Octonion

In Mathematical, the octonions or octaves is an not-associative extension of the Quaternion S. They form an algebra with 8 Dimension S on the Réel S. the Algèbre of octonions is generally noted $\ mathbb {O}$ .

By losing the important property of Associativeness, the octonions received less attention than the Quaternion S. In spite of that, the octonions keep their importance in Algèbre and Géométrie, in particular among the groups of Dregs.

History

The octonions were discovered in 1843 per John T. Serious, a friend of William Hamilton, which called them octaves . They were independently discovered by Arthur Cayley, which published the first article on the subject in 1845. They are often called octaves of Cayley or algebra of Cayley .

Definition

Each octonion is a linear Combinaison with real coefficients S of octonions unit $\ {\ 1, \ I, \ J, \ K, \ L, \ Li, \ lj, \ lk \ \}$ .

In other words, each octonion $\ X \$ can be written in the form

$x \ = \ x_0 \ + \ x_1 \. \ I \ + \ x_2 \. \ J \ + \ x_3 \. \ K \ + \ x_4 \. \ L \ + \ x_5 \. \ Li \ + \ x_6 \. \ lj \ + \ x_7 \. \ lk$ ,

with real coefficients

\ x_n \

. The whole of these linear combinations is a vector Space noted

\ mathbb {O}

, isomorphous with

\ mathbb {R} ^8

Addition

The Addition of octonions is carried out by adding the corresponding coefficients, as for the complex numbers and the Quaternion S:

(x_0 \ + \ x_1 \. \ I \ + \ x_2 \. \ J \ + \ x_3 \. \ K \ + \ x_4 \. \ L \ + \ x_5 \. \ Li \ + \ x_6 \. \ lj \ + \ x_7 \. \ lk) \ +

(y_0 \ + \ y_1 \. \ I \ + \ y_2 \. \ J \ + \ y_3 \. \ K \ + \ y_4 \. \ L \ + \ y_5 \. \ Li \ + \ y_6 \. \ lj \ + \ y_7 \. \ lk) \ =

(x_0+y_0) \ + \ (x_1+y_1) .i \ + \ (x_2+y_2) .j \ + \ (x_3+y_3) .k \ + \ (x_4+y_4) .l \ + \ (x_5+y_5) .li \ + \ (x_6+y_6) .lj \ + \ (x_7+y_7) .lk

Properties

The addition of the octonions is commutative:

$x \ + \ there \ = \ there \ + \ x$ ,

associative:

$x \ + \ (there + \ Z) \ = \ (X \ + \ there) \ + \ z$ ,

and has a neutral element, zero, noted

\ 0 \

$x \ + \ 0 \ = \ 0 + \ X \ = \ x$ .

For all octonion $\ X \$ exists a single octonion , noted $\ - X \$ , such as their nap is null:

$\ X \ + \ - X \ = \ 0$ .
This octonion, named opposite , is obtained simply by taking the opposite of the real coefficients of $\ X \$ .

Thus the whole of the octonions provided with the addition and of the opposite commutative group is a .

Subtraction

The subtraction of the octonions is then the operation simply defined by:

$\ X \ - \ there \ = \ X \ + \ there$ .

Multiplication

The Multiplication of the octonions then is completely determined by the property of Distributivité on the right and on the left:

$a\ .\ (B \ + \ c) \ = \ has \. \ B \ + \ has \. \ c$

$(has \ + \ b) \. \ C \ = \ has \. \ C \ + \ B \. \ c$

where $a, \ B, \ c$ are octonions unspecified, and the zero element absortant , and by the multiplication table of the octonions unit below:

In the table above, the operand of left is indicated in the first column, and the operand of right-hand side is in the first line. The table is not symmetrical, which means that this Multiplication is not commutative.
The multiplication table can be entirely defined by the remarkable identity:

$i^2 \ = \ j^2 \ = \ k^2 \ = \ l^2 \ = \ ijk \ = \ jki \ = \ kij \ = \ -1$ .

