- Split-octonion
In

mathematics , the**split-octonions**are anonassociative extension of thequaternion s (or thesplit-quaternion s). They differ from theoctonion s in the signature ofquadratic form : the split-octonions have a split-signature (4,4) whereas the octonions have a positive-definite signature (8,0).The split-octonions form the unique

split octonion algebra over the real numbers. There are corresponding algebras over any field "F".**Definition****Cayley-Dickson construction**The octonions and the split-octonions can be obtained from the

Cayley-Dickson construction by defining a multiplication on pairs of quaternions. We introduce a new imaginary unit ℓ and write a pair of quaterions ("a", "b") in the form "a" + ℓ"b". The product is defined by the rule::$(a\; +\; ell\; b)(c\; +\; ell\; d)\; =\; (ac\; +\; lambda\; dar\; b)\; +\; ell(ar\; a\; d\; +\; c\; b)$where:$lambda\; =\; ell^2.$If λ is chosen to be −1, we get the octonions. If, instead, it is taken to be +1 we get the split-octonions. One can also obtain the split-octonions via a Cayley-Dickson doubling of thesplit-quaternion s. Here either choice of λ (±1) gives the split-octonions. See alsosplit-complex numbers in general.**Multiplication table**A basis for the split-octonions is given by the set {1, "i", "j", "k", ℓ, ℓ"i", ℓ"j", ℓ"k"}. Every split-octonion "x" can be written as a

linear combination of the basis elements,:$x\; =\; x\_0\; +\; x\_1,i\; +\; x\_2,j\; +\; x\_3,k\; +\; x\_4,ell\; +\; x\_5,ell\; i\; +\; x\_6,ell\; j\; +\; x\_7,ell\; k,$with real coefficients "x"_{"a"}. By linearity, multiplication of split-octonions is completely determined by the followingmultiplication table :**Conjugate, norm and inverse**The "conjugate" of a split-octonion "x" is given by:$ar\; x\; =\; x\_0\; -\; x\_1,i\; -\; x\_2,j\; -\; x\_3,k\; -\; x\_4,ell\; -\; x\_5,ell\; i\; -\; x\_6,ell\; j\; -\; x\_7,ell\; k$just as for the octonions. The

quadratic form (or "square norm") on "x" is given by:$N(x)\; =\; ar\; x\; x\; =\; (x\_0^2\; +\; x\_1^2\; +\; x\_2^2\; +\; x\_3^2)\; -\; (x\_4^2\; +\; x\_5^2\; +\; x\_6^2\; +\; x\_7^2)$This norm is the standard pseudo-Euclidean norm on**R**^{4,4}. Due to the split signature the norm "N" is isotropic, meaning there are nonzero "x" for which "N"("x") = 0. An element "x" has an (two-sided) inverse "x"^{−1}if and only if "N"("x") ≠ 0. In this case the inverse is given by:$x^\{-1\}\; =\; frac\{ar\; x\}\{N(x)\}.$**Properties**The split-octonions, like the octonions, are noncommutative and nonassociative. Also like the octonions, they form a

composition algebra since the quadratic form "N" is multiplicative. That is,:$N(xy)\; =\; N(x)N(y).,$The split-octonions satisfy theMoufang identities and so form analternative algebra . Therefore, byArtin's theorem , the subalgebra generated by any two elements is associative. The set of all invertible elements (i.e. those elements for which "N"("x") ≠ 0) form aMoufang loop .**Zorn's vector-matrix algebra**Since the split-octonions are nonassociative they cannot be represented by ordinary matrices (matrix multiplication is always associative). Zorn found a way to represent them as "matrices" containing both scalars and vectors using a modified version of matrix multiplication. Specifically, define a "vector-matrix" to be a 2×2 matrix of the form:$egin\{bmatrix\}a\; mathbf\; v\backslash \; mathbf\; w\; bend\{bmatrix\}$where "a" and "b" are real numbers and

**v**and**w**and vectors in**R**^{3}. Define multiplication of these matrices by the rule:$egin\{bmatrix\}a\; mathbf\; v\backslash \; mathbf\; w\; bend\{bmatrix\}\; egin\{bmatrix\}a\text{'}\; mathbf\; v\text{'}\backslash \; mathbf\; w\text{'}\; b\text{'}end\{bmatrix\}\; =\; egin\{bmatrix\}aa\text{'}\; +\; mathbf\; vcdotmathbf\; w\text{'}\; amathbf\; v\text{'}\; +\; b\text{'}mathbf\; v\; +\; mathbf\; w\; imes\; mathbf\; w\text{'}\backslash \; a\text{'}mathbf\; w\; +\; bmathbf\; w\text{'}\; -\; mathbf\; v\; imesmathbf\; v\text{'}\; bb\text{'}\; +\; mathbf\; v\text{'}cdotmathbf\; w\; end\{bmatrix\}$where · and × are the ordinarydot product andcross product of 3-vectors. With addition and scalar multiplication defined as usual the set of all such matrices forms a nonassociative unital 8-dimensional algebra over the reals, called**Zorn's vector-matrix algebra**.Define the "

determinant " of a vector-matrix by the rule:$detegin\{bmatrix\}a\; mathbf\; v\backslash \; mathbf\; w\; bend\{bmatrix\}\; =\; ab\; -\; mathbf\; vcdotmathbf\; w$.This determinant is a quadratic form on the Zorn's algebra which satisfies the composition rule::$det(AB)\; =\; det(A)det(B).,$Zorn's vector-matrix algebra is, in fact, isomorphic to the algebra of split-octonions. Write an octonion "x" in the form:$x\; =\; (a\; +\; mathbf\; a)\; +\; ell(b\; +\; mathbf\; b)$where "a" and "b" are real numbers and

**a**and**b**are pure quaternions regarded as vectors in**R**^{3}. The isomorphism from the split-octonions to the Zorn's algebra is given by:$xmapsto\; phi(x)\; =\; egin\{bmatrix\}a\; +\; b\; mathbf\; a\; +\; mathbf\; b\; \backslash \; -mathbf\; a\; +\; mathbf\; b\; a\; -\; bend\{bmatrix\}.$This isomorphism preserves the norm since $N(x)\; =\; det(phi(x))$.**Applications**Split-octonions are used in the description of physical law. For example, (a) the

Dirac equation in physics (the equation of motion of a free spin 1/2 particle, like e.g. an electron or a proton) can be expressed on native split-octonion arithmetic, (b) the supersymmetric quantum mechanics has an octonionic extension (see references below; split-octonions are isomorphic to hyperbolic octonions fromMusean hypernumber s).**References***cite book

first = F. Reese

last = Harvey

year = 1990

title = Spinors and Calibrations

publisher = Academic Press

location = San Diego

id = ISBN 0-12-329650-1

*cite book

first = T. A.

last = Springer

coauthors = F. D. Veldkamp

year = 2000

title = Octonions, Jordan Algebras and Exceptional Groups

publisher = Springer-Verlag

id = ISBN 3-540-66337-1For physics on native split-octonion arithmetic see e.g.

* M. Gogberashvili, Octonionic Electrodynamics, "J. Phys. A: Math. Gen." 39 (2006) 7099-7104. [

*http://dx.doi.org/10.1088/0305-4470/39/22/020 doi:10.1088/0305-4470/39/22/020*]

* J. Köplinger, Dirac equation on hyperbolic octonions. "Appl. Math. Computation" (2006) [*http://dx.doi.org/10.1016/j.amc.2006.04.005 doi:10.1016/j.amc.2006.04.005*]

* V. Dzhunushaliev, Non-associativity, supersymmetry and hidden variables, "J. Math. Phys." 49, 042108 (2008); doi:10.1063/1.2907868; arXiv:0712.1647 [quant-ph] .

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