is effective if and
only if the corresponding representation is faithful. A linear action can never be free or transitive, because
the origin is always an orbit by itself. The space of orbits has singularities corresponding to orbits with
larger stabilizer. Consider for example the fundamental representation of SO(3) on R3 . All orbits are two
dimensional spheres, except for the origin, which is a point. The stabilizer of the spherical orbits is SO(2),
while the stabilizer of the origin is SO(3). The orbits can be parametrized by the radius and the space
of orbits R3 /SO(3) is a half-line, with zero corresponding to the exceptional orbit. In this case the space
of orbits is a one dimensional manifold with boundary. In the case of more complicated representations
there can be many classes of orbits with different stabilizers, and the space of orbits is correspondingly more
complicated.
If a group G acts on a manifold M , one can define a homomorphism of L(G) to the algebra of vectorfields
X(M ). We assume without loss of generality that the action is from the left. Let v ∈ L(G) be the generator
of a one-parameter subgroup c(t). For each x ∈ M we define Kv (x) ∈ Tx M to be the vector tangent to the
curve Lc(t) (x). We thus obtain a vectorfield Kv ∈ X(M ). By construction, the flow of this vectorfield is the
left action of c(t), so Kv is called a generator of the action of the group. Note that Kv vanishes at the fixed
points of the action.
5.4 Homogeneous spaces and principal bundles
In this section we discuss in more detail the transitive and the free actions. A group acts on itself by
multiplication both from the left and from the right. These two actions commute and are both free and
transitive. Now let H be a proper subgroup of G (by proper it is meant that H is not G itself, and it
does not consist only of the identity). The subgroup H acts on G both from the left and from the right by
multiplication. We focus on the right action. It is again a free action, but it is not transitive. The orbits
of H in G are called right cosets; the space of orbits G/H is called a right coset space. A coset space is an
example of a space of orbits which is a smooth manifold. The reason for this is that each orbit of H in G
has the same stabilizer, namely the trivial group consisting only of the identity.
There is a left action L̄ of G on G/H, defined by L̄g (g ′ H) = (Lg (g ′ ))H, where we denote gH the coset
of g and L the left action of G on itself. This definition is independent of the representative g ′ in the coset
because the left action of G on itself and the right action of H on G commute. The coset space G/H has a
preferred point which is the coset eH. We will call this point the “origin” of G/H. The action L̄ is transitive
and effective. The stabilizer of the origin is the group H. We shall now see that up to diffeomorphisms,
every transitive action is of this type.
6
A manifold N is said to be homogeneous if there is a group G acting on N transitively. For example,
each orbit of an action of G on some space M is homogeneous. We assume without loss of generality that
the action is from the left. Let x0 be any point of N and H0 be the corresponding stabilizer. Then N is
diffeomorphic to the coset space G/H0 . We give here the bijective correspondence between points of N and
points of G/H0 , without discussing its differentiability. Let L denote the action of G on N and define a map
β : G/H0 → N by
β(gH0 ) = Lg (x0 ) .
First of all we have to prove that this map is well defined, i.e. that the r.h.s. is independent of the choice
of g in the coset gH0 . If g ′ is another element of the coset, g ′ = gh for some h ∈ H0 , and therefore the
r.h.s. is Lg′ (x0 ) = Lgh (x0 ) = Lg (Lh (x0 )) = Lg (x0 ). The map β is surjective, because L is transitive. The
map β is injective: if β(gH0 ) = β(g ′ H0 ), also Lg (x0 ) = Lg′ (x0 ) and therefore g ′ = gh for h ∈ H0 . But then
gH0 = g ′ H0 . Q.E.D.
In the special case when H = {1} we obtain the following corollary: if a group G acts on a manifold M
transitively and freely, M is diffeomorphic to G.
We now discuss general free actions (without assuming transitivity). Given x, y ∈ M , if there is some
g ∈ G such that y = Lg (x) (i.e. if x and y belong to the same orbit), then this g is unique. To see this
suppose that there is another g ′ such that y = Lg′ (x). Then x = L−1
g′ ◦ Lg (x) = Lg′−1 g (x), but since the
action is free, g ′ = g.
Consider a space P and a free right action of a group G on P (the choice of right action is conventional).
Since G acts transitively on the orbits, and the action is free, each orbit is diffeomorphic to G. In this case
the quotient is well defined. Let M = P/G, i.e. the space of orbits of this action, and π : P → M be the
natural projection that associates to each point its orbit. We make the further technical assumption that π
admits local sections, i.e. that for each point of M there is a neighbourhood U ⊂ M and a map s : U → P
such that π(s(q)) = q for q ∈ U . In this case the space P is said to be a principal bundle over M with
structure group G.
A local section s can be used to define a diffeomorphism Ψ : π −1 (U ) → U × G as follows. Let p ∈ P such
that π(p) = x; p is in the same orbit as s(x), therefore there is a unique group element a such that p = s(x)a.
We define Ψ(p) = (x, a); Ψ is called a local trivialization of P . Conversely, given a local trivialization Ψ
one can define a local section s by s(x) = Ψ(x, e). Therefore, in a principal bundle, there is a one-to-one
correspondence between local sections and local trivializations.
A particular class of right free actions is given by the right action of a Lie group on itself by multiplication.
Let H be a Lie subgroup of G. There is a natural projection π : G → G/H that maps a group element to
its coset. One can prove that these actions admit local sections, so G is a principal H-bundle over G/H.
Since the projection π : G → G/H factors out the right action of H on G, the right invariant vectorfields
{Ra } on G project onto vectorfields T π(Ra ) on G/H. (In this case the issue mentioned before equation (3.3.7)
does not arise.) By (3.3.7) their Lie algebra is isomorphic to (5.1.3). In order to have an algebra that is
isomorphic to L(G) it is convenient to define Ka = −T π(Ra ); then
[Ka , Kb ] = fab c Kc .
(5.4.1)
These vectorfields generate the left action of G on G/H, in the sense that their flow consists of G-actions.
Example 5.4.1. Let U (1) ≡ S 1 be the subgroup of SU (2) ≡ S 3 consisting of matrices of the
form U (0, 0, Ψ) with 0 < Ψ ≤ 4π. Since U (Θ, Φ, Ψ)U (0, 0, α) = U (Θ, Φ, Ψ + α), the space of
right cosets SU (2)/U (1) can be identified with S 2 and the natural projection h : SU (2) → S 2
maps U (Θ, Φ, Ψ) to the point with spherical coordinates Θ, Φ. This is called the Hopf bundle.
It is a principal U (1) bundle with totalspace S 3 ≡ SU (2), base space S 2 and projection h. The
Euler angles are coordinates in S 3 adapted to the bundle structure, in the sense that Θ and Φ are
coordinates in the base space S 2 and Ψ is a coordinate in the typical fiber S 1 .
Example 5.4.2. In the special case of the two–dimensional sphere one can obtain the generators
of SU (2) by projecting the invariant vectorfields given in example 5.1.2. Since the Euler coordinates
are adapted to the Hopf fibration, it is enough to delete the components in the direction ψ, which
7
are tangent to the fibers:
∂
∂
+ cot Θ cos Φ
,
∂Θ
∂Φ
∂
∂
K2 = − cos Φ
+ cot Θ sin Φ
,
∂Θ
∂Φ
∂
.
K3 = −
∂Φ
K1 = sin Φ
8
(5.4.2.1a)
(5.4.2.1b)
(5.4.2.1c)