homogeneous space

homogeneous space

Overview and definition.

Let

G

be a group acting transitively on a set

X

. In other words,
we consider a

homomorphism

ϕ

:

G

→

Perm

⁡

(

X

)

,

where the latter
denotes the group of all

bijections

of

X

. If we consider

G

as
being, in some sense, the

automorphisms

of

X

, the

transitivity

assumption

means that it is impossible to distinguish a particular
element of

X

from any another

element

. Since the elements of

X

are

indistinguishable

, we call

X

a

homogeneous space

.
Indeed, the

concept

of a homogeneous space, is

logically equivalent

to
the concept of a

transitive group action

.

Action on cosets.

Let

G

be a group,

H

<

G

a

subgroup

, and let

G

/

H

denote the set of

left cosets

, as above. For every

g

∈

G

we consider the

mapping

ψ

H

⁢

(

g

)

:

G

/

H

→

G

/

H

with action

a

⁢

H

→

g

⁢

a

⁢

H

,

a

∈

G

.

Proposition 1

The mapping

ψ

H

⁢

(

g

)

is a bijection. The
corresponding mapping

ψ

H

:

G

→

Perm

⁡

(

G

/

H

)

is a

group homomorphism

,
specifying a transitive group action of

G

on

G

/

H

.

Thus,

G

/

H

has the natural

structure

of a homogeneous space. Indeed,
we shall see that every homogeneous space

X

is

isomorphic

to

G

/

H

,
for some subgroup

H

.

N.B. In geometric applications, the want the homogeneous space

X

to
have some extra structure, like a

topology

or a differential
structure. Correspondingly, the group of automorphisms is either a

continuous group

or a Lie group. In order for the

quotient space

X

to have a

Hausdorff topology

, we need to assume that the subgroup

H

is closed in

G

.

The isotropy subgroup and the basepoint identification.

Let

X

be a homogeneous space. For

x

∈

X

, the subgroup

H

x

=

{

h

∈

G

:

h

⁢

x

=

x

}

,

consisting of all

G

-actions that fix

x

, is called the

isotropy subgroup

at the

basepoint

x

. We
identify the space of cosets

G

/

H

x

with the homogeneous space by
means of the mapping

τ

x

:

G

/

H

x

→

X

, defined by

τ

x

⁢

(

a

⁢

H

x

)

=

a

⁢

x

,

a

∈

G

.

Proposition 2

The above mapping is a well-defined bijection.

To show that

τ

x

is well defined, let

a

,

b

∈

G

be members of
the same left coset, i.e. there exists an

h

∈

H

x

such that

b

=

a

⁢

h

. Consequently

b

⁢

x

=

a

⁢

(

h

⁢

x

)

=

a

⁢

x

,

as desired. The mapping

τ

x

is onto because the action of

G

on

X

is
assumed to be

transitive

. To show that

τ

x

is

one-to-one

, consider
two cosets

a

⁢

H

x

,

b

⁢

H

x

,

a

,

b

∈

G

such that

a

⁢

x

=

b

⁢

x

. It follows that

b

-

1

⁢

a

fixes

x

, and hence is an
element of

H

x

. Therefore

a

⁢

H

x

and

b

⁢

H

x

are the same coset.

The homogeneous space as a quotient.

Next, let us show
that

τ

x

is

equivariant

relative to the action of

G

on

X

and the action of

G

on the

quotient

G

/

H

x

.

Proposition 3

We have that

ϕ

⁢

(

g

)

∘

τ

x

=

τ

x

∘

ψ

H

x

⁢

(

g

)

for all

g

∈

G

.

To prove this, let

g

,

a

∈

G

be given, and note that

ψ

H

x

⁢

(

g

)

⁢

(

a

⁢

H

x

)

=

g

⁢

a

⁢

H

x

.

The latter coset corresponds under

τ

x

to the
point

g

⁢

a

⁢

x

, as desired.

Finally, let us note that

τ

x

identifies the point

x

∈

X

with
the coset of the

identity element

e

⁢

H

x

, that is to say, with the
subgroup

H

x

itself. For this reason, the point

x

is often called
the basepoint of the identification

τ

x

:

G

/

H

x

→

X

.

The choice of basepoint.

Next, we consider the effect of the choice of basepoint on the

quotient structure

of a homogeneous space. Let

X

be a homogeneous
space.

Proposition 4

The set of all isotropy subgroups

{

H

x

:

x

∈

X

}

forms a
single

conjugacy class

of subgroups in

G

.

To show this, let

x

0

,

x

1

∈

X

be given. By the transitivity of the action we may choose a

g

^

∈

G

such that

x

1

=

g

^

⁢

x

0

. Hence, for all

h

∈

G

satisfying

h

⁢

x

0

=

x

0

, we have

(

g

^

⁢

h

⁢

g

^

-

1

)

⁢

x

1

=

g

^

⁢

(

h

⁢

(

g

^

-

1

⁢

x

1

)

)

=

g

^

⁢

x

0

=

x

1

.

Similarly, for all

h

∈

H

x

1

we have that

g

^

-

1

⁢

h

⁢

g

^

fixes

x

0

.
Therefore,

g

^

⁢

(

H

x

0

)

⁢

g

^

-

1

=

H

x

1

;

or what is

equivalent

, for all

x

∈

X

and

g

∈

G

we have

g

⁢

H

x

⁢

g

-

1

=

H

g

⁢

x

.

Equivariance.

Since we can identify a homogeneous space

X

with

G

/

H

x

for every
possible

x

∈

X

, it stands to reason that there exist equivariant
bijections between the different

G

/

H

x

. To describe these, let

H

0

,

H

1

<

G

be

conjugate subgroups

with

H

1

=

g

^

⁢

H

0

⁢

g

^

-

1

for some fixed

g

^

∈

G

. Let us set

X

=

G

/

H

0

,

and let

x

0

denote the

identity

coset

H

0

, and

x

1

the coset

g

^

⁢

H

0

. What is the subgroup of

G

that fixes

x

1

?
In other words, what are all the

h

∈

G

such that

h

⁢

g

^

⁢

H

0

=

g

^

⁢

H

0

,

or what is equivalent, all

h

∈

G

such that

g

^

-

1

⁢

h

⁢

g

^

∈

H

0

.

The

collection

of all such

h

is
precisely the subgroup

H

1

. Hence,

τ

x

1

:

G

/

H

1

→

G

/

H

0

is
the desired equivariant bijection. This is a well defined mapping
from the set of

H

1

-cosets to the set of

H

0

-cosets, with action
given by

τ

x

1

⁢

(

a

⁢

H

1

)

=

a

⁢

g

^

⁢

H

0

,

a

∈

G

.

Let

ψ

0

:

G

→

Perm

⁡

(

G

/

H

0

)

and

ψ

1

:

G

→

Perm

⁡

(

G

/

H

1

)

denote the
corresponding coset

G

-actions.

Proposition 5

For all

g

∈

G

we have that

τ

x

1

∘

ψ

1

⁢

(

g

)

=

ψ

0

⁢

(

g

)

∘

τ

x

1

.

Title

homogeneous space

Canonical name

HomogeneousSpace

Date of creation

2013-03-22 13:28:07

Last modified on

2013-03-22 13:28:07

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

6

Author

rmilson (146)

Entry type

Definition

Classification

msc 20A05

Defines

action on cosets

Defines

isotropy subgroup

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