2Lie groups

III Symmetries, Fields and Particles

2.1 Definitions

A Lie group is a group and a manifold, where the group operations define smooth

maps. We already know what a group is, so let’s go into a bit more details about

what a manifold is.

The canonical example of a manifold one might want to keep in mind is the

sphere

S

2

. Having lived on Earth, we all know that near any point on Earth,

S

2

looks like

R

2

. But we also know that

S

2

is not

R

2

. We cannot specify a point

on the surface of the Earth uniquely by two numbers. Longitude/latitude might

be a good start, but adding 2

π

to the longitude would give you the same point,

and things also break down at the poles.

Of course, this will not stop us from producing a map of Cambridge. One

can reasonably come up with consistent coordinates just for Cambridge itself.

So what is true about

S

2

is that near each point, we can come up with some

coordinate system, but we cannot do so for the whole space itself.

Definition

(Smooth map)

.

We say a map

f

:

R

n

→ R

m

is smooth if all partial

derivatives of all orders exist.

Definition

(Manifold)

.

A manifold (of dimension

n

) is a set

M

together with

the following data:

(i) A collection U

α

of subsets of M whose union is M;

(ii)

A collection of bijections

ϕ

α

:

U

α

→ V

α

, where

V

α

is an open subset of

R

n

.

These are known as charts.

The charts have to satisfy the following compatibility condition:

–

For all

α, β

, we have

ϕ

α

(

U

α

∩U

β

) is open in

R

n

, and the transition function

ϕ

α

◦ ϕ

−1

β

: ϕ

β

(U

α

∩ U

β

) → ϕ

α

(U

α

∩ U

β

)

is smooth.

We write (M, ϕ

α

) for the manifold if we need to specify the charts explicitly.

U

β

U

α

ϕ

β

ϕ

α

ϕ

α

ϕ

−1

β

Definition

(Smooth map)

.

Let

M

be a manifold. Then a map

f

:

M → R

is smooth if it is smooth in each coordinate chart. Explicitly, for each chart

(U

α

, ϕ

α

), the composition f ◦ ϕ

−1

α

: ϕ

α

(U

α

) → R is smooth (in the usual sense)

p

U

ϕ(p)

f ◦ ϕ

−1

R

f

ϕ

More generally, if

M, N

are manifolds, we say

f

:

M → N

is smooth if for

any chart (

U, ϕ

) of

M

and any chart (

V, ξ

) of

N

, the composition

ξ ◦f ◦ ϕ

−1

:

ϕ(U) → ξ(V ) is smooth.

ϕ

ξ

f

ξ ◦f ◦ ϕ

−1

Finally, we note that if

M, N

are manifolds of dimensions

m

and

n

respectively,

then

M × N

is a manifold of dimension

m

+

n

, where charts (

U, ϕ

:

U →

R

m

)

,

(

V, ξ

:

V → R

n

) of

M

and

N

respectively give us a chart

ϕ × ξ

:

U × V →

R

m

× R

n

= R

m+n

of M × N .

All those definitions were made so that we can define what a Lie group is:

Definition

(Lie group)

.

A Lie group is a group

G

whose underlying set is given

a manifold structure, and such that the multiplication map

m

:

G ×G → G

and

inverse map

i

:

G → G

are smooth maps. We sometimes write

M

(

G

) for the

underlying manifold of G.

Example. The unit 2-sphere

S

2

= {(x, y, z) ∈ R

3

: x

2

+ y

2

+ z

2

= 1}

is a manifold. Indeed, we can construct a coordinate patch near

N

= (0

,

0

,

1).

Near this point, we have

z

=

p

1 − x

2

− y

2

. This works, since near the north pole,

the

z

-coordinate is always positive. In this case,

x

and

y

are good coordinates

near the north pole.

However, it is a fact that the 2-sphere S

2

has no Lie group structure.

In general, most of our manifolds will be given by subsets of Euclidean space

specified by certain equations, but note that not all subsets given by equations

are manifolds! It is possible that they have some singularities.

Definition

(Dimension of Lie group)

.

The dimension of a Lie group

G

is the

dimension of the underlying manifold.

Definition

(Subgroup)

.

A subgroup

H

of

G

is a subset of

G

that is also a group

under the same operations. We write H ≤ G if H is a subgroup of G.

The really interesting thing is when the subgroup is also a manifold!

Definition

(Lie subgroup)

.

A subgroup is a Lie subgroup if it is also a manifold

(under the induced smooth structure).

We now look at some examples.

Example.

Let

G

= (

R

D

,

+) be the

D

-dimensional Euclidean space with addition

as the group operation. The inverse of a vector

x

is

−x

, and the identity is

0

.

This is obviously locally homeomorphic to

R

D

, since it is

R

D

, and addition and

negation are obviously smooth.

This is a rather boring example, since

R

D

is a rather trivial manifold, and

the operation is commutative. The largest source of examples we have will be

matrix Lie groups.