Mnemotechnical plan of Fano

An average mnemotechnics to remember the products of the octonions unit is given by the diagram opposite.
This diagram on 7 points and 7 lines (the circle passing by $\ I \$ , $\ J \$ and $\ K \$ is regarded as a line) is called the plane Fano . Let us note that the lines are directed in this diagram. The 7 points correspond to the 7 basic elements of $\ mathbb {O}$ . Let us note that each couple of distinct points is on a single line and that each line crosses 3 points exactly.
Either $(has, \ B, \ c)$ an ordered triplet of points located on a line given with the order given by the direction of the arrow. The multiplication is given by:

$a\ .\ B = \ c$ and
$b \. \ has = \ - c$
with cyclic permutations. Those operate in the following way:

$\ 1 \$ is the neutral element for the multiplication,

$e^2 \ = \ -1$ for each point $\ E \$ of the diagram completely defines the algebraic structure of the octonions .

Let us note that each of the 7 lines generates an isomorphous subalgebra of $\ mathbb {O}$ with the Quaternion S $\ mathbb {H}$ .

Combined

The Combined of a octonion

$x \ = \ x_0 \ + \ x_1 \. \ I \ + \ x_2 \. \ J \ + \ x_3 \. \ K \ + \ x_4 \. \ L \ + \ x_5 \. \ Li \ + \ x_6 \. \ lj \ + \ x_7 \. \ lk$ ,

by

$x^ {*} \ = \ x_0 \ is given - \ x_1 \. \ I \ - \ x_2 \. \ J \ - \ x_3 \. \ K \ - \ x_4 \. \ L \ - \ x_5 \. \ Li \ - \ x_6 \. \ lj \ - \ x_7 \. \ lk$ .

The conjugation is a Involution of $\ mathbb {O}$ and satisfied

$(X \. \ there) ^ {*} \ = \ y^ {*} \. \ x^ {*}$

(let us note the change in the order of succession).

Parts real and imaginary

The real Partie the octonion $\ X \$ is defined as follows

$Re (X) \ = \ \ frac {X \ + \ x^ {*}} {2} \ = \ x_0$

and the imaginary Left

$Im (X) \ = \ \ frac {X \ - \ x^ {*}} {2} \ = \ x_1 \. \ I \ + \ x_2 \. \ J \ + \ x_3 \. \ K \ + \ x_4 \. \ L \ + \ x_5 \. \ Li \ + \ x_6 \. \ lj \ + \ x_7 \. \ lk$

so that for all octonion $\ X \$ ,

$Re (X) \ + \ Im (X) \ = \ x$ ,

$Re (x^ {*}) \ = \ Re (X)$ ,

$Im (x^ {*}) \ = \ - Im (X)$ .

The whole of all the octonions purely imaginary (whose real left is null) form under space to 7 Dimension S on the real S of $\ mathbb {O}$ , noted $Im (\ mathbb {O})$ , isomorph with $\ mathbb {R} ^7$ . It is not under Algèbre because the multiplication of octonions purely imaginary can be a reality.
The whole of all the octonions purely real (whose imaginary left is null) form under Algèbre to 1 Dimension of $\ mathbb {O}$ , noted $Re (\ mathbb {O})$ , isomorph with $\ mathbb {R}$ .

Normalizes

The Norme of a octonion $\ X \$ is defined as follows

$\|X \|\ = \ \ sqrt {X \. \ x^ {*}}$

This square Racine is well a Real number Positif:

$\|X \|^2 \ = \ X \. \ x^ {*} = x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 + x_7^2$

This standard corresponds with the Euclidean standard on $\ mathbb {R} ^8$ .
One also has:

$\|X \|\ = \ \ sqrt {^2 \ - \ ^2}$ ,

$Re (X) \ = \ \ pm \ sqrt {^2 + \|X \|^2}$ ,

$^2 \ = \ ^2 - \|X \|^2$ (the square of the imaginary part is a reality).

