Sedenion

In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers; they are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic to a subalgebra of the sedenions. Unlike the octonions, the sedenions are not an alternative algebra. Applying the Cayley–Dickson construction to the sedenions yields a 32-dimensional algebra, sometimes called the ''32-ions'' or ''trigintaduonions''. It is possible to continue applying the Cayley–Dickson construction arbitrarily many times.

The term ''sedenion'' is also used for other 16-dimensional algebraic structures, such as a tensor product of two copies of the biquaternions, or the algebra of 4 × 4 matrices over the real numbers, or that studied by .

Arithmetic

Like octonions, multiplication of sedenions is neither commutative nor associative.
But in contrast to the octonions, the sedenions do not even have the property of being alternative.
They do, however, have the property of power associativity, which can be stated as that, for any element x of $\mathbb$, the power $x^n$ is well defined. They are also flexible.

Every sedenion is a linear combination of the unit sedenions $e_0$, $e_1$, $e_2$, $e_3$, ..., $e_$,
which form a basis of the vector space of sedenions. Every sedenion can be represented in the form

:$x = x_0 e_0 + x_1 e_1 + x_2 e_2 + \cdots + x_ e_ + x_ e_.$

Addition and subtraction are defined by the addition and subtraction of corresponding coefficients and multiplication is distributive over addition.

Like other algebras based on the Cayley–Dickson construction, the sedenions contain the algebra they were constructed from. So, they contain the octonions (generated by $e_0$ to $e_7$ in the table below), and therefore also the quaternions (generated by $e_0$ to $e_3$), complex numbers (generated by $e_0$ and $e_1$) and real numbers (generated by $e_0$).

The sedenions have a multiplicative identity element $e_0$ and multiplicative inverses, but they are not a division algebra because they have zero divisors. This means that two non-zero sedenions can be multiplied to obtain zero: an example is $(e_3 + e_)(e_6 - e_)$. All hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.

A sedenion multiplication table is shown below:

Sedenion properties

From the above table, we can see that:

:$e_0e_i = e_ie_0 = e_i \, \text \, i,$

:$e_ie_i = -e_0 \,\, \text\,\, i \neq 0,$ and

:$e_ie_j = -e_je_i \,\, \text\,\, i \neq j \,\,\text\,\, i,j \neq 0.$

Anti-associative

The sedenions are not fully anti-associative. Choose any four generators, $i,j,k$ and $l$. The following 5-cycle shows that these five relations cannot all be anti-associative.

$(ij)(kl) = -((ij)k)l = (i(jk))l = -i((jk)l) = i(j(kl)) = -(ij)(kl) = 0$

In particular, in the table above, using $e_1,e_2,e_4$ and $e_8$ the last expression associates.
$(e_1e_2)e_ = e_1(e_2e_) = -e_$

Quaternionic subalgebras

The 35 triads that make up this specific sedenion multiplication table with the 7 triads of the octonions used in creating the sedenion through the Cayley–Dickson construction shown in bold:

The binary representations of the indices of these triples bitwise XOR to 0.

The list of 84 sets of zero divisors $\$, where
$(e_a + e_b) \circ (e_c + e_d) = 0$:

$\begin \text \quad \ \\ \text ~ (e_a + e_b) \circ (e_c + e_d) = 0 \\ \begin 1 \leq a \leq 6, & c > a, & 9 \leq b \leq 15 \\ 9 \leq c \leq 15 & & -9 \geq d \geq -15 \end \\ \begin \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \\ \ & \ \end \end$

Applications

showed that the space of pairs of norm-one sedenions that multiply to zero is homeomorphic to the compact form of the exceptional Lie group G₂. (Note that in his paper, a "zero divisor" means a ''pair'' of elements that multiply to zero.)

Sedenion neural networks provide a means of efficient and compact expression in machine learning applications and have been used in solving multiple time-series and traffic forecasting problems.

See also

* Hypercomplex number
* Pfister's sixteen-square identity
* Split-complex number

Notes

References

*
*
*
*
*
*
*
*L. S. Saoud and H. Al-Marzouqi, "Metacognitive Sedenion-Valued Neural Network and its Learning Algorithm," in IEEE Access, vol. 8, pp. 144823-144838, 2020, doi
10.1109/ACCESS.2020.3014690

{{Number systems

Hypercomplex numbers
Non-associative algebras