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Split-octonion

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In mathematics, the split-octonions are a nonassociative extension of the quaternions (or the split-quaternions). They differ from the octonions in the signature of quadratic form: the split-octonions have a split-signature (4,4) whereas the octonions have a positive-definite signature (8,0).

Contents

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 + ℓb. The product is defined by the rule:

[(a + \ell b)(c + \ell d) = (ac + \lambda d\bar b) + \ell(\bar 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 the split-quaternions. Here either choice of λ (±1) gives the split-octonions. See also split-complex numbers in general.

Multiplication table

A basis for the split-octonions is given by the set . 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 xa. By linearity, multiplication of split-octonions is completely determined by the following multiplication table:

[1\,] [i\,] [j\,] [k\,] [\ell\,] [\ell i\,] [\ell j\,] [\ell k\,]
[i\,] [-1\,] [k\,] [-j\,] [-\ell i\,] [\ell\,] [-\ell k\,] [\ell j\,]
[j\,] [-k\,] [-1\,] [i\,] [-\ell j\,] [\ell k\,] [\ell\,] [-\ell i\,]
[k\,] [j\,] [-i\,] [-1\,] [-\ell k\,] [-\ell j\,] [\ell i\,] [\ell\,]
[\ell\,] [\ell i\,] [\ell j\,] [\ell k\,] [1\,] [i\,] [j\,] [k\,]
[\ell i\,] [-\ell\,] [-\ell k\,] [\ell j\,] [-i\,] [1\,] [k\,] [-j\,]
[\ell j\,] [\ell k\,] [-\ell\,] [-\ell i\,] [-j\,] [-k\,] [1\,] [i\,]
[\ell k\,] [-\ell j\,] [\ell i\,] [-\ell\,] [-k\,] [j\,] [-i\,] [1\,]

Conjuage, norm and inverse

The conjugate of a split-octonion x is given by

[\bar 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) = \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 norm is the standard pseudo-Euclidean norm on R4,4. 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^ = \frac.]

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 the Moufang identities and so form an alternative algebra. Therefore, by Artin'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 a Moufang loop.

Hyperbolic Octonions

Split-octonions are computationally equivalent to hyperbolic octonions from the hypernumbers program. The above case of [\lambda = \ell^2 = +1] corresponds to [\epsilon] arithmetic, which is denoted the third hypernumber level (out of 10; with the first level being real numbers, and imaginary numbers the second).

Split-Octonions in Physics

Split-octonions are used in the description of physical law, e.g. in String theory. Also, 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 (see references below).

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

[\begina & \mathbf v\\ \mathbf w & b\end]
where a and b are real numbers and v and w and vectors in R3. Define multiplication of these matrices by the rule
[\begina & \mathbf v\\ \mathbf w & b\end \begina' & \mathbf v'\\ \mathbf w' & b'\end = \beginaa' + \mathbf v\cdot\mathbf w' & a\mathbf v' + b'\mathbf v + \mathbf w \times \mathbf w'\\ a'\mathbf w + b\mathbf w' - \mathbf v\times\mathbf v' & bb' + \mathbf v'\cdot\mathbf w \end]
where · and × are the ordinary dot product and cross 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

[\det\begina & \mathbf v\\ \mathbf w & b\end = ab - \mathbf v\cdot\mathbf w].
This determinant is a quadratic form on the Zorn's algebra which satisfies the composition rule:
[\det(AB) = \det(A)\det(B).\,]
AZorn'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 R3. The isomorphism from the split-octonions to the Zorn's algebra is given by
[x\mapsto \phi(x) = \begina + b & \mathbf a + \mathbf b \\ -\mathbf a + \mathbf b & a - b\end.]
This isomorphism preserves the norm since [N(x) = \det(\phi(x))].

References

For physics on native split-octonion arithmetic see e.g.

 

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