Opposite

The existence of a standard on $\ mathbb {O}$ implies the existence of a Inverse for each element distinct from zero in $\ mathbb {O}$ . The reverse of all $\ X \$ different from zero is given by

$x^ {- 1} \ = \ \|X \|^ {- 2} \. \ x^ {*}$

That satisfied

$x \. \ x^ {- 1} \ = \ x^ {- 1} \. \ X \ = \ 1$ .

The unit $\ mathbb {O} ^ {*}$ of octonions nonnull, provided with the multiplication and of the reverse, commutative magma not is a and not associative.

Division

The division of the octonions $\ X \$ and $\ \$ is then defined there by the following equality:

$\ frac {X} {there} = \ X \. \ y^ {- 1} \ = \ {\|there \|} ^ {- 2} \. \ X \. \ y^ {*}$ , with $\ there \$ different from zero.

Construction of Cayley-Dickson

Following the example Quaternion S compared to the couples of complex numbers (and complex numbers compared to the couples of real numbers), the octonions can be treated in the form of couples of Quaternion S.
The Addition of couples of Quaternion S $(has, \ b)$ and $(C, \ d)$ is defined by:

$(has, \ b) \ + \ (C, \ d) \ = \ (has \ + \ C, \ B \ + \ d)$

The Multiplication of 2 couples of Quaternion S $(has, \ b)$ and $(C, \ d)$ is defined as follows:

$(has, \ b) \. \ (C, \ d) \ = \ (has \. \ C \ - \ D \. \ b^ {*}, \ a^ {*} \. \ D \ + \ C \. \ b)$

where $\ z^ {*} \$ is the Conjugué of the Quaternion $\ Z \$ .
The Multiplication of a real number $\ has \$ by a couple of Quaternion S $(C, \ d)$ is defined by:

$a\ .\ (C, \ d) \ = \ (has, \ 0) \. \ (C, \ d)$ , from where

$a \. \ (C, \ d) \ = \ (has \. \ C, \ has \. \ d)$

One can then define the algebra of the couples of Quaternion S by the unit $\ mathbb {H} ^2$ of the linear combinations in real coefficients of the unit couples of Quaternion S following:

$(1, \ 0) \; \ (I, \ 0) \; \ (J, \ 0) \; \ (K, \ 0) \;$

$(0, \ 1) \; \ (0, \ I) \; \ (0, \ J) \; \ (0, \ K)$ .

This unit, provided with the operations above forms an algebra with 2 Dimension S on the whole of the Quaternion S, and with 8 Dimension S on the whole of the real numbers.
Either $\ I_0 \$ the invertible operation which associates with all Quaternion real coordinates $(has, \ B, \ C, \ d)$ the octonion of same coordinates in the subalgebra generated by octonions unit ${\ 1, \ I, \ J, \ K \}$ .
It is shown easily that the following operation $I$ , which associates any couple of Quaternion S $(C, \ d)$ of $\ mathbb {H} ^2$ with a octonion of $\ mathbb {O}$ such as:

$I (C, \ d) \ = \ I_0 (c) \ + \ I_0 (d) \. \ l$ is bijective.

It follows that $\ mathbb {H} ^2$ is isomorphous with $\ mathbb {O}$ .
It is shown whereas the additions and multiplications of octonions $\ o_1 \$ and $\ o_2 \$ in $\ mathbb {O}$ is equivalent to the operations above of couples of quaternions $(a_1, \ b_1)$ and $(a_2, \ b_2)$ in $\ mathbb {H} ^2$ :

$I^ {- 1} (I (a_1, \ b_1) \ + \ I (a_2, \ b_2))\ = \ (a_1, \ b_1) \ + \ (a_2, \ b_2)$ ,

$I^ {- 1} (I (a_1, \ b_1) \. \ I (a_2, \ b_2))\ = \ (a_1, \ b_1) \. \ (a_2, \ b_2)$ ,

$I (I^ {- 1} (o_1) \ + \ I^ {- 1} (o_2))\ = \ o_1 \ + \ o_2$ ,

$I (I^ {- 1} (o_1) \. \ I^ {- 1} (o_2))\ = \ o_1 \. \ o_2$ .

