Split-octonion

In

Up to isomorphism, the octonions and the split-octonions are the only two 8-dimensional composition algebras over the real numbers. They are also the only two octonion algebras over the real numbers. Split-octonion algebras analogous to the split-octonions can be defined over any field.

Definition

Cayley–Dickson construction

The octonions and the split-octonions can be obtained from the

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.

Multiplication table

A basis for the split-octonions is given by the set ${\displaystyle \{\ 1,\ i,\ j,\ k,\ \ell ,\ \ell i,\ \ell j,\ \ell k\ \}}$.

Every split-octonion ${\displaystyle x}$ can be written as a linear combination of the basis elements,

${\displaystyle x=x_{0}+x_{1}\,i+x_{2}\,j+x_{3}\,k+x_{4}\,\ell +x_{5}\,\ell i+x_{6}\,\ell j+x_{7}\,\ell k,}$

with real coefficients ${\displaystyle x_{a}}$.

By linearity, multiplication of split-octonions is completely determined by the following multiplication table:

		multiplier
		${\displaystyle 1}$	${\displaystyle i}$	${\displaystyle j}$	${\displaystyle k}$	${\displaystyle \ell }$	${\displaystyle \ell i}$	${\displaystyle \ell j}$	${\displaystyle \ell k}$
multiplicand	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle i}$	${\displaystyle j}$	${\displaystyle k}$	${\displaystyle \ell }$	${\displaystyle \ell i}$	${\displaystyle \ell j}$	${\displaystyle \ell k}$
	${\displaystyle i}$	${\displaystyle i}$	${\displaystyle -1}$	${\displaystyle k}$	${\displaystyle -j}$	${\displaystyle -\ell i}$	${\displaystyle \ell }$	${\displaystyle -\ell k}$	${\displaystyle \ell j}$
	${\displaystyle j}$	${\displaystyle j}$	${\displaystyle -k}$	${\displaystyle -1}$	${\displaystyle i}$	${\displaystyle -\ell j}$	${\displaystyle \ell k}$	${\displaystyle \ell }$	${\displaystyle -\ell i}$
	${\displaystyle k}$	${\displaystyle k}$	${\displaystyle j}$	${\displaystyle -i}$	${\displaystyle -1}$	${\displaystyle -\ell k}$	${\displaystyle -\ell j}$	${\displaystyle \ell i}$	${\displaystyle \ell }$
	${\displaystyle \ell }$	${\displaystyle \ell }$	${\displaystyle \ell i}$	${\displaystyle \ell j}$	${\displaystyle \ell k}$	${\displaystyle 1}$	${\displaystyle i}$	${\displaystyle j}$	${\displaystyle k}$
	${\displaystyle \ell i}$	${\displaystyle \ell i}$	${\displaystyle -\ell }$	${\displaystyle -\ell k}$	${\displaystyle \ell j}$	${\displaystyle -i}$	${\displaystyle 1}$	${\displaystyle k}$	${\displaystyle -j}$
	${\displaystyle \ell j}$	${\displaystyle \ell j}$	${\displaystyle \ell k}$	${\displaystyle -\ell }$	${\displaystyle -\ell i}$	${\displaystyle -j}$	${\displaystyle -k}$	${\displaystyle 1}$	${\displaystyle i}$
	${\displaystyle \ell k}$	${\displaystyle \ell k}$	${\displaystyle -\ell j}$	${\displaystyle \ell i}$	${\displaystyle -\ell }$	${\displaystyle -k}$	${\displaystyle j}$	${\displaystyle -i}$	${\displaystyle 1}$

A convenient mnemonic is given by the diagram at the right, which represents the multiplication table for the split-octonions. This one is derived from its parent octonion (one of 480 possible), which is defined by:

${\displaystyle e_{i}e_{j}=-\delta _{ij}e_{0}+\varepsilon _{ijk}e_{k},\,}$

where ${\displaystyle \delta _{ij}}$ is the Kronecker delta and ${\displaystyle \varepsilon _{ijk}}$ is the Levi-Civita symbol with value ${\displaystyle +1}$ when ${\displaystyle ijk=123,154,176,264,257,374,365,}$ and:

${\displaystyle e_{i}e_{0}=e_{0}e_{i}=e_{i};\,\,\,\,e_{0}e_{0}=e_{0},}$

with ${\displaystyle e_{0}}$ the scalar element, and ${\displaystyle i,j,k=1...7.}$

The red arrows indicate possible direction reversals imposed by negating the lower right quadrant of the parent creating a split octonion with this multiplication table.

Conjugate, norm and inverse

The conjugate of a split-octonion x is given by

${\displaystyle {\bar {x}}=x_{0}-x_{1}\,i-x_{2}\,j-x_{3}\,k-x_{4}\,\ell -x_{5}\,\ell i-x_{6}\,\ell j-x_{7}\,\ell k,}$

just as for the octonions.

The quadratic form on x is given by

${\displaystyle N(x)={\bar {x}}x=(x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2})-(x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}).}$

This quadratic form N(x) is an isotropic quadratic form since there are non-zero split-octonions x with N(x) = 0. With N, the split-octonions form a pseudo-Euclidean space of eight dimensions over R, sometimes written R^4,4 to denote the signature of the quadratic form.

If N(x) ≠ 0, then x has a (two-sided) multiplicative inverse x⁻¹ given by

${\displaystyle x^{-1}=N(x)^{-1}{\bar {x}}.}$

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,

${\displaystyle N(xy)=N(x)N(y).}$

The split-octonions satisfy the

The automorphism group of the split-octonions is a 14-dimensional Lie group, the split real form of the exceptional simple Lie group G₂.

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.^[2] Specifically, define a vector-matrix to be a 2×2 matrix of the form^[3][4][5][6]

${\displaystyle {\begin{bmatrix}a&\mathbf {v} \\\mathbf {w} &b\end{bmatrix}},}$

where a and b are real numbers and v and w are vectors in R³. Define multiplication of these matrices by the rule

${\displaystyle {\begin{bmatrix}a&\mathbf {v} \\\mathbf {w} &b\end{bmatrix}}{\begin{bmatrix}a'&\mathbf {v} '\\\mathbf {w} '&b'\end{bmatrix}}={\begin{bmatrix}aa'+\mathbf {v} \cdot \mathbf {w} '&a\mathbf {v} '+b'\mathbf {v} +\mathbf {w} \times \mathbf {w} '\\a'\mathbf {w} +b\mathbf {w} '-\mathbf {v} \times \mathbf {v} '&bb'+\mathbf {v} '\cdot \mathbf {w} \end{bmatrix}}}$

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

${\displaystyle \det {\begin{bmatrix}a&\mathbf {v} \\\mathbf {w} &b\end{bmatrix}}=ab-\mathbf {v} \cdot \mathbf {w} }$.

This determinant is a quadratic form on Zorn's algebra which satisfies the composition rule:

${\displaystyle \det(AB)=\det(A)\det(B).\,}$

Zorn's vector-matrix algebra is, in fact, isomorphic to the algebra of split-octonions. Write an octonion ${\displaystyle x}$ in the form

${\displaystyle x=(a+\mathbf {v} )+\ell (b+\mathbf {w} )}$

where ${\displaystyle a}$ and ${\displaystyle b}$ are real numbers and v and w are pure imaginary quaternions regarded as vectors in R³. The isomorphism from the split-octonions to Zorn's algebra is given by

${\displaystyle x\mapsto \phi (x)={\begin{bmatrix}a+b&\mathbf {v} +\mathbf {w} \\-\mathbf {v} +\mathbf {w} &a-b\end{bmatrix}}.}$

This isomorphism preserves the norm since ${\displaystyle N(x)=\det(\phi (x))}$.

Applications

Split-octonions are used in the description of physical law. For example:

• 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.^[7]
• Supersymmetric quantum mechanics has an octonionic extension.^[8]
• The Zorn-based split-octonion algebra can be used in modeling local gauge symmetric SU(3) quantum chromodynamics.^[9]
• The problem of a ball rolling without slipping on a ball of radius 3 times as large has the split real form of the exceptional group G₂ as its symmetry group, owing to the fact that this problem can be described using split-octonions.^[10]

References