quadratic space

quadratic space

A

quadratic space

(over a field) is a

vector space

V

equipped with a

quadratic form

Q

on

V

. It is denoted by

(

V

,

Q

)

. The

dimension

of the quadratic space is the dimension of the underlying vector space. Any vector space admitting a

bilinear form

has an induced quadratic form and thus is a quadratic space.

Two quadratic spaces

(

V

1

,

Q

1

)

and

(

V

2

,

Q

2

)

are said to be

isomorphic

if there exists an isomorphic

linear transformation

T

:

V

1

→

V

2

such that for any

v

∈

V

1

,

Q

1

⁢

(

v

)

=

Q

2

⁢

(

T

⁢

v

)

. Since

T

is easily seen to be an

isometry

between

V

1

and

V

2

(over the

symmetric bilinear forms

induced by

Q

1

and

Q

2

respectively), we also say that

(

V

1

,

Q

1

)

and

(

V

2

,

Q

2

)

are isometric.

A quadratic space equipped with a

regular quadratic form

is called a

regular quadratic space

.

Example of a Qudratic Space.

The

Generalized Quaternion Algebra

.

Let

F

be a field and

a

,

b

∈

F

˙

:=

F

-

{

0

}

. Let

H

be the algebra over

F

generated by

i

,

j

with the following

defining relations

:

1.

i

2

=

a

,

2.

j

2

=

b

, and

3.

i

⁢

j

=

-

j

⁢

i

.

Then

{

1

,

i

,

j

,

k

}

, where

k

:=

i

⁢

j

, forms a basis for the vector space

H

over

F

. For a direct proof, first note

(

i

⁢

j

)

2

=

(

i

⁢

j

)

⁢

(

i

⁢

j

)

=

i

⁢

(

j

⁢

i

)

⁢

j

=

i

⁢

(

-

i

⁢

j

)

⁢

j

=

-

a

⁢

b

≠

0

, so that

k

∈

F

˙

. It’s also not hard to show that

k

anti-commutes with both

i

,

j

:

i

⁢

k

=

-

k

⁢

i

and

j

⁢

k

=

-

k

⁢

j

. Now, suppose

0

=

r

+

s

⁢

i

+

t

⁢

j

+

u

⁢

k

. Multiplying both sides of the

equation

on the right by

i

gives

0

=

r

⁢

i

+

s

⁢

a

+

t

⁢

j

⁢

i

+

u

⁢

k

⁢

i

. Multiplying both sides on the left by

i

gives

0

=

r

⁢

i

+

s

⁢

a

+

t

⁢

i

⁢

j

+

u

⁢

i

⁢

k

. Adding the two results and reduce, we have

0

=

r

⁢

i

+

s

⁢

a

. Multiplying this again by

i

gives us

0

=

r

⁢

a

+

s

⁢

a

⁢

i

, or

0

=

r

+

s

⁢

i

. Similarly, one shows that

0

=

r

+

t

⁢

j

, so that

s

⁢

i

=

t

⁢

j

. This leads to two equations,

s

⁢

a

=

t

⁢

i

⁢

j

and

s

⁢

a

=

t

⁢

j

⁢

i

, if one multiplies it on the left and right by

i

. Adding the results then dividing by 2 gives

s

⁢

a

=

0

. Since

a

≠

0

,

s

=

0

. Therefore,

0

=

r

+

s

⁢

i

=

r

. Same

argument

shows that

t

=

u

=

0

as well.

Next, for any

element

α

=

r

+

s

⁢

i

+

t

⁢

j

+

u

⁢

k

∈

H

, define its

conjugate

α

¯

by

r

-

s

⁢

i

-

t

⁢

j

-

u

⁢

k

. Note that

α

=

α

¯

iff

α

∈

F

. Also, it’s not hard to see that

•

α

¯

¯

=

α

,

•

α

+

β

¯

=

α

¯

+

β

¯

,

•

α

⁢

β

¯

=

β

¯

⁢

α

¯

,

We next define the norm

N

on

H

by

N

⁢

(

α

)

=

α

⁢

α

¯

. Since

N

⁢

(

α

)

¯

=

α

⁢

α

¯

¯

=

α

¯

¯

⁢

α

¯

=

α

⁢

α

¯

=

N

⁢

(

α

)

,

N

⁢

(

α

)

∈

F

. It’s easy to see that

N

⁢

(

r

⁢

α

)

=

r

2

⁢

N

⁢

(

α

)

for any

r

∈

F

.

Finally, if we define the trace

T

on

H

by

T

⁢

(

α

)

=

α

+

α

¯

, we have that

N

⁢

(

α

+

β

)

-

N

⁢

(

α

)

-

N

⁢

(

β

)

=

T

⁢

(

α

⁢

β

¯

)

is bilinear (linear each in

α

and

β

).

Therefore,

N

defines a quadratic form on

H

(

N

is commonly called a

norm form

), and

H

is thus a quadratic space over

F

.

H

is denoted by

(

a

,

b

F

)

.

It can be shown that

H

is a

central simple algebra

over

F

. Since

H

is four dimensional over

F

, it is a

quaternion algebra

. It is a direct

generalization

of the quaternions

ℍ

over the reals

(

-

1

,

-

1

ℝ

)

.

In fact, every quaternion algebra (over a field

F

) is of the form

(

a

,

b

F

)

for some

a

,

b

∈

F

.

Title

quadratic space

Canonical name

QuadraticSpace

Date of creation

2013-03-22 15:05:55

Last modified on

2013-03-22 15:05:55

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A63

Classification

msc 11E88

Synonym

non-degenerate quadratic space

Related topic

QuadraticForm

Related topic

QuaternionAlgebra

Defines

norm form

Defines

isomorphic quadratic spaces

Defines

isometric quadratic spaces

Defines

generalized quaternion algebra

Defines

regular quadratic space

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