You can check your answers by comparing your results

(You can check your answers by comparing your results with Example
15.3 on page 410 in the following chapter.)

15
Homogeneous Spaces
In this section we will explore the concept of symmetries even further. We in-
troduced some of the basics in chapter 6, and we will pursue the ideas further
here. In doing so, we will generalise the FRW models to the Bianchi models
which are in general homogeneous but not necessarily isotropic.
15.1 Lie groups and Lie algebras
First we will introduce some very important concepts used in mathematics
and physics. Whenever we talk about continuous symmetries, the words Lie
groups and Lie algebra are usually mentioned.
We saw earlier that the Killing vectors generate a special class of diffeo-
morphisms; Killing vectors generate isometries . The isometries of a space form
a group. For instance, let us take the sphere, S 2 , with the usual round met-
ric. The isometry group of the sphere is all the rotations in three-dimensional
space that leaves the sphere invariant. These (orientation-preserving 1 ) rota-
tions form the group SO (3) . What is so special about this group, is that the
group itself, can be considered as a manifold! Since the dimension of the
group is three, the group SO (3) can be considered as a three-dimensional
manifold. The group SO (3) is an example of a Lie group . We define Lie groups
as follows.
Definition: Lie Group. A Lie group, G , is a topological space that has the
following properties:
1. G is a manifold.
2. The group multiplication m : G × G mapsto−→ G is smooth.
3. Inversion i : G mapsto−→ G is smooth.
1 We will always assume that we are talking about orientation-preserving isometries, unless
stated otherwise.

402 Homogeneous Spaces
To show that SO (3) has these properties is not difficult. We already know
that multiplication and inversion are continuous operations. Each element in
SO (3) corresponds to a rotation, and rotations are continuous operations. We
can show that SO (3) is actually equal to the manifold P
3
and hence, SO (3) is
a manifold.
Let us also define what we mean with Lie algebra .
Definition: Lie Algebra. A real (or complex) Lie algebra, g , is a (finite di-
mensional) vector space equipped with a bilinear map [ − , − ] : g × g mapsto−→ g
which satisfies the following properties:
1. [ X , X ] = 0 for all X ∈ g .
2. Jacobi’s identity:
[ X , [ Y , Z ]] + [ Y , [ Z , X ]] + [ Z , [ X , Y ]] = 0 (15.1)
for all X , Y , Z ∈ g .
Note that 1. implies that the bilinear map [ − , − ] is skew-symmetric:
[ X , Y ] = − [ Y , X ] (15.2)
An example of a Lie algebra is the space of all n × n matrices gl ( n ) . The bracket
[ − , − ] is in this case simply defined by
[ A , B ] = AB − BA (15.3)
for all matrices A and B . The bracket is in this case the usual commutator
multiplication of matrices.
There is actually a deep connection between these two concepts. A Lie
algebra is a vector space, while a Lie group is a group manifold. Amazingly
we have the following theorem.
Theorem: Let G be a Lie group. The tangent space of G at the identity
element, T
e
G , is a Lie algebra. Hence,
g = T
e
G.