Consequently, one will be able simply to define the octonions by means of couples of Quaternion S, by including the quaternions in the whole of octonions provided with the operations of the construction of Cayley-Dickinson and the following equalities:

$(1, \ 0) \ = \ 1 \; \ (I, \ 0) \ = \ I \; \ (J, \ 0) \ = \ J \; \ (K, \ 0) \ = \ K \;$

$(0, \ 1) \ = \ L \; \ (0, \ I) \ = \ it \; \ (0, \ J) \ = \ jl \; \ (0, \ K) \ = \ kl.$

(in this case, the Isomorphisme $I$ above which becomes a simple identity.)

Properties

The multiplication of the octonions is not nor commutative :

$i\ .\ J \ = \ - \ J \. \ i$

nor associative :

$(I \. \ J) \. \ L \ = \ - I \. \ (J \. \ L)$ .

It satisfies a form weaker than the Associativité: the Alternativité. That means that under Algèbre generated by 2 unspecified elements $(has, \ b)$ is associative:

$(has \. \ b) \. \ B \ = \ has \. \ (B \. \ b)$ .

One can show that under Algèbre generated by 2 unspecified elements of $\ mathbb {O}$ is isomorphous with $\ mathbb {R}$ , $\ mathbb {C}$ , or $\ mathbb {H}$ , which is all associative.
The octonions share an important property with $\ mathbb {R}$ , $\ mathbb {C}$ , and $\ mathbb {H}$ : the Standard on $\ mathbb {O}$ which satisfies

$\|X \. \ there \|\ = \ \|X \|\. \ \|there \|$

That implies that the octonions form a associative division normalized not . The Algebra S of higher Dimension S defined by the construction of Cayley - Dickson (for example the Sédénion S) does not satisfy this property: they have all of the Diviseurs of zero and their multiplications do not satisfy any more the conservation of the standards.
It proves that the only algebras of division normalized on the real S are $\ mathbb {R}$ , $\ mathbb {C}$ , $\ mathbb {H}$ and $\ mathbb {O}$ . These 4 algebras form also the only alternative algebras of division, of Dimension finished on the real S.
The Multiplication of the octonions not being associative, the elements of $\ mathbb {O}$ distinct from zero do not form a algebraic group, nor a body or a ring. They form a Quasigroupe or additive group .

Automorphisms

A Automorphisme $\ has \$ octonions is a linear Transformation invertible of $\ mathbb {O}$ on itself which checks

$A (X \. \ there) \ = \ has (X) \. \ Has (there)$ .

The whole of the automorphisms of $\ mathbb {O}$ form a group noted $\ \ mathbb {G} _2 \$ . The group $\ \ mathbb {G} _2 \$ is a real Groupe of Dregs Simplement related and compact, of Dimension 14. This group is smallest of the 5 exceptional groups of Dregs.

Particular subalgebras

It is checked easily that all the operations in the subalgebra of the octonions of which the imaginary part is null are equivalent to the operations in the algebra of the real Of the same S. the subalgebra of the octonions of which all real dimensions except the 2 first are null is equivalent to the algebra of the complex . In the same way the subalgebra of the octonions of which all real dimensions except the 4 first are null is equivalent to the algebra of the Quaternion S.
Consequently one will identify the numbers real S, complex S and Quaternion S as of the octonions particular, than one will note in the same way: $\ mathbb {R} \ \ subset \ \ mathbb {C} \ \ subset \ \ mathbb {H} \ \ subset \ \ mathbb {O}$ .

Dependant subjects

Adolf Hurwitz

Number hypercomplexe

Quaternion S

Biquaternion S

Sédénion S

External bonds and references

The Octonions - year article by John C. Baez

Octonion Fractals - fractals generated using octonion mathematics

OCTONIONS: Dictionary of the numbers.

Zh-classical: 八元數
Random links: Saint-bridge | Eugene Bethmont | Rio Biguaçu | Melissa Mazzotta | List people born in Aix-en-Provence | Comté_de_Crawford,_la_Géorgie