octonion

octonion

Let

ℍ

be the

quaternions

over the reals

ℝ

. Apply the

Cayley-Dickson construction

to

ℍ

once, and we obtain an

algebra

,
variously called

Cayley algebra

,

the octonion algebra

, or simply

the octonions

, over

ℝ

. Specifically the construction is carried out as follows:

1.

Form the

vector space

𝕆

=

ℍ

⊕

ℍ

⁢

𝐤

; any element of

𝕆

can be written as

a

+

b

⁢

𝐤

, where

a

,

b

∈

ℍ

;

2.

Define a

binary operation

on

𝕆

called

the multiplication in

O

by

(

a

+

b

⁢

𝐤

)

⁢

(

c

+

d

⁢

𝐤

)

:=

(

a

⁢

c

-

d

¯

⁢

b

)

+

(

d

⁢

a

+

b

⁢

c

¯

)

⁢

𝐤

,

where

a

,

b

,

c

,

d

∈

ℍ

, and

c

¯

is the quaternionic

conjugation

of

c

∈

ℍ

. When

b

=

d

=

0

, the multiplication is

reduced

the
multiplication in

ℍ

. In addition, the multiplication rule above
imply the following:

a

⁢

(

d

⁢

𝐤

)

=

(

d

⁢

a

)

⁢

𝐤

(1)

(

b

⁢

𝐤

)

⁢

c

=

(

b

⁢

c

¯

)

⁢

𝐤

(2)

(

b

⁢

𝐤

)

⁢

(

d

⁢

𝐤

)

=

-

d

¯

⁢

b

.

(3)

In particular, in the last

equation

, if

b

=

d

=

1

,

𝐤

2

=

-

1

.

3.

Define a unary operation on

𝕆

called

the octonionic
conjugation in

O

by

a

+

b

⁢

𝐤

¯

:=

a

¯

-

b

⁢

𝐤

,

where

a

,

b

∈

ℍ

.
Clearly, the octonionic conjugation is an

involution

(

http://planetmath.org/Involution2

)
(

x

¯

¯

=

x

).

4.

Finally, define a unary operation

N

on

𝕆

called

the norm in

O

by

N

⁢

(

x

)

:=

x

⁢

x

¯

, where

x

∈

𝕆

. Write

x

=

a

+

b

⁢

𝐤

, then

N

⁢

(

x

)

=

(

a

+

b

⁢

𝐤

)

⁢

(

a

¯

-

b

⁢

𝐤

)

=

(

a

⁢

a

¯

+

b

¯

⁢

b

)

+

(

-

b

⁢

a

+

b

⁢

a

¯

¯

)

⁢

𝐤

=

a

⁢

a

¯

+

b

⁢

b

¯

≥

0

.

It is not hard to see that

N

⁢

(

x

)

=

0

iff

x

=

0

.

The above four (actually, only the first two suffice) steps makes

𝕆

into an

8

-dimensional algebra over

ℝ

such that

ℍ

is
embedded as a

subalgebra

.

With the last two steps, one can define the

inverse

of a non-zero
element

x

∈

𝕆

by

x

-

1

:=

x

¯

N

⁢

(

x

)

so that

x

⁢

x

-

1

=

x

-

1

⁢

x

=

1

. Since

x

is arbitrary,

𝕆

has no

zero
divisors

. Upon checking that

x

-

1

⁢

(

x

⁢

y

)

=

y

=

(

y

⁢

x

)

⁢

x

-

1

, the

non-associative algebra

𝕆

is turned into a

division algebra

.

Since

N

⁢

(

x

)

≥

0

for any

x

∈

𝕆

, we can define a non-negative
real-valued function

∥

⋅

∥

on

𝕆

by

∥

x

∥

=

N

⁢

(

x

)

. This is clearly well-defined and

∥

x

∥

=

0

iff

x

=

0

. In addition, it is not hard to see that, for any

r

∈

ℝ

and

x

∈

𝕆

,

∥

r

⁢

x

∥

=

|

r

|

⁢

∥

x

∥

, and that

∥

⋅

∥

satisfies the triangular inequality. This makes

𝕆

into a normed
division algebra.

Since the multiplication in

ℍ

is noncommutative,

𝕆

is
noncommutative. In fact, if we write

ℍ

=

ℂ

⊕

ℂ

⁢

𝐣

,
where

ℂ

are the

complex numbers

and

𝐣

2

=

-

1

, then

B

=

{

1

,

𝐢

,

𝐣

,

𝐢𝐣

}

is a basis
for the vector space

ℍ

over

ℝ

. With the introduction
of

𝐤

∈

𝕆

, we quickly check that

𝐤

anti-commute with the non-real basis elements in

B

:

𝐢𝐤

=

-

𝐤𝐢

,

𝐣𝐤

=

-

𝐤𝐣

,

(

𝐢𝐣

)

⁢

𝐤

=

-

𝐤

⁢

(

𝐢𝐣

)

.

Furthermore, one checks that

𝐢

⁢

(

𝐣𝐤

)

=

(

𝐣𝐢

)

⁢

𝐤

=

-

(

𝐢𝐣

)

⁢

𝐤

, so that

𝕆

is not

associative

.

Since

𝕆

=

ℍ

⊕

ℍ

⁢

𝐤

, the set

{

1

,

𝐢

,

𝐣

,

𝐢𝐣

,

𝐤

,

𝐢𝐤

,

𝐣𝐤

,

(

𝐢𝐣

)

⁢

𝐤

}

(

=

B

∪

B

⁢

𝐤

) is a
basis for

𝕆

over

ℝ

. A less messy way to represent these basis
elements is done the following assignment:

Title

octonion

Canonical name

Octonion

Date of creation

2013-03-22 15:21:42

Last modified on

2013-03-22 15:21:42

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 17A75

Classification

msc 17D05

Synonym

Cayley algebra

Related topic

TheoremsOnSumsOfSquares

Related topic

DivisionAlgebra

Defines

octonion algebra

\@unrecurse

Generated on Fri Feb 9 18:48:20 2018 by

LaTeXML