quadratic map

quadratic map

Given a

commutative ring

K

and two

K

-modules

M

and

N

then a map

q

:

M

→

N

is called

quadratic

if

1.

q

⁢

(

α

⁢

x

)

=

α

2

⁢

q

⁢

(

x

)

for all

x

∈

M

and

α

∈

K

.

2.

b

⁢

(

x

,

y

)

:=

q

⁢

(

x

+

y

)

-

q

⁢

(

x

)

-

q

⁢

(

y

)

, for

x

,

y

∈

M

, is a bilinear map.

The only

difference

between

quadratic maps

and

quadratic forms

is the insistence on the

codomain

N

instead of a

K

. So in this way every quadratic form is a special case of a quadratic map. Most of the properties for quadratic forms apply to quadratic maps as well. For

instance

, if

K

has no 2-torsion (

2

⁢

x

=

0

implies

x

=

0

) then

2

⁢

c

⁢

(

x

,

y

)

=

q

⁢

(

x

+

y

)

-

q

⁢

(

x

)

-

q

⁢

(

y

)

.

defines a

symmetric

K

-bilinear map

c

:

M

×

M

→

N

with

c

⁢

(

x

,

x

)

=

q

⁢

(

x

)

. In particular if

1

/

2

∈

K

then

c

⁢

(

x

,

y

)

=

1

2

⁢

b

⁢

(

x

,

y

)

.
This definition is one instance of a

polarization

(i.e.: substituting a single
variable in a

formula

with

x

+

y

and comparing the result with the formula over

x

and

y

separately.) Continuing
without

2

-

torsion

, if

b

is a symmetric

K

-bilinear map (perhaps not a form) then defining

q

b

⁢

(

x

)

=

b

⁢

(

x

,

x

)

determines a quadratic map since

q

b

⁢

(

α

⁢

x

)

=

b

⁢

(

α

⁢

x

,

α

⁢

x

)

=

α

2

⁢

b

⁢

(

x

,

x

)

=

α

2

⁢

q

⁢

(

x

)

and

q

b

⁢

(

x

+

y

)

-

q

b

⁢

(

x

)

-

q

b

⁢

(

y

)

=

b

⁢

(

x

+

y

,

x

+

y

)

-

b

⁢

(

x

,

x

)

-

b

⁢

(

y

,

y

)

=

b

⁢

(

x

,

y

)

+

b

⁢

(

y

,

x

)

=

2

⁢

b

⁢

(

x

,

y

)

.

Have have no

2

-torsion we can recover

b

form

q

b

. So in odd and 0

characteristic

rings we find symmetric
bilinear maps and quadratic maps are in

1-1 correspondence

.

An alternative understanding of

b

is to treat this as the obstruction to

q

being an

additive

homomorphism

. Thus a

submodule

T

of

M

for which

b

⁢

(

T

,

T

)

=

0

is a submodule of

M

on which

q

|

T

is an additive homomorphism.
Of course because of the first condition,

q

is semi-linear on

T

only when

α

↦

α

2

is an

automorphism

of

K

, in particular, if

K

has characteristic 2. When the characteristic of

K

is odd or 0 then

q

⁢

(

T

)

=

0

if and only if

b

⁢

(

T

,

T

)

=

0

simply because

q

⁢

(

x

)

=

b

⁢

(

x

,

x

)

(or up to a

1

/

2

multiple

depending on conventions). However, in characteristic 2 it is
possible for

b

⁢

(

T

,

T

)

=

0

yet

q

⁢

(

T

)

≠

0

. For instance, we can have

q

⁢

(

x

)

≠

0

yet

b

⁢

(

x

,

x

)

=

q

⁢

(

2

⁢

x

)

-

q

⁢

(

x

)

-

q

⁢

(

x

)

=

0

. This is summed up in the following
definition:

A

subspace

T

of

M

is called

totally singular

if

q

⁢

(

T

)

=

0

and
totally isotropic if

b

⁢

(

T

,

T

)

=

0

. In odd or 0 characteristic, totally singular
subspaces are precisely totally isotropic subspaces.

Title

quadratic map

Canonical name

QuadraticMap

Date of creation

2013-03-22 16:27:55

Last modified on

2013-03-22 16:27:55

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

9

Author

Algeboy (12884)

Entry type

Derivation

Classification

msc 11E08

Classification

msc 11E04

Classification

msc 15A63

Related topic

QuadraticJordanAlgebra

Related topic

IsotropicQuadraticSpace

Defines

quadratic map

Defines

totally singular

Defines

totally isotropic

Defines

polarization formula

Defines

polarization identity

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