polarities and forms

polarities and forms

Through out this article we assume

dim

⁡

V

≠

2

. This is not a true constraint as there are only

trivial

dualities

for

dim

⁡

V

≤

2

.

Proposition 1

.

Every duality gives rise to a

non-degenerate

sesquilinear form

,
and visa-versa.

Proof.

To see this, let

d

:

P

⁢

G

⁢

(

V

)

→

P

⁢

G

⁢

(

V

)

be a duality. We may express this as an order preserving map

d

:

P

⁢

G

⁢

(

V

)

→

P

⁢

G

⁢

(

V

*

)

. Then by the

fundamental theorem of projective geometry

it follows

d

is

induced

by a

bijective

semi-linear transformation

d

^

:

V

→

V

*

.

An semi-linear

isomorphism

of

V

to

V

*

is

equivalent

to specifying a non-degenerate sesquilinear form. In particular, define the form

b

:

V

×

V

→

k

by

b

⁢

(

v

,

w

)

=

(

v

)

⁢

(

w

⁢

d

^

)

(notice

w

⁢

d

^

∈

V

*

so

w

⁢

d

^

:

V

→

k

).

Now, if

b

:

V

×

V

→

k

is a non-degenerate sesquilinear form. Then
define

b

^

:

V

→

V

*

:

v

↦

b

⁢

(

-

,

v

)

:

V

→

k

which is semi-linear, as

b

is sesquilinear, and bijective, since

b

is
non-degenerate. Therefore

b

^

induces an order preserving

bijection

P

⁢

G

⁢

(

V

)

→

P

⁢

G

⁢

(

V

*

)

, that is, a duality.
∎

We write

W

⟂

for the image of the induced duality of a non-degenerate
sesquilinear form

b

. Notice that

W

⟂

=

{

w

∈

V

:

b

⁢

(

v

,

W

)

=

0

}

. (Although the form may not be

reflexive

, we still use the

⟂

notation, but we now demonstrate that we can indeed specialize to the reflexive case.)
Notice then that

dim

⁡

W

⟂

=

dim

⁡

V

-

dim

⁡

W

.

Corollary 2

.

Every

polarity

gives rise to a reflexive non-degenerate sesquilinear form,
and visa-versa.

Proof.

Let

b

be the sesquilinear form induced by the polarity

p

. Then suppose we
have

v

,

w

∈

V

such that

0

=

b

⁢

(

v

,

w

)

=

(

v

)

⁢

(

w

⁢

p

^

)

. So

⟨

v

⟩

≤

⟨

w

⟩

⟂

=

⟨

w

⟩

⁢

p

. But

p

has order 2 so

⟨

v

⟩

⟂

=

⟨

v

⟩

⁢

p

≥

⟨

w

⟩

. But this implies

b

⁢

(

w

,

v

)

=

0

so

b

is reflexive.

Likewise, given a reflexive non-degenerate sesquilinear form

b

it gives rise
do a duality

d

induced by

b

^

. By the

reflexivity

,

b

⁢

(

W

,

W

⟂

)

=

0

implies

b

⁢

(

W

⟂

,

W

)

=

0

also. As

(

W

⟂

)

⟂

=

{

v

∈

V

:

b

⁢

(

v

,

(

W

⟂

)

⟂

)

=

0

}

it follows

W

≤

(

W

⟂

)

⟂

. But by

dimension

arguments

:

dim

⁡

(

W

⟂

)

⟂

=

dim

⁡

V

-

dim

⁡

W

⟂

=

dim

⁡

V

-

(

dim

⁡

V

-

dim

⁡

W

)

=

dim

⁡

W

we conclude

W

=

(

W

⟂

)

⟂

. Thus

d

is a polarity.
∎

From the fundamental theorem of projective geometry it follows if

dim

⁡

V

≠

2

then every order preserving map is induced by a semi-linear transformation of

V

. In

similar

fashion we have

Proposition 3

.

P

⁢

Γ

⁢

L

*

⁢

(

V

)

=

P

⁢

Γ

⁢

L

⁢

(

V

)

⋊

ℤ

2

, meaning that every order
reversing map

f

:

P

⁢

G

⁢

(

V

)

→

P

⁢

G

⁢

(

V

)

can be decomposed as a

f

=

s

⁢

r

where

s

is induced from a semi-linear transformation and

r

is a polarity.

Proof.

Let

d

be any duality of

P

⁢

G

⁢

(

V

)

. Then

d

2

is order preserving. Thus

d

2

is a

projectivity

so by the fundamental theorem of projective geometry

d

2

is induced by a semi-linear transformation

s

. Therefore

P

⁢

Γ

⁢

L

⁢

(

V

)

has index 2 in

P

⁢

Γ

⁢

L

*

⁢

(

V

)

. Finally it suffices to
provide any polarity of

P

⁢

G

⁢

(

V

)

to prove

P

⁢

Γ

⁢

L

*

⁢

(

V

)

=

P

⁢

Γ

⁢

L

⁢

(

V

)

⋊

ℤ

2

.
For this use any reflexive non-degenerate sesquilinear form.
∎

Remark 4

.

The group

P

⁢

Γ

⁢

L

*

⁢

(

V

)

is the

automorphism group

of

P

⁢

S

⁢

L

⁢

(

V

)

.
In particular, the polarities account for the

graph automorphisms

of the
Dynkin diagram of

A

d

-

1

,

d

=

dim

⁡

V

. When

dim

⁡

V

=

2

there is no
graph automorphism, just as there are no dualities (points are

hyperplanes

when

dim

⁡

V

=

2

.)

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).

2

Kantor, W. M.

Lectures notes on

Classical Groups

.

Title

polarities and forms

Canonical name

PolaritiesAndForms

Date of creation

2013-03-22 15:58:13

Last modified on

2013-03-22 15:58:13

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

7

Author

Algeboy (12884)

Entry type

Topic

Classification

msc 51A05

Related topic

polarity

Related topic

Projectivity

Related topic

ProjectiveGeometry

Related topic

Isometry2

Related topic

ProjectiveGeometry3

Related topic

ClassicalGroups

Related topic

Polarity2

Related topic

DualityWithRespectToANonDegenerateBilinearForm

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