semilinear transformation

semilinear transformation

Let

K

be a field and

k

its

prime subfield

. For example, if

K

is

ℂ

then

k

is

ℚ

, and if

K

is the

finite field

of order

q

=

p

i

, then

k

is

ℤ

p

.

Definition 1

.

Given a

field automorphism

θ

of

K

, a function

f

:

V

→

W

between two

K

vector spaces

V

and

W

is

θ

-semilinear, or simply semilinear, if for all

x

,

y

∈

V

and

l

∈

K

it follows: (shown here first in left hand notation and then in the preferred right hand notation.)

1.

f

⁢

(

x

+

y

)

=

f

⁢

(

x

)

+

f

⁢

(

y

)

, (in right hand notation:

(

x

+

y

)

⁢

f

=

x

⁢

f

+

y

⁢

f

.)

2.

f

⁢

(

l

⁢

x

)

=

l

θ

⁢

f

⁢

(

x

)

, (in right hand notation:

(

l

⁢

x

)

⁢

f

=

l

θ

⁢

x

⁢

f

.)

where

l

θ

denotes the image of

l

under

θ

.

Remark 2

.

θ

must be a field automorphism for

f

to remain additive, for example,

θ

must fix the prime subfield as

n

θ

⁢

x

⁢

f

=

(

n

⁢

x

)

⁢

f

=

(

x

+

⋯

+

x

)

⁢

f

=

n

⁢

(

x

⁢

f

)

.

Also

(

l

1

+

l

2

)

θ

⁢

x

⁢

f

=

(

(

l

1

+

l

2

)

⁢

x

)

⁢

f

=

(

l

1

⁢

x

)

⁢

f

+

(

l

2

⁢

x

)

⁢

f

=

(

l

1

θ

+

l

2

θ

)

⁢

x

⁢

f

so

(

l

1

+

l

2

)

θ

=

l

1

θ

+

l

2

θ

. Finally,

(

l

1

l

2

)

θ

x

f

=

(

(

l

1

l

2

x

)

f

=

l

1

θ

(

l

2

x

)

f

=

l

1

θ

l

2

θ

x

f

.

Every

linear transformation

is semilinear, but the converse is generally not true. If we treat

V

and

W

as vector spaces over

k

, (by considering

K

as vector space over

k

first) then every

θ

-

semilinear map

is a

k

-linear map, where

k

is the prime subfield of

K

.

Example

•

Let

K

=

ℂ

,

V

=

ℂ

n

with

standard basis

e

1

,

…

,

e

n

. Define the map

f

:

V

→

V

by

f

⁢

(

∑

i

=

1

z

i

⁢

e

i

)

=

∑

i

=

1

n

z

¯

i

⁢

e

i

.

f

is semilinear (with respect to the

complex conjugation

field automorphism)
but not linear.

•

Let

K

=

G

⁢

F

⁢

(

q

)

– the Galois field of order

q

=

p

i

,

p

the

characteristic

.
Let

l

θ

=

l

p

, for

l

∈

K

. By the Freshman’s dream it is known that this is a field automorphism. To every linear map

f

:

V

→

W

between
vector spaces

V

and

W

over

K

we can establish a

θ

-semilinear map

(

∑

i

=

1

l

i

⁢

e

i

)

⁢

f

~

=

∑

i

=

1

n

l

i

θ

⁢

e

i

⁢

f

.

□

Indeed every linear map can be converted into a semilinear map in such a way.
This is part of a general observation collected into the following result.

Definition 3

.

Given a vector space

V

, the set of all

invertible

semilinear maps (over all field automorphisms) is the group

Γ

⁢

L

⁢

(

V

)

.

Proposition 4

.

Given a vector space

V

over

K

, and

k

the prime subfield of

K

, then

Γ

⁢

L

⁢

(

V

)

decomposes as the

semidirect product

Γ

⁢

L

⁢

(

V

)

=

G

⁢

L

⁢

(

V

)

⋊

G

⁢

a

⁢

l

⁢

(

K

/

k

)

where

G

⁢

a

⁢

l

⁢

(

K

/

k

)

is the

Galois group

of

K

/

k

.

Remark 5

.

We identify

G

⁢

a

⁢

l

⁢

(

K

/

k

)

with a

subgroup

of

Γ

⁢

L

⁢

(

V

)

by fixing a basis

B

for

V

and defining the semilinear maps:

∑

b

∈

B

l

b

⁢

b

↦

∑

b

∈

B

l

b

σ

⁢

b

for any

σ

∈

G

⁢

a

⁢

l

⁢

(

K

/

k

)

. We shall denoted this subgroup by

G

⁢

a

⁢

l

⁢

(

K

/

k

)

B

. We also see these

complements

to

G

⁢

L

⁢

(

V

)

in

Γ

⁢

L

⁢

(

V

)

are acted on regularly by

G

⁢

L

⁢

(

V

)

as they correspond to a

change of basis

.

Proof.

Every linear map is semilinear thus

G

⁢

L

⁢

(

V

)

≤

Γ

⁢

L

⁢

(

V

)

. Fix a basis

B

of

V

. Now given any semilinear map

f

with respect to a field automorphism

σ

∈

G

⁢

a

⁢

l

⁢

(

K

/

k

)

, then define

g

:

V

→

V

by

(

∑

b

∈

B

l

b

⁢

b

)

⁢

g

=

∑

b

∈

B

(

l

b

σ

-

1

⁢

b

)

⁢

f

=

∑

b

∈

B

l

b

⁢

(

b

)

⁢

f

.

As

(

B

)

⁢

f

is also a basis of

V

, it follows

g

is simply a basis exchange
of

V

and so linear and invertible:

g

∈

G

⁢

L

⁢

(

V

)

.

Set

h

:=

g

-

1

⁢

f

. For every

v

=

∑

b

∈

B

l

b

≠

0

in

V

,

v

⁢

h

=

v

⁢

g

-

1

⁢

f

=

∑

b

∈

B

l

b

σ

⁢

b

thus

h

is in the

G

⁢

a

⁢

l

⁢

(

K

/

k

)

subgroup relative to the fixed basis

B

.
This factorization is unique to the fixed basis

B

. Furthermore,

G

⁢

L

⁢

(

V

)

is normalized by the action of

G

⁢

a

⁢

l

⁢

(

K

/

k

)

B

, so

Γ

⁢

L

⁢

(

V

)

=

G

⁢

L

⁢

(

V

)

⋊

G

⁢

a

⁢

l

⁢

(

K

/

k

)

.
∎

The

Γ

⁢

L

⁢

(

V

)

groups extend the typical

classical groups

in

G

⁢

L

⁢

(

V

)

. The importance in considering such maps follows from the consideration of

projective geometry

.

The projective geometry of a vector space

V

, denoted

P

⁢

G

⁢

(

V

)

, is the lattice of all

subspaces

of

V

. Although the typical semilinear map is not a linear map, it does follow that every semilinear map

f

:

V

→

W

induces an order-preserving map

f

:

P

⁢

G

⁢

(

V

)

→

P

⁢

G

⁢

(

W

)

. That is, every semilinear map induces a

projectivity

. The converse of this observation is the

Fundamental Theorem of Projective Geometry

. Thus semilinear maps are useful because they define the

automorphism group

of the projective geometry of a vector space.

References

1

Gruenberg, K. W. and Weir, A.J.

Linear

Geometry

2nd Ed.

(English)
[B] Graduate Texts in Mathematics. 49. New York - Heidelberg - Berlin: Springer-Verlag. X, 198 p. DM 29.10; $ 12.80 (1977).

Title

semilinear transformation

Canonical name

SemilinearTransformation

Date of creation

2013-03-22 15:51:06

Last modified on

2013-03-22 15:51:06

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

20

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 15A04

Synonym

semilinear map

Synonym

semilinear transform

Synonym

semi-linear transformation

Synonym

semi-linear map

Related topic

ClassicalGroups

Related topic

ProjectiveSpace

Defines

semilinear transform

Defines

Gamma L

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