topological group representation

topological group representation

1

Finite Dimensional Representations

Let

G

be a

topological group

and

V

a

finite-dimensional

normed vector space

. We denote by

G

⁢

L

⁢

(

V

)

the

general linear group

of

V

, endowed with the

topology

coming from the

operator norm

.

Regarding only the group

structure

of

G

, recall that a

representation

of

G

in

V

is a

group homomorphism

π

:

G

⟶

G

⁢

L

⁢

(

V

)

.

Definition -

A

representation

of the topological group

G

in

V

is a

continuous group

homomorphism

π

:

G

⟶

G

⁢

L

⁢

(

V

)

, i.e. is a continuous representation of the

abstract group

G

in

V

.

We have the following

equivalent

definitions:

•

A representation of

G

in

V

is a group homomorphism

π

:

G

⟶

G

⁢

L

⁢

(

V

)

such that the

mapping

G

×

V

⟶

V

defined by

(

g

,

v

)

↦

π

⁢

(

g

)

⁢

v

is

continuous

.

•

A representation of

G

in

V

is a group homomorphism

π

:

G

⟶

G

⁢

L

⁢

(

V

)

such that, for every

v

∈

V

, the mapping

G

⟶

V

defined by

g

↦

π

⁢

(

g

)

⁢

v

is continuous.

2

Representations in Hilbert Spaces

Let

G

be a topological group and

H

a

Hilbert space

. We denote by

B

⁢

(

H

)

the

algebra

of

bounded operators

endowed with the

strong operator topology

(this topology does not coincide with the norm topology unless

H

is finite-dimensional). Let

𝒢

⁢

(

H

)

the set of

invertible

operators

in

B

⁢

(

H

)

endowed with the

subspace topology

.

Definition -

A

representation

of the topological group

G

in

H

is a continuous group homomorphism

π

:

G

⟶

𝒢

⁢

(

H

)

, i.e. is a continuous representation of the abstract group

G

in

H

.

We denote by

r

⁢

e

⁢

p

⁢

(

G

,

H

)

the set of all representations of

G

in the Hilbert space

H

.

We have the following equivalent definitions:

•

A representation of

G

in

H

is a group homomorphism

π

:

G

⟶

𝒢

⁢

(

H

)

such that the mapping

G

×

H

⟶

H

defined by

(

g

,

v

)

↦

π

⁢

(

g

)

⁢

v

is continuous.

•

A representation of

G

in

H

is a group homomorphism

π

:

G

⟶

𝒢

⁢

(

H

)

such that, for every

v

∈

H

, the mapping

G

⟶

H

defined by

g

↦

π

⁢

(

g

)

⁢

v

is continuous.

Remark -

The 3rd definition is exactly the same as the 1st definition, just written in other .

3

Representations as G-modules

Recall that, for an abstract group

G

, it is the same to consider a representation of

G

or to consider a

G

-module (

http://planetmath.org/GModule

), i.e. to each representation of

G

corresponds a

G

-module and vice-versa.

For a topological group

G

, representations of

G

satisfy

some

continuity

. Thus, we are not interested in all

G

-modules, but rather in those which are

compatible

with the continuity conditions.

Definition -

Let

G

be a topological group. A

G

-module

is a normed vector space (or a Hilbert space)

V

where

G

acts continuously, i.e. there is a continuous action

ψ

:

G

×

V

⟶

V

.

To give a representation of a topological group

G

is the same as giving a

G

-module (in the sense described above).

4

Special Kinds of Representations

•

•

Let

π

∈

r

⁢

e

⁢

p

⁢

(

G

,

H

)

. We say that a

subspace

V

⊆

H

is by

π

if

V

is

invariant

under every operator

π

⁢

(

s

)

with

s

∈

G

.

•

•

A of a representation

π

∈

r

⁢

e

⁢

p

⁢

(

G

,

H

)

is a representation

π

0

∈

r

⁢

e

⁢

p

⁢

(

G

,

H

0

)

obtained from

π

by restricting to a closed subspace

H

0

⊆

H

.

•

•

A representation

π

∈

r

⁢

e

⁢

p

⁢

(

G

,

H

)

is said to be if the only closed subspaces of

H

are the

trivial

ones,

{

0

}

and

H

.

•

Two representations

π

1

:

G

⟶

G

⁢

L

⁢

(

V

1

)

and

π

2

:

G

⟶

G

⁢

L

⁢

(

V

2

)

of a topological group

G

are said to be

equivalent

if there exists an invertible linear transformation

T

:

V

1

⟶

V

2

such that for every

g

∈

G

one has

π

1

⁢

(

g

)

=

T

-

1

⁢

π

2

⁢

(

g

)

⁢

T

.

The definition is

similar

for Hilbert spaces, by taking

T

as an invertible

bounded linear operator

.

Title

topological group representation

Canonical name

TopologicalGroupRepresentation

Date of creation

2013-03-22 18:02:18

Last modified on

2013-03-22 18:02:18

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

8

Author

asteroid (17536)

Entry type

Definition

Classification

msc 43A65

Classification

msc 22A25

Classification

msc 22A05

Synonym

representation of topological groups

Defines

equivalent representations of topological groups

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