invariant differential form

invariant differential form

1

Lie Groups

Let

G

be a Lie group and

g

∈

G

.

Let

L

g

:

G

⟶

G

and

R

g

:

G

⟶

G

be the

functions

of left and right

multiplication

by

g

(respectively). Let

C

g

:

G

⟶

G

be the function of

conjugation

by

g

, i.e.

C

g

⁢

(

h

)

:=

g

⁢

h

⁢

g

-

1

.

A differential

k

-form (

http://planetmath.org/DifferentialForms

)

ω

on

G

is said to be

•

left

invariant

if

L

g

*

⁢

ω

=

ω

for every

g

∈

G

, where

L

g

*

is the

pullback

induced

L

g

.

•

right invariant

if

R

g

*

⁢

ω

=

ω

for every

g

∈

G

, where

R

g

*

is the pullback induced

R

g

.

•

invariant

or

bi-invariant

if it is both left invariant and right invariant.

•

adjoint

invariant

if

C

g

*

⁢

ω

=

ω

for every

g

∈

G

, where

C

g

*

is the pullback induced by

C

g

.

Much like left invariant vector fields (

http://planetmath.org/LieGroup

),

left invariant forms

are uniquely determined by their values in

T

e

⁢

(

G

)

, the tangent space at the

identity element

e

∈

G

, i.e. a left invairant form

ω

is uniquely determined by the values

w

e

⁢

(

X

1

,

…

,

X

k

)

,

X

1

,

…

,

X

k

∈

T

e

⁢

(

G

)

This means that left invariant forms are uniquely determined by their values on the Lie algebra of

G

.

Under this setting, the space

Ω

L

k

⁢

(

G

)

of left invariant

k

-forms can be identified with

H

⁢

o

⁢

m

⁢

(

Λ

k

⁢

𝔤

,

ℝ

)

, the space of

homomorphisms

from the

k

-th

exterior power

of

𝔤

to

ℝ

, where

𝔤

denotes the Lie algebra of

G

.

-

Let

Ω

k

⁢

(

G

)

be the space of

k

-forms in

G

. The

exterior derivative

d

:

Ω

k

⁢

(

G

)

⟶

Ω

k

+

1

⁢

(

G

)

takes left invariant forms to left invariant forms. Moreover, the

formula

for exterior derivative for left invariant forms simplifies to

d

⁢

ω

⁢

(

X

0

,

…

,

X

k

)

=

∑

i

<

j

(

-

1

)

i

+

j

⁢

ω

⁢

(

[

X

i

,

X

j

]

,

X

0

,

…

,

X

i

^

,

…

,

X

j

^

,

…

,

X

k

)

where

ω

∈

Ω

k

⁢

(

G

)

and

X

0

,

…

,

X

k

are left invariant vector fields in

G

.

Hence, the exterior derivative induces a map

d

:

Ω

L

k

⁢

(

G

)

⟶

Ω

L

k

+

1

⁢

(

G

)

and

(

Ω

L

*

⁢

(

G

)

,

d

)

forms a

chain complex

. Thus, we can talk about the

cohomology groups

of left invariant forms.

Similar

results hold for right invariant forms.

2

Manifolds

Suppose a Lie group

G

acts smoothly (

http://planetmath.org/

Manifold

) on a differential manifold

M

and let

(

g

,

x

)

⟼

t

g

⁢

(

x

)

,

g

∈

G

,

x

∈

M

denote the action of

G

.

A differential

k

-form

ω

in

M

is said to be

invariant

if

t

g

*

⁢

ω

=

ω

for every

g

∈

G

, where

t

g

*

denotes the pullback induced by

t

g

.

This definition reduces to the previous ones when we take

M

as the group

G

itself and when the action is

•

the action of

G

on itself by left multiplication.

•

the action of

G

on itself by right multiplication.

•

the action of

G

×

G

on

G

defined by

t

(

g

,

h

)

⁢

(

k

)

:=

g

⁢

k

⁢

h

-

1

.

•

the action of

G

on itself by conjugation.

3

Compact Lie Group Actions

We now consider actions of a

compact Lie group

G

on a manifold

M

. Let

Ω

k

⁢

(

M

)

the space of

k

-forms in

M

and

Ω

G

k

⁢

(

M

)

the space of invariant

k

-forms in

M

. Let

μ

be the Haar measure of

G

.

From each

k

-form in

M

we can

construct

an invariant form by taking on its ””. Following this idea we define a map

J

:

Ω

k

⁢

(

M

)

⟶

Ω

G

k

⁢

(

M

)

by

J

⁢

(

ω

)

⁢

(

X

1

,

…

,

X

k

)

:=

1

μ

⁢

(

G

)

⁢

∫

G

t

g

*

⁢

ω

⁢

(

X

1

,

…

,

X

k

)

⁢

𝑑

μ

⁢

(

g

)

where

ω

∈

Ω

k

⁢

(

M

)

and

X

1

,

…

,

X

k

are vector fields of

M

.

