2Lie groups
III Symmetries, Fields and Particles
2.3 Properties of Lie groups
The first thing we want to consider is when two Lie groups are “the same”. We
take the obvious definition of isomorphism.
Definition
(Homomorphism of Lie groups)
.
Let
G, H
be Lie groups. A map
J : G → H is a homomorphism if it is smooth and for all g
1
, g
2
∈ G, we have
J(g
1
g
2
) = J(g
1
)J(g
2
).
(the second condition says it is a homomorphism of groups)
Definition
(Isomorphic Lie groups)
.
An isomorphism of Lie groups is a bijective
homomorphism whose inverse is also a homomorphism. Two Lie groups are
isomorphic if there is an isomorphism between them.
Example. We define the map J : U(1) → SO(2) by
J(e
iθ
) 7→
cos θ −sin θ
sin θ cos θ
∈ SO(2).
This is easily seen to be a homomorphism, and we can construct an inverse
similarly.
Exercise. Show that M(SU(2))
∼
=
S
3
.
We now look at some words that describe manifolds. Usually, manifolds that
satisfy these properties are considered nice.
The first notion is the idea of compactness. The actual definition is a bit
weird and takes time to get used to, but there is an equivalent characterization
if the manifold is a subset of R
n
.
Definition
(Compact)
.
A manifold (or topological space)
X
is compact if every
open cover of X has a finite subcover.
If the manifold is a subspace of some
R
n
, then it is compact iff it is closed
and bounded.
Example.
The sphere
S
2
is compact, but the hyperboloid given by
x
2
−y
2
−z
2
=
1 (as a subset of R
3
) is not.
Example.
The orthogonal groups are compact. Recall that the definition of
an orthogonal matrix requires
M
T
M
=
I
. Since this is given by a polynomial
equation in the entries, it is closed. It is also bounded since each entry of
M
has
magnitude at most 1.
Similarly, the special orthogonal groups are compact.
Example.
Sometimes we want to study more exciting spaces such as Minkowski
spaces. Let
n
=
p
+
q
, and consider the matrices that preserve the metric on
R
n
of signature (p, q), namely
O(p, q) = {M ∈ GL(n, R) : M
T
ηM = η},
where
η =
I
p
0
0 −I
q
.
For
p, q
both non-zero, this group is non-compact. For example, if we take
SO(1, 1), then the matrices are all of the form
M =
cosh θ sinh θ
sinh θ cosh θ
,
where θ ∈ R. So this space is homeomorphic to R, which is not compact.
Another common example of a non-compact group is the Lorentz group
O(3, 1).
Another important property is simply-connectedness.
Definition
(Simply connected)
.
A manifold
M
is simply connected if it is
connected (there is a path between any two points), and every loop
l
:
S
1
→ M
can be contracted to a point. Equivalently, any two paths between any two
points can be continuously deformed into each other.
Example.
The circle
S
1
is not simply connected, as the loop that goes around
the circle once cannot be continuously deformed to the loop that does nothing
(this is non-trivial to prove).
Example.
The 2-sphere
S
2
is simply connected, but the torus is not.
SO
(3) is
also not simply connected. We can define the map by
l(θ) =
(
θn θ ∈ [0, π)
−(2π − θ)n θ ∈ [π, 2π)
This is indeed a valid path because we identify antipodal points.
The failure of simply-connectedness is measured by the fundamental group.
Definition
(Fundamental group/First homotopy group)
.
Let
M
be a manifold,
and
x
0
∈ M
be a preferred point. We define
π
1
(
M
) to be the equivalence classes
of loops starting and ending at
x
0
, where two loops are considered equivalent if
they can be continuously deformed into each other.
This has a group structure, with the identity given by the “loop” that stays
at x
0
all the time, and composition given by doing one after the other.
Example. π
1
(
S
2
) =
{
0
}
and
π
1
(
T
2
) =
Z ×Z
. We also have
π
1
(
SO
(3)) =
Z/
2
Z
.
We will not prove these results.