sedenion

{{Infobox number system

| official_name = Sedenions

| symbol = $\mathbb S$

| units = e₀, ..., e₁₅

| identity = e₀

| properties = {{Plainlist|

}}

In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers, usually represented by the capital letter S, boldface {{math|S}} or blackboard bold $\mathbb S$ .

The sedenions are obtained by applying the Cayley–Dickson construction to the octonions, which can be mathematically expressed as $\mathbb{S}=\mathcal{CD}(\mathbb{O},1)$ .{{cite web|url=https://mathsci.kaist.ac.kr/~tambour/fichiers/publications/Ensembles_de_nombres.pdf|date=6 September 2011|title=Ensembles de nombre|publisher=Forum Futura-Science|access-date=11 October 2024|language=fr}} 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, called the trigintaduonions or sometimes the 32-nions.Raoul E. Cawagas, et al. (2009). [https://arxiv.org/abs/0907.2047 "THE BASIC SUBALGEBRA STRUCTURE OF THE CAYLEY-DICKSON ALGEBRA OF DIMENSION 32 (TRIGINTADUONIONS)"].

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 {{harvtxt|Smith |1995}}.

Arithmetic

File:Sedenion-Fano Tesseract.gif,{{Harv|Baez|2002|loc=p. 6}} showing the 35 triads as hyperplanes through the real $(e_0)$ vertex of the sedenion example given]]

Every sedenion is a linear combination of the unit sedenions $e_0$ , $e_1$ , $e_2$ , $e_3$ , ..., $e_{15}$ ,

which form a basis of the vector space of sedenions. Every sedenion can be represented in the form

: $x = x_0 e_0 + x_1 e_1 + x_2 e_2 + \cdots + x_{14} e_{14} + x_{15} e_{15}.$

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 $e_0$ ).

= Multiplication =

Like octonions, multiplication of sedenions is neither commutative nor associative. However, 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 $\mathbb{S}$ , the power $x^n$ is well defined. They are also flexible.

The sedenions have a multiplicative identity element $e_0$ and multiplicative inverses, but they are not a division algebra because they have zero divisors: two nonzero sedenions can be multiplied to obtain zero, for example $(e_3 + e_{10})(e_6 - e_{15})$ . All hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.

The sedenion multiplication table is shown below:

