Split Octonion

In Mathematical, the octonions split are an associative extension not Quaternion S (or split quaternions). They differ from the Octonion S by the signature of the quadratic Forme: octonions split have a signature of slit (4,4) where octonions them have a positive definite signature (8,0).

Definition

The construction of Cayley-Dickson

Octonions and octonions them split can be obtained by the Construction of Cayley-Dickson by defining a multiplication on the pairs of quaternions. We introduce a new imaginary unit ℓ and we write a pair of quaternions ( has , B ) in the form has + ℓ B . The product is defined by the following rule:

$(has\; +\; \backslash \; ell\; b)\; (C\; +\; \backslash \; ell\; d)\; =\; (ac\; +\; \backslash \; lambda\; D\; \backslash \; bar\; b)\; +\; \backslash \; ell\; (\backslash \; bar\; has\; D\; +\; C\; b)$

where

$\backslash \; lambda\; =\; \backslash \; ell^2.$

If $\backslash \; lambda\; \backslash ,$ is selected equal to - 1, we obtain octonions them. If, in the place, it is selected equal to + 1, we obtain octonions them split. One can also obtain octonions them split via a doubling of Cayley-Dickson of the split quaternions. Here, whatever the choice of $\backslash \; lambda\; \backslash ,$ (±1), that will give octonions them split. See also the split complex numbers in general.

The multiplication table

A bases for octonions split is given by the unit {1, I , J , K , ℓ, ℓ I , ℓ J , ℓ K }. Each octonion split X can be written as a linear Combinaison of the elements of the base,

$x\; =\; x\_0\; +\; x\_1\; \backslash ,\; I\; +\; x\_2\; \backslash ,\; J\; +\; x\_3\; \backslash ,\; K\; +\; x\_4\; \backslash ,\; \backslash \; ell\; +\; x\_5\; \backslash ,\; \backslash \; ell\; I\; +\; x\_6\; \backslash ,\; \backslash \; ell\; J\; +\; x\_7\; \backslash ,\; \backslash \; ell\; K,$

with real coefficients X _has . By linearity, the multiplication of octonions split is completely determined by the following Multiplication table:

Combined, the standard and the reverse

The combined of a octonion split X is given by

$\backslash \; bar\; X\; =\; x\_0\; -\; x\_1\; \backslash ,\; I\; -\; x\_2\; \backslash ,\; J\; -\; x\_3\; \backslash ,\; K\; -\; x\_4\; \backslash ,\; \backslash \; ell\; -\; x\_5\; \backslash ,\; \backslash \; ell\; I\; -\; x\_6\; \backslash ,\; \backslash \; ell\; J\; -\; x\_7\; \backslash ,\; \backslash \; ell\; k$

as for octonions. The quadratic Forme (or square standard ) on X is given by

$N\; (X)\; =\; \backslash \; bar\; X\; X\; =\; (x\_0^2\; +\; x\_1^2\; +\; x\_2^2\; +\; x\_3^2)\; -\; (x\_4^2\; +\; x\_5^2\; +\; x\_6^2\; +\; x\_7^2)$

This standard is the standard pseudo-Euclidean standard on $\backslash \; mathbb\; \{R\}\; ^\; \{4,4\}\; \backslash ,$. Because of the signature of slit, the standard NR is isotropic, which means that there exist elements X different from zero for which NR ( X ) = 0. An element X has an opposite (with two faces) $x^\; \{-\; 1\}\; \backslash ,$ if and only if NR ( X ) ≠ 0. In this case, the reverse is given by

$x^\; \{-\; 1\}\; =\; \backslash \; frac\; \{\backslash \; bar\; X\}\; \{NR\; (X)\}\backslash ,$.

Properties

Octonions split, as octonions them, are not commutative nor associative. As octonions them, also, they form a Algèbre of composition since the quadratic form NR is multiplicative. I.e.,

$N\; (xy)\; =\; NR\; (X)\; NR\; (there)\; \backslash ,$.

Octonions split satisfy the identities of Moufang and thus form a alternative algebra. Consequently, by the Theorem of Artin, the subalgebra generated by two unspecified elements is associative. The whole of all the invertible elements (i.e these elements for which NR ( X ) ≠ 0) form a Boucle of Moufang.

Octonions hyperbolic

Octonions split are in a calculative way, equivalent to octonions hyperbolic starting from the program of the hypernombres. The case above of $\backslash \; lambda\; =\; \backslash \; ell^2\; =\; +1\; \backslash ,$ corresponds to the $\backslash \; epsilon\; \backslash ,$ arithmetic, which is indicated by the third level of the hypernombres (on 10; the first corresponds to the real numbers, and the second with the imaginary numbers).

Octonions split in physics

Octonions split are used in the description of a physical law, e.g in Théorie of the cords. The equation of Dirac in physics (the equation of motion of a free particle of spin 1/2, like an electron or a proton) can be expressed with the arithmetic one of octonions split (see the references below).

Matric-vectorial algebra of Zorn

Since octonions them split are not associative, they cannot be represented by the ordinary matrices (the matric multiplication is always associative). Zorn found a manner of representing them in the form of " matrices" containing at the same time scalars and vectors by using a modified version of the matric multiplication. More precisely, let us define that a matrix-vector is a matrix 2 X 2 of the form

where has and B is real numbers and v and W of the vectors in $\backslash \; mathbb\; \{R\}\; ^3\; \backslash ,$. Let us define the multiplication of these matrices by the following rule

where. is the scalar Produit and X the vector Product ordinary of 3 vectors. With the addition and the definite scalar multiplication as usual in the whole of all the matrices of this kind forms a unit algebra with eight dimensions nonassociative on realities, called matric-vectorial algebra of Zorn .

Let us define the " Determinant " of a matrix vector by the rule

.

This determinant is a quadratic form of the algebra of Zorn which satisfies the law of composition:

$\backslash \; det\; (AB)\; =\; \backslash \; det\; (A)\; \backslash \; det\; (B)\; \backslash ,$.

The matric-vectorial algebra of Zorn is, in fact, isomorph with the algebra of octonions split. Let us write a octonion X in the form

$x\; =\; (has\; +\; \backslash \; mathbf\; a)\; +\; \backslash \; ell\; (B\; +\; \backslash \; mathbf\; b)\; \backslash ,$

where has and B is real numbers, has and B is pure quaternions which are seen like vectors in $\backslash \; mathbb\; \{R\}\; ^3\; \backslash ,$. The isomorphism of octonions split towards the algebra of Zorn is given by

.

This isomorphism preserves the standard since $N\; (X)\; =\; \backslash \; det\; (\backslash \; phi\; (X))\backslash ,$.

References

For physics on the arithmetic one of octonions split, to see

• Mr. Gogberashvili, Octonionic Electrodynamics, J. Phys. a: Maths. Gen. 39 (2006) 7099-7104. DOI: 10.1088/0305-4470/39/22/020

• J. Köplinger, Dirac equation one hyperbolic octonions. Appl. Maths. Computation (2006) DOI: 10.1016/j.amc.2006.04.005

Random links:

Codificación del carácter | November 15th | Mauricio Merleau-Ponty | Saint-Ouen-of-double sacks | Manikongo | 1854 in science | Bill Tamplin | Université_de_Paul_Quinn

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org