The image of the map

J

is indeed in

Ω

G

k

⁢

(

M

)

since for every

h

∈

G

:

t

h

⁢

(

J

⁢

(

ω

)

)

⁢

(

X

1

,

…

,

X

k

)

=

J

⁢

(

ω

)

⁢

(

(

t

h

)

*

⁢

X

1

,

…

,

(

t

h

)

*

⁢

X

k

)

=

1

μ

⁢

(

G

)

⁢

∫

G

ω

⁢

(

(

t

g

)

*

⁢

(

t

h

)

*

⁢

X

1

,

…

,

(

t

g

)

*

⁢

(

t

h

)

*

⁢

X

k

)

⁢

𝑑

μ

⁢

(

g

)

=

1

μ

⁢

(

G

)

⁢

∫

G

ω

⁢

(

(

t

g

⁢

h

)

*

⁢

X

1

,

…

,

(

t

g

⁢

h

)

*

⁢

X

k

)

⁢

𝑑

μ

⁢

(

g

)

=

1

μ

⁢

(

G

)

⁢

∫

G

ω

⁢

(

(

t

g

)

*

⁢

X

1

,

…

,

(

t

g

)

*

⁢

X

k

)

⁢

𝑑

μ

⁢

(

g

)

=

J

⁢

(

ω

)

⁢

(

X

1

,

…

,

X

k

)

Moreover,

J

is the

identity

for invariant

k

-forms. Suppose

ω

∈

Ω

G

k

⁢

(

M

)

, then

J

⁢

(

ω

)

⁢

(

X

1

,

…

,

X

k

)

=

1

μ

⁢

(

G

)

⁢

∫

G

t

g

*

⁢

(

ω

)

⁢

(

X

1

,

…

,

X

k

)

⁢

𝑑

μ

⁢

(

g

)

=

1

μ

⁢

(

G

)

⁢

∫

G

ω

⁢

(

X

1

,

…

,

X

k

)

⁢

𝑑

μ

⁢

(

g

)

=

ω

⁢

(

X

1

,

…

,

X

k

)

Theorem

-

The map

J

is a

chain map

, i.e.

d

⁢

J

=

J

⁢

d

, where

d

is the exterior derivative of a form.

From the previous observations we can see that the exterior derivative takes invariant forms to invariant forms, inducing a map

d

:

Ω

G

k

⁢

(

M

)

⟶

Ω

G

k

+

1

⁢

(

M

)

. Hence,

(

Ω

G

*

⁢

(

M

)

,

d

)

is a chain complex and we can talk about the cohomology groups of invariant forms in

M

.

4

Cohomology of Manifolds

Let

G

be a compact Lie group that acts smoothly on a manifold

M

(again, with the action denoted by

t

g

).

Since

t

g

is a

diffeomorphism

of

M

it induces an automorphism

t

g

*

on the cohomology groups

H

k

⁢

(

M

;

ℝ

)

. Hence,

G

acts as a group of automorphisms on

H

k

⁢

(

M

;

ℝ

)

. Let

H

k

⁢

(

M

;

ℝ

)

G

be the

fixed point

set of this action.

Theorem -

The

inclusion

I

:

Ω

G

k

⁢

(

M

)

⟶

Ω

k

⁢

(

M

)

induces an

isomorphism

\xymatrix

⁢

I

*

:

H

k

⁢

(

Ω

G

⁢

(

M

)

)

⁢

\ar

⁢

[

r

]

≃

⁢

&

⁢

H

k

⁢

(

M

;

ℝ

)

G

If in

G

is connected, then

t

g

and the identity

1

M

are

homotopic

,

t

g

≃

1

M

, for every

g

∈

G

. This implies that the induced automorphisms are the same, i.e.

t

g

*

=

I

⁢

d

, where

I

⁢

d

is the identity on

H

k

⁢

(

M

;

ℝ

)

. Hence, the fixed point set is the whole

H

k

⁢

(

M

;

ℝ

)

and there is an isomorphism

\xymatrix

⁢

I

*

:

H

k

⁢

(

Ω

G

⁢

(

M

)

)

⁢

\ar

⁢

[

r

]

≃

⁢

&

⁢

H

k

⁢

(

M

;

ℝ

)

Thus, the cohomology groups of a manifold where a compact connected Lie group acts are just the cohomology groups defined by the invariant forms on

M

. This means we can ”forget” the whole of

differential forms

in

M

and regard only those who are invariant.

Title

invariant differential form

Canonical name

InvariantDifferentialForm

Date of creation

2013-03-22 17:48:31

Last modified on

2013-03-22 17:48:31

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

25

Author

asteroid (17536)

Entry type

Definition

Classification

msc 58A10

Classification

msc 57T10

Classification

msc 57S15

Classification

msc 22E30

Classification

msc 22E15

Synonym

invariant form

Synonym

bi-invariant form

Synonym

bi-invariant differential form

Related topic

CohomologyOfCompactConnectedLieGroups

Defines

left invariant differential form

Defines

left invariant form

Defines

right invariant differential form

Defines

right invariant form

Defines

adjoint invariant form

Defines

adjoint invariant differential form

Defines

chain complex of invariant forms

Defines

cohomology

of manifolds with a Lie group action

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