composition algebra

composition algebra

1

Definition

The classical definition of a

composition algebra

is a

non-associative algebra

C

over a field

k

where

1.

C

admits a

non-degenerate quadratic form

q

:

C

→

k

, such
that

2.

q

is

multiplicative

:

q

⁢

(

x

⁢

y

)

=

q

⁢

(

x

)

⁢

q

⁢

(

y

)

.

We also say

q

permits

composition

or that it obeys the composition law.

This definition is geometric in that

quadratic forms

give rise to geometric atributes
for a

vector space

such as length, distance and orthogonality. Indeed, originally created
over the

real numbers

such properties seem appropriate for an

algebra

; however,

concepts

of length and distance are less appropriate over arbitrary fields and encourage a second

equivalent

definition based solely on the

algebraic

aspect of such algebras.

Alternatively, a composition algebra can be defined as a unital

alternative algebra

C

over a field

k

with an

involution

x

↦

x

¯

, that is an

anti-isomorphism

of order at most 2, such that:

1.

C

has no non-zero absolute zero

divisors

(that is,

(

x

⁢

a

)

⁢

x

=

0

for all

a

∈

C

implies

x

=

0

);

2.

the norm

N

⁢

(

x

)

:=

x

⁢

x

¯

is a

scalar multiple

of

1

, that is,

N

:

C

→

k

⁢

1

.

The first definition makes the composition property part of the definition but obscures
the alternative

multiplication

as well as the

existence

of an involutary anti-isomorphism
for the algebra. The second definition makes both of these properties evident but obscures
the composition property of the norm, and also hides the property that

N

is a quadratic form.
However both definitions have merit, the first captures the classical view of an algebra
respecting a certain geometric condition while the second, introduced by Jacobson, promotes
a purely algebraic treatment. In our examples and constructions to follows we attempt to exhibit
both aspects by supplying the norm, the involution, and the

product

.

Both definitions can be generalized to algebras over

commutative

unital rings

k

.

Recall that a quadratic form gives rise to a

symmetric

bilinear

from

b

:

C

×

C

→

k

by

b

⁢

(

x

,

y

)

=

q

⁢

(

x

+

y

)

-

q

⁢

(

x

)

-

q

⁢

(

y

)

, for all

x

,

y

∈

C

. Some of the immediate properties include:

1.

b

⁢

(

x

,

x

)

=

2

⁢

q

⁢

(

x

)

,

2.

b

⁢

(

x

⁢

z

,

y

⁢

z

)

=

b

⁢

(

x

,

y

)

⁢

q

⁢

(

z

)

,

3.

b

⁢

(

x

⁢

y

,

z

⁢

w

)

+

b

⁢

(

x

⁢

w

,

z

⁢

y

)

=

b

⁢

(

x

,

y

)

⁢

b

⁢

(

z

,

w

)

.

These strongly limit the

structure

of composition algebras
and leads to the celebrated

theorem

of Hurwitz (see Theorem

4

)
which suitably classifies the composition algebras over

ℝ

. The work of
many others including Albert, Dickson, Jacobson, and Kaplansky extended the essential
conclussion of Hurwitz to all fields and the resulting

generalization

is still
refered to as Hurwitz’s theorem.

There are other algebras

A

with norms

q

:

A

→

k

which permit composition
in the sense that

q

⁢

(

x

⁢

y

)

=

q

⁢

(

x

)

⁢

q

⁢

(

y

)

. For example, alternative algebras with involutions.
However, the distinguishing property of composition algebras is that

q

is a quadratic form.
Classifications for such norms have been caried out by Schafer and McCrimmon.

2

Norms

Originally, composition algebras were created over the real numbers

k

=

ℝ

.
Here the usual

positive definite

norm on the

real vector space

was used instead of
the quadratic form (the square of the norm is the quadratic form).

The first non-trivial example is the set of complex numbers

ℂ

with where

z

=

a

+

b

⁢

i

∈

ℂ

is assigned:

z

¯

=

a

-

b

⁢

i

;

|

z

|

=

|

a

+

b

⁢

i

|

=

a

2

+

b

2

=

z

⁢

z

¯

.

More interesting is the non-commutative algebra of

Hamiltonians

ℍ

, created by Hamilton, where each

x

∈

ℋ

has the form

x

=

a

+

b

⁢

i

+

c

⁢

j

+

d

⁢

k

and

x

¯

=

a

-

b

⁢

i

-

c

⁢

j

-

d

⁢

k

;

|

x

|

=

|

a

+

b

⁢

i

+

c

⁢

j

+

d

⁢

k

|

=

a

2

+

b

2

+

c

2

+

d

2

=

x

⁢

x

¯

.

The last

addition

to the list was the non-associative algebra of

octonions

initially created by Cayley and the norm is simply

x

¯

=

a

-

b

⁢

i

-

c

⁢

j

-

d

⁢

k

-

f

⁢

i

⁢

l

-

g

⁢

j

⁢

l

-

h

⁢

k

⁢

l

;

|

x

|

=

|

a

+

b

⁢

i

+

c

⁢

j

+

d

⁢

k

+

e

⁢

l

+

f

⁢

i

⁢

l

+

g

⁢

j

⁢

l

+

h

⁢

k

⁢

l

|

=

a

2

+

b

2

+

c

2

+

d

2

+

e

2

+

f

2

+

g

2

+

h

2

=

x

⁢

x

¯

.

Because general fields do not

sufficient

squareroots, the use of norms
in the classical

Euclidean

sense is replaced by the use of quadratic forms.
Furthermore, the lack of ordering a field, such as a

finite field

, introduces the
need to use

non-degenerate

rather than positive definite conditions. Under these
generalizations composition algebras can be redefined form the classical context
of composition algebras over

ℝ

to general composition algebras over
arbitrary fields, as done by our original definitions above. In this context,
there are three further composition algebras over

ℝ

.

Example 1

.

Let

C

=

R

⊕

R

with

q

⁢

(

x

,

y

)

=

x

⁢

y

for all

(

x

,

y

)

∈

C

. Then

C

is a composition algebra.

Proof.

Evidently

q

⁢

(

a

⁢

x

,

a

⁢

y

)

=

a

2

⁢

q

⁢

(

x

,

y

)

and the

polarization

of

q

is the

symmetric bilinear
form

b

⁢

(

(

x

,

y

)

,

(

z

,

w

)

)

=

x

⁢

z

-

y

⁢

w

for all

(

x

,

y

)

,

(

z

,

w

)

∈

C

(so the

signature

is

(

1

,

-

1

)

).
Thus

q

is a quadratic form.

To check that

q

has the compositional property let

(

x

,

y

)

,

(

z

,

w

)

∈

C

. Then

q

⁢

(

(

x

,

y

)

⁢

(

z

,

w

)

)

=

q

⁢

(

x

⁢

z

,

y

⁢

w

)

=

(

x

⁢

z

)

⁢

(

y

⁢

w

)

=

(

x

⁢

y

)

⁢

(

z

⁢

w

)

=

q

⁢

(

x

,

y

)

⁢

q

⁢

(

z

,

w

)

.

Note also that by defining

(

x

,

y

)

¯

=

(

y

,

x

)

then

(

x

,

y

)

⁢

(

x

,

y

)

¯

=

(

x

⁢

y

,

y

⁢

x

)

=

q

⁢

(

x

,

y

)

⁢

(

1

,

1

)

and

b

⁢

(

(

x

,

y

)

,

(

z

,

w

)

)

⁢

(

1

,

1

)

=

(

x

,

y

)

⁢

(

z

,

w

)

+

(

x

,

y

)

⁢

(

z

,

w

)

¯

.
∎

Example 2

.

Let

C

=

M

2

⁢

(

R

)

with

q

⁢

(

X

)

=

det

⁡

X

for all

X

∈

C

.
Then

C

is a composition algebra.

Proof.

Let

X

∈

C

and

a

∈

ℝ

. Then

q

⁢

(

a

⁢

X

)

=

det

⁡

(

a

⁢

X

)

=

det

⁡

(

a

⁢

I

2

)

⁢

det

⁡

X

=

a

2

⁢

det

⁡

X

=

a

2

⁢

q

⁢

(

X

)

.
It is also evident that if

X

=

[

a

b

c

d

]

then
setting

X

¯

=

[

d

-

b

-

c

a

]

makes

(

det

⁡

X

)

⁢

I

2

=

X

⁢

X

¯

and also

T

⁡

(

X

)

⁢

I

2

=

X

+

X

¯

, where

T

⁡

(

X

)

is the trace of

X

. Hence

b

⁢

(

X

,

Y

)

=

(

q

⁢

(

X

+

Y

)

-

q

⁢

(

X

)

-

q

⁢

(

Y

)

)

⁢

I

2

=

(

det

⁡

(

X

+

Y

)

)

⁢

I

2

-

(

det

⁡

X

)

⁢

I

2

-

(

det

⁡

Y

)

⁢

I

2

=

(

X

+

Y

)

⁢

(

X

+

Y

)

¯

-

X

⁢

X

¯

-

Y

⁢

Y

¯

=

Y

⁢

X

¯

+

X

⁢

Y

¯

=

T

⁡

(

X

⁢

Y

¯

)

⁢

I

2

.

Therefore,

b

⁢

(

X

,

Y

)

=

T

⁡

(

X

⁢

Y

¯

)

. Since

T

⁡

(

X

⁢

Y

¯

)

=

T

⁡

(

Y

¯

⁢

X

)

=

T

⁡

(

X

⁢

Y

¯

)

, it follows
that

b

is a symmetric bilinear form and so

q

is quadratic form.

Finally, for composition note

q

⁢

(

X

⁢

Y

)

=

det

⁡

(

X

⁢

Y

)

=

d

⁢

e

⁢

t

⁢

X

⁢

det

⁡

Y

=

q

⁢

(

X

)

⁢

q

⁢

(

Y

)

.

Therefore

C

is a composition algebra.
∎

This gives two new composition algebras over

ℝ

and indeed there is a third, constructed
below as the algebra

(

1

,

1

,

1

ℝ

)

, which is 8-dimensional and

non-associative

but
unlike the octonions, it has non-trivial zero-divisors.

Definition 3

.

A composition algebra is

split

if the quadratic form is isotropic.

The example of

ℝ

⊕

ℝ

and

M

2

⁢

(

ℝ

)

just given are both examples of
split composition algebras.

3

Involution

Define

x

¯

:=

b

⁢

(

x

,

1

)

⁢

1

-

x

,

∀

x

∈

C

.

Immediately it follows that: for all

x

,

y

∈

C

,

1.

x

¯

¯

=

x

,

2.

x

+

y

¯

=

x

¯

+

y

¯

,

3.

1

¯

=

1

.

Define the

trace

of

x

as

T

⁢

(

x

)

=

x

+

x

¯

and
the

norm

of

x

as

N

⁢

(

x

)

=

x

¯

⁢

x

. Then it follows that:

x

2

-

T

⁢

(

x

)

⁢

x

+

N

⁢

(

x

)

=

0

,

∀

x

∈

C

.

So

C

is a

quadratic

algebra since every

element

in

x

has at most
a quadratic

minimal polynomial

. In fact

N

⁢

(

x

)

is a quadratic form
allowing composition.

4

Constructing composition algebras

All of the following are composition algebras.

[

5

, III.4]

dim

⁡

1

:

k

, with

trivial

involution

x

=

x

¯

for all

x

in

k

.

dim

⁡

2

:

For any

α

∈

k

, a

quadratic

extension

of

k

, that is

(

α

k

)

=

⟨

1

,

i

|

i

2

=

α

⟩

.

Here

{

1

,

i

}

is a basis and has an involution defined by

1

¯

=

1

and

i

¯

=

-

i

.

dim

⁡

4

:

For any

α

,

β

∈

k

, a

quaternion

algebra over

k

defined as

(

α

,

β

k

)

=

⟨

1

,

i

,

j

|

i

2

=

α

,

j

2

=

β

,

i

⁢

j

=

-

j

⁢

i

⟩

Then

{

1

,

i

,

j

,

i

⁢

j

}

forms a basis.

1

1

It is common to use

k

for

i

⁢

j

, but

k

here is used
exclusively for the underlying field.

An involution is defined by

1

¯

=

1

,

i

¯

=

-

i

,

j

¯

=

-

j

and extended linearly.

dim

⁡

8

:

For any

α

,

β

,

γ

∈

k

, an

octonion

algebra over

k

:

(

α

,

β

,

γ

k

)

=

⟨

1

,

i

,

j

,

l

|

⁢

i

2

=

α

,

j

2

=

β

,

l

2

=

γ

,

i

⁢

j

=

-

j

⁢

i

,

i

⁢

l

=

-

l

⁢

i

,

j

⁢

l

=

-

l

⁢

j

,

i

(

l

j

)

=

-

l

(

i

j

)

,

(

l

i

)

j

=

l

(

j

i

)

,

(

l

i

)

(

l

j

)

=

-

γ

j

i

⟩

.

The set

{

1

,

i

,

j

,

i

⁢

j

,

l

,

i

⁢

l

,

j

⁢

l

,

i

⁢

j

⁢

l

}

is a basis.
An involution is defined by

1

¯

=

1

,

i

¯

=

-

i

,

j

¯

=

-

j

,

l

¯

=

-

l

and extended linearly.

Each of these algebras can be realized by the Cayley-Dickson method which takes

C

an associative

k

-algebra with involution and produces for each

α

∈

C

-

{

0

}

a new
algebra

(

α

C

)

on the vector space

C

⊕

C

with product

(

a

,

b

)

⁢

(

c

,

d

)

=

(

a

⁢

c

+

α

⁢

d

⁢

b

¯

,

a

¯

⁢

d

+

c

⁢

b

)

.

Set the involution on

(

α

C

)

to be

(

a

,

b

)

¯

=

(

a

¯

,

-

b

)

.

The algebras are equipped with a

trace

T

⁢

r

⁢

(

x

)

=

x

+

x

¯

,
and

norm

N

⁢

(

x

)

=

x

⁢

x

¯

. This norm serves as the

quadratic map

to establish
these algebras a composition algebras. The images of the trace and norm lie in

k

.

The new algebra is associative only if

C

is commutative, otherwise it is alternative.
This means that

k

,

(

α

k

)

,

(

α

,

β

k

)

are
the associative composition algebras.

An algebra is a

division

algebra if the only zero-divisor is

0

[

5

, II.2]

. A

central

simple composition algebra with a non-trivial
zero-divisor is called a

split

composition algebra.

Finite dimensional

split
central simple composition algebras are unique up to

isomorphism

to one of

k

,

(

1

k

)

≅

k

⊕

k

,

(

1

,

1

k

)

≅

M

2

⁢

(

k

)

,

(

1

,

1

,

1

k

)

.

5

Classification theorem

Theorem 4

.

[

2

, Theorem 6.2.3]

A composition algebra

C

over a field

k

with quadratic form

q

⁢

(

x

)

=

x

⁢

x

¯

is isomorphic
to one of the following:

(i)

A

purely inseparable extension

field

E

/

F

of

characteristic

2

and

exponent

1

(trivial
involution) so

q

⁢

(

x

)

=

x

2

.

(ii)

k

with trivial involution, so

q

⁢

(

x

)

=

x

2

,

(iii)

Quadratic composition algebra:

(

α

k

)

for

α

∈

k

,

(iv)

Quaternion algebra

:

(

α

,

β

k

)

for

α

,

β

∈

k

,

(v)

Octonion algebra:

(

α

,

β

,

γ

k

)

for

α

,

β

,

γ

∈

k

.

In particular, all composition algebras over

k

, save perhaps those of type

(

i

)

, are finite
dimensional and of

dimension

1

,

2

,

4

or

8

.

References

1

T.Y. Lam:

Introduction

to Quadratic Forms over Fields

, AMS, Providence (2004).

2

N. Jacobson

Structure theory of

Jordan algebras

, The University of Arkansas lecture
notes in mathematics, vol. 5, Fayetteville, 1981.

3

K. McCrimmon:

A Taste of Jordan Algebras

, Springer, New York (2004).

4

J.H. Conway, D.A. Smith:

On Quaternions and Octonions, Their Geometry,

Arithmetic

, and

Symmetry

, AK Peters, Natick, Mass (2003)

5

Richard D. Schafer,

An introduction to

nonassociative algebras

, Pure and
Applied Mathematics, Vol. 22, Academic Press, New York, 1966.

Title

composition algebra

Canonical name

CompositionAlgebra

Date of creation

2013-03-22 15:11:33

Last modified on

2013-03-22 15:11:33

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

14

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 17A75

Related topic

TheoremsOnSumsOfSquares

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