# Oktave (mathematics)

the Oktaven, also the Oktonionen or Cayleyzahlen, are a Verallgemeinerung of the quaternions and possess the quantity symbol [itex] \ mathbb {O}< /math>. They result from the application of the doubling procedure from the quaternions.

## history

the Oktonionen in the year 1843 by John Graves in a letter at William Rowan Hamilton was for the first time described. Independently of itthey were published 1845 by Arthur Cayley.

## multiplication table

the Oktonionen are a 8-dimensionaler vector space over the realNumbers with addition and multiplication. The multiplication is -- with the basis (1, i, j, k, l, m, n, o) -- given as follows:

[itex] \ begin {matrix}

``` i^2=j^2=k^2=l^2=m^2=n^2=o^2=-1 \ \ i=jk=lm=on=-kj=-ml=-no \ \ j=ki=ln=mo=-ik=-nl=-om \ \ k=ij=lo=nm=-ji=-ol=-mn \ \ l=mi=nj=ok=-im=-jn=-ko \ \ m=il=oj=kn=-li=-jo=-nk \ \ n=jl=io=mk=-lj=-oi=-km \ \ o=ni=jm=kl=-in=-mj=-lk
```

\ end {to matrix}< /math>

## characteristics

The Oktonionen is a division algebra with one element.

They do not form an inclined body (and concomitantly no body), because it hurt

the associative law of the multiplication: [itex] A \ cdot (b \ cdot C) = (A \ cdot b) \ cdot C< /math>.

It applies however to all Oktaven A and b:

[itex] A \ cdot (A \ cdot b) = (A \ cdot A) \ cdot b< /math> and [itex] A \ cdot (b \ cdot b) = (A \ cdot b) \ cdot b< /math>.

This characteristic is called Alternativität. The Oktonionen forms an alternative body.

From the Alternativität follows the relationship

[itex] A \ cdot (b \ cdot A) = (A \ cdot b) \ cdot A< /math>.

This relationship is called also flexibility law.

In addition the Oktonionen fulfills the sharper Moufang identities

< math> [A \ cdot (b \ cdot A)] \ C = A cdot \ cdot [b \ cdot (A\ C cdot)] [/itex]

and

[itex] (A \ cdot b) \ cdot (C \ cdot A) = A \ [(b \ cdot C) \ A cdot] /math< cdot>

## more

each Oktave can be represented…

... as eight tuple of real numbers: (r 1, r 2,…, r 8)
… as 4er-Tupel of complex numbers: (C 1 , C 2, C 3, C 4)
… as arranged pair of quaternions: (h 1 , h 2)

the body of the real numbers [itex] \ mathbb {R}< /math> can as sub-structure of [itex] \ mathbb {O}< /math> are regarded:

For all numbers of r out [itex] \ mathbb {R}< /math> applies: r corresponds (r, 0,… , 0)

The body of the complex numbers [itex] \ mathbb {C}< /math> /math can as sub-structure <of> math \ mathbb {O<}> are regarded:

For all numbers of C out [itex] \ mathbb {C}< /math> applies: C corresponds (C, 0, 0, 0)

the inclined body of the quaternions [itex] \ mathbb {H}< /math> /math can as sub-structure <of> math \ mathbb {O<}> are regarded:

For all numbers of h out [itex] \ mathbb {H}< /math> applies: h corresponds (h, 0)

for the Oktaven is in such a way defined to addition and multiplication that they are downward compatible, i.e....

... to all real numbers of r and s applies:
[itex] r + s = (r, 0,… , 0) + (s, 0,… , 0)< /math>
[itex] r \ cdot s = (r, 0,… , 0) \ cdot (s, 0,… , 0)< /math>
... for all complex numbers of C and D applies:
[itex] C + D = (C, 0, 0, 0) + (D, 0, 0, 0)< /math>
[itex] C \ cdot D = (C, 0, 0, 0) \ cdot (D, 0, 0, 0)< /math>
... to all quaternions h and i applies:
[itex] h + i = (h, 0) + (i,0)< /math>
[itex] h \ cdot i = (h, 0) \ cdot (i, 0)< /math>