central collineations

central collineations

Definitions and general properties

Definition 1

.

A

collineation

of a finite dimensional

projective geometry

is a

central collineation

if there is a

hyperplane

of points fixed by the collineation.

Recall that collineations send any three

collinear points

to three collinear points. Thus
if a collineation fixes more than a hyperplane of points then it in fact fixes all the points
of the

geometry

and so it is the

identity map

. Therefore a central collineation can be
viewed the simplest of the non-identity collineations.

Theorem 2

.

Every collineation of a finite dimensional projective geometry of

dimension

n

>

1

is
a

product

of at most

n

central collineations. In particular, the

automorphism group

of a projective geometry of dimension

n

>

1

is generated by central collineations.

Suppose that a central collineation is not the

identity

. Then the hyperplane of

fixed points

is unique and receives the title of

the axis of the central collineation

. There is
one further important result which justifies the name “central”.

Proposition 3

.

Given a non-identity central collineation

f

, there is a
unique point

C

such that for all other points

P

, it follows that

C

,

P

and

P

⁢

f

are

collinear

.

The point

C

determined by

Proposition

3

is called the

center

of the non-identity central collineation. It is possible for the center to lie on the axis.

Central collineations in coordinates

Suppose we have a projective geometry of dimension

n

>

2

, that is, we exclude now
the case of

projective lines

and planes. The the geometry can be coordinatized through
so that we may regard the projective geometry as the

lattice

of

subspaces

of a

vector space

V

of dimension

n

+

1

over a

division ring

Δ

.
Following the

fundamental theorem of projective geometry

we further know that
every collineation is induced by a

semi-linear transformation

of

V

. So it is possible
to explore central collineations as semi-linear transformations.

Every hyperplane is a kernel of some

linear functional

, so we let

φ

:

V

→

Δ

be a linear functional of

V

with

H

⁢

φ

=

0

. Furthermore, we fix

v

∈

V

so that

v

⁢

φ

=

1

(which implieas also that

v

∉

H

). Hence, for each

u

∈

V

,

u

=

(

u

-

(

u

⁢

φ

)

⁢

v

)

+

(

u

⁢

φ

)

⁢

v

where

u

-

(

u

⁢

φ

)

⁢

v

∈

H

and

(

u

⁢

φ

)

⁢

v

∈

⟨

v

⟩

.

Let

f

∈

GL

Δ

⁡

(

V

)

such that

f

induces a central collineation

f

~

on

P

⁢

G

⁢

(

V

)

with
axis

H

≤

V

. As every

scalar multiple

of

f

induces the same collineation of

P

⁢

G

⁢

(

V

)

, we
may assume that

f

is the identity on

H

. Using the

decomposition

given by

φ

we have

u

⁢

f

=

(

(

u

-

(

u

⁢

φ

)

⁢

v

)

+

(

u

⁢

φ

)

⁢

v

)

⁢

f

=

(

u

-

(

u

⁢

φ

)

⁢

v

)

+

(

u

⁢

φ

)

⁢

v

⁢

f

,

u

∈

V

.

Hence

u

⁢

f

=

u

+

(

u

⁢

φ

)

⁢

v

^

,

v

^

:=

v

⁢

f

-

v

.

Suppose instead that

φ

is any linear functional of

V

. Then select some

v

^

∈

V

such that

v

^

⁢

φ

≠

-

1

. Then

u

⁢

g

:=

u

+

(

u

⁢

φ

)

⁢

v

^

fixes all the points of

ker

⁡

φ

so

g

induces a central collineation.

If we wish to do the same without appealing to linear functionals, we may select a basis

{

v

1

,

…

,

v

n

+

1

}

such that

H

=

⟨

v

1

,

…

,

v

n

⟩

and

v

n

+

1

⁢

φ

=

1

. As

f

is selected to be the identity on

H

we have so far specified

f

by the matrix:

[

1

⋯

0

0

⋮

⋱

0

1

a

1

a

2

⋯

a

n

+

1

]

in the basis

{

v

1

,

…

,

v

n

,

v

n

+

1

}

.

Title

central collineations

Canonical name

CentralCollineations

Date of creation

2013-03-22 16:03:13

Last modified on

2013-03-22 16:03:13

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

8

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 51A10

Classification

msc 51A05

Related topic

Perspectivity

Defines

transvection

Defines

center

Defines

axis

Defines

central collineation

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