class="wikitable" style="margin:1em auto; text-align: center;" !colspan="2" rowspan="2"\| $e_ie_j$ !colspan="16" \| $e_j$
$e_0$ ! $e_1$ ! $e_2$ ! $e_3$ ! $e_4$ ! $e_5$ ! $e_6$ ! $e_7$ ! $e_8$ ! $e_9$ ! $e_{10}$ ! $e_{11}$ ! $e_{12}$ ! $e_{13}$ ! $e_{14}$ ! $e_{15}$
! rowspan="16" \| $e_i$ ! width="30pt" \| $e_0$ \| width="30pt" \| $e_0$ \| width="30pt" \| $e_1$ \| width="30pt" \| $e_2$ \| width="30pt" \| $e_3$ \| width="30pt" \| $e_4$ \| width="30pt" \| $e_5$ \| width="30pt" \| $e_6$ \| width="30pt" \| $e_7$ \| width="30pt" \| $e_8$ \| width="30pt" \| $e_9$ \| width="30pt" \| $e_{10}$ \| width="30pt" \| $e_{11}$ \| width="30pt" \| $e_{12}$ \| width="30pt" \| $e_{13}$ \| width="30pt" \| $e_{14}$ \| width="30pt" \| $e_{15}$
$e_1$ \| $e_1$ \| $-e_0$ \| $e_3$ \| $-e_2$ \| $e_5$ \| $-e_4$ \| $-e_7$ \| $e_6$ \| $e_9$ \| $-e_8$ \| $-e_{11}$ \| $e_{10}$ \| $-e_{13}$ \| $e_{12}$ \| $e_{15}$ \| $-e_{14}$
$e_2$ \| $e_2$ \| $-e_3$ \| $-e_0$ \| $e_1$ \| $e_6$ \| $e_7$ \| $-e_4$ \| $-e_5$ \| $e_{10}$ \| $e_{11}$ \| $-e_8$ \| $-e_9$ \| $-e_{14}$ \| $-e_{15}$ \| $e_{12}$ \| $e_{13}$
$e_3$ \| $e_3$ \| $e_2$ \| $-e_1$ \| $-e_0$ \| $e_7$ \| $-e_6$ \| $e_5$ \| $-e_4$ \| $e_{11}$ \| $-e_{10}$ \| $e_9$ \| $-e_8$ \| $-e_{15}$ \| $e_{14}$ \| $-e_{13}$ \| $e_{12}$
$e_4$ \| $e_4$ \| $-e_5$ \| $-e_6$ \| $-e_7$ \| $-e_0$ \| $e_1$ \| $e_2$ \| $e_3$ \| $e_{12}$ \| $e_{13}$ \| $e_{14}$ \| $e_{15}$ \| $-e_8$ \| $-e_9$ \| $-e_{10}$ \| $-e_{11}$
$e_5$ \| $e_5$ \| $e_4$ \| $-e_7$ \| $e_6$ \| $-e_1$ \| $-e_0$ \| $-e_3$ \| $e_2$ \| $e_{13}$ \| $-e_{12}$ \| $e_{15}$ \| $-e_{14}$ \| $e_9$ \| $-e_8$ \| $e_{11}$ \| $-e_{10}$
$e_6$ \| $e_6$ \| $e_7$ \| $e_4$ \| $-e_5$ \| $-e_2$ \| $e_3$ \| $-e_0$ \| $-e_1$ \| $e_{14}$ \| $-e_{15}$ \| $-e_{12}$ \| $e_{13}$ \| $e_{10}$ \| $-e_{11}$ \| $-e_8$ \| $e_9$
$e_7$ \| $e_7$ \| $-e_6$ \| $e_5$ \| $e_4$ \| $-e_3$ \| $-e_2$ \| $e_1$ \| $-e_0$ \| $e_{15}$ \| $e_{14}$ \| $-e_{13}$ \| $-e_{12}$ \| $e_{11}$ \| $e_{10}$ \| $-e_9$ \| $-e_8$
$e_8$ \| $e_8$ \| $-e_9$ \| $-e_{10}$ \| $-e_{11}$ \| $-e_{12}$ \| $-e_{13}$ \| $-e_{14}$ \| $-e_{15}$ \| $-e_0$ \| $e_1$ \| $e_2$ \| $e_3$ \| $e_4$ \| $e_5$ \| $e_6$ \| $e_7$
$e_9$ \| $e_9$ \| $e_8$ \| $-e_{11}$ \| $e_{10}$ \| $-e_{13}$ \| $e_{12}$ \| $e_{15}$ \| $-e_{14}$ \| $-e_1$ \| $-e_0$ \| $-e_3$ \| $e_2$ \| $-e_5$ \| $e_4$ \| $e_7$ \| $-e_6$
$e_{10}$ \| $e_{10}$ \| $e_{11}$ \| $e_8$ \| $-e_9$ \| $-e_{14}$ \| $-e_{15}$ \| $e_{12}$ \| $e_{13}$ \| $-e_2$ \| $e_3$ \| $-e_0$ \| $-e_1$ \| $-e_6$ \| $-e_7$ \| $e_4$ \| $e_5$
$e_{11}$ \| $e_{11}$ \| $-e_{10}$ \| $e_9$ \| $e_8$ \| $-e_{15}$ \| $e_{14}$ \| $-e_{13}$ \| $e_{12}$ \| $-e_3$ \| $-e_2$ \| $e_1$ \| $-e_0$ \| $-e_7$ \| $e_6$ \| $-e_5$ \| $e_4$
$e_{12}$ \| $e_{12}$ \| $e_{13}$ \| $e_{14}$ \| $e_{15}$ \| $e_8$ \| $-e_9$ \| $-e_{10}$ \| $-e_{11}$ \| $-e_4$ \| $e_5$ \| $e_6$ \| $e_7$ \| $-e_0$ \| $-e_1$ \| $-e_2$ \| $-e_3$
$e_{13}$ \| $e_{13}$ \| $-e_{12}$ \| $e_{15}$ \| $-e_{14}$ \| $e_9$ \| $e_8$ \| $e_{11}$ \| $-e_{10}$ \| $-e_5$ \| $-e_4$ \| $e_7$ \| $-e_6$ \| $e_1$ \| $-e_0$ \| $e_3$ \| $-e_2$
$e_{14}$ \| $e_{14}$ \| $-e_{15}$ \| $-e_{12}$ \| $e_{13}$ \| $e_{10}$ \| $-e_{11}$ \| $e_8$ \| $e_9$ \| $-e_6$ \| $-e_7$ \| $-e_4$ \| $e_5$ \| $e_2$ \| $-e_3$ \| $-e_0$ \| $e_1$
$e_{15}$ \| $e_{15}$ \| $e_{14}$ \| $-e_{13}$ \| $-e_{12}$ \| $e_{11}$ \| $e_{10}$ \| $-e_9$ \| $e_8$ \| $-e_7$ \| $e_6$ \| $-e_5$ \| $-e_4$ \| $e_3$ \| $e_2$ \| $-e_1$ \| $-e_0$

class="wikitable" style="margin:1em auto; text-align: center;"

!colspan="2" rowspan="2"| $e_ie_j$

!colspan="16" | $e_j$

e_0

! $e_1$

! $e_2$

! $e_3$

! $e_4$

! $e_5$

! $e_6$

! $e_7$

! $e_8$

! $e_9$

! $e_{10}$

! $e_{11}$

! $e_{12}$

! $e_{13}$

! $e_{14}$

! $e_{15}$

! rowspan="16" | $e_i$

! width="30pt" | $e_0$

| width="30pt" | $e_0$

| width="30pt" | $e_1$

| width="30pt" | $e_2$

| width="30pt" | $e_3$

| width="30pt" | $e_4$

| width="30pt" | $e_5$

| width="30pt" | $e_6$

| width="30pt" | $e_7$

| width="30pt" | $e_8$

| width="30pt" | $e_9$

| width="30pt" | $e_{10}$

| width="30pt" | $e_{11}$

| width="30pt" | $e_{12}$

| width="30pt" | $e_{13}$

| width="30pt" | $e_{14}$

| width="30pt" | $e_{15}$

e_1

| $e_1$

| $-e_0$

| $e_3$

| $-e_2$

| $e_5$

| $-e_4$

| $-e_7$

| $e_6$

| $e_9$

| $-e_8$

| $-e_{11}$

| $e_{10}$

| $-e_{13}$

| $e_{12}$

| $e_{15}$

| $-e_{14}$

e_2

| $e_2$

| $-e_3$

| $-e_0$

| $e_1$

| $e_6$

| $e_7$

| $-