Cayley-Dickson construction

Cayley-Dickson construction

In the foregoing discussion, an

algebra

shall mean a

non-associative algebra

.

Let

A

be a normed

*

-algebra, an algebra admitting an

involution

(

http://planetmath.org/Involution2

)

*

, over a

commutative ring

R

with

1

≠

0

. The

Cayley-Dickson construction

is a way of enlarging

A

to a new algebra,

K

⁢

D

⁢

(

A

)

, extending the

*

as well as the norm operations in

A

, such that

A

is a

subalgebra

of

K

⁢

D

⁢

(

A

)

.

Define

K

⁢

D

⁢

(

A

)

to be the module (external)

direct sum

of

A

with itself:

K

⁢

D

⁢

(

A

)

:=

A

⊕

A

.

Therefore,

addition

in

K

⁢

D

⁢

(

A

)

is defined by addition componentwise in each copy of

A

. Next, let

λ

be a unit in

R

and define three additional operations:

1.

(

Multiplication

)

(

a

⊕

b

)

⁢

(

c

⊕

d

)

:=

(

a

⁢

c

+

λ

⁢

d

*

⁢

b

)

⊕

(

d

⁢

a

+

b

⁢

c

*

)

, where

*

is the involution on

A

,

2.

(Extended involution)

(

a

⊕

b

)

*

:=

a

*

⊕

(

-

b

)

, and

3.

(Extended Norm)

N

⁢

(

a

⊕

b

)

:=

(

a

⊕

b

)

⁢

(

a

⊕

b

)

*

.

One readily checks that the multiplication is

bilinear

, since the involution

*

(on

A

) is linear. Therefore,

K

⁢

D

⁢

(

A

)

is an algebra.

Furthermore, since the extended involution

*

is clearly bijective and linear, and that

(

a

⊕

b

)

*

*

=

(

a

*

⊕

(

-

b

)

)

*

=

a

*

*

⊕

b

=

a

⊕

b

,

this extended involution is well-defined and so

K

⁢

D

⁢

(

A

)

is in addition a

*

-algebra.

Finally, to see that

K

⁢

D

⁢

(

A

)

is a normed

*

-algebra, we identify

A

as the first

component

of

K

⁢

D

⁢

(

A

)

, then

A

becomes a subalgebra of

K

⁢

D

⁢

(

A

)

and elements of the form

a

⊕

0

can now be written simply as

a

. Now, the extended norm

N

⁢

(

a

⊕

b

)

=

(

a

⊕

b

)

⁢

(

a

*

⊕

(

-

b

)

)

=

(

a

⁢

a

*

-

λ

⁢

b

*

⁢

b

)

⊕

0

=

N

⁢

(

a

)

-

λ

⁢

N

⁢

(

b

)

∈

A

,

where

N

in the subsequent terms of the above

equation

array is the norm on

A

given by

N

⁢

(

a

)

=

a

⁢

a

*

. The fact that the

N

:

K

⁢

D

⁢

(

A

)

→

A

, together with the equality

N

⁢

(

0

⊕

0

)

=

0

show that the extended norm

N

on

K

⁢

D

⁢

(

A

)

is well-defined. Thus,

K

⁢

D

⁢

(

A

)

is a normed

*

-algebra.

The normed

*

-algebra

K

⁢

D

⁢

(

A

)

, together with the

invertible element

λ

∈

R

, is called the

Cayley-Dickson algebra

,

K

⁢

D

⁢

(

A

,

λ

)

,

obtained from

A

.

If

A

has a unity 1, then so does

K

⁢

D

⁢

(

A

,

λ

)

and its unity is

1

⊕

0

. Furthermore, write

i

=

0

⊕

1

, we check that,

i

⁢

a

=

(

0

⊕

1

)

⁢

(

a

⊕

0

)

=

0

⊕

a

*

=

(

a

*

⊕

0

)

⁢

(

0

⊕

1

)

=

a

*

⁢

i

. Therefore,

i

⁢

A

=

A

⁢

i

and we can identify the second component of

K

⁢

D

⁢

(

A

,

λ

)

with

A

⁢

i

and write elements of

A

⁢

i

as

a

⁢

i

for

a

∈

A

.

It is not hard to see that

A

⁢

(

A

⁢

i

)

=

(

A

⁢

i

)

⁢

A

⊆

A

⁢

i

and

(

A

⁢

i

)

⁢

(

A

⁢

i

)

⊆

A

. We are now able to write

K

⁢

D

⁢

(

A

,

λ

)

=

A

⊕

A

⁢

i

,

where each element

x

∈

K

⁢

D

⁢

(

A

,

λ

)

has a unique expression

x

=

a

+

b

⁢

i

.

Properties

. Let

x

,

y

,

z

will be general elements of

K

⁢

D

⁢

(

A

,

λ

)

.

1.

(

x

⁢

y

)

*

=

y

*

⁢

x

*

,

2.

x

+

x

*

∈

A

,

3.

N

⁢

(

x

⁢

y

)

=

N

⁢

(

x

)

⁢

N

⁢

(

y

)

.

Examples

. All examples considered below have ground ring the reals

ℝ

.

•

K

⁢

D

⁢

(

ℝ

,

-

1

)

=

ℂ

, the

complex numbers

.

•

K

⁢

D

⁢

(

ℂ

,

-

1

)

=

ℍ

, the

quaternions

.

•

K

⁢

D

⁢

(

ℍ

,

-

1

)

=

𝕆

, the

octonions

.

•

K

⁢

D

⁢

(

𝕆

,

-

1

)

=

𝕊

, which are called the

sedenions

, an algebra of

dimension

16 over

ℝ

.

Remarks

.

1.

Starting from

ℝ

, notice each stage of Cayley-Dickson construction produces a new algebra that loses some intrinsic properties of the previous one:

ℂ

is no longer orderable (or

formally real

);

commutativity

is lost in

ℍ

;

associativity

is gone from

𝕆

; and finally,

𝕊

is not even a

division algebra

anymore!

2.

More generally, given any field

k

, any algebra obtained by applying the Cayley-Dickson construction twice to

k

is called a

quaternion algebra

over

k

, of which

ℍ

is an example. In other words, a quaternion algebra has the form

K

⁢

D

⁢

(

K

⁢

D

⁢

(

k

,

λ

1

)

,

λ

2

)

,

where each

λ

i

∈

k

*

:=

k

-

{

0

}

. Any algebra obtained by applying the Cayley-Dickson construction three times to

k

is called a

Cayley algebra

, of which

𝕆

is an example. In other words, a Cayley algebra has the form

K

⁢

D

⁢

(

K

⁢

D

⁢

(

K

⁢

D

⁢

(

k

,

λ

1

)

,

λ

2

)

,

λ

3

)

,

where each

λ

i

∈

k

*

. A Cayley algebra is an

octonion algebra

when

λ

1

=

λ

2

=

λ

3

=

-

1

.

References

1

Richard D. Schafer,

An Introduction to

Nonassociative Algebras

, Dover Publications, (1995).

Title

Cayley-Dickson construction

Canonical name

CayleyDicksonConstruction

Date of creation

2013-03-22 14:54:11

Last modified on

2013-03-22 14:54:11

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

25

Author

CWoo (3771)

Entry type

Definition

Classification

msc 17A99

Synonym

Cayley-Dickson process

Synonym

doubling process

Synonym

octonion algebra

Related topic

TheoremsOnSumsOfSquares

Defines

Cayley-Dickson algebra

Defines

sedenion

Defines

quaternion algebra

Defines

Cayley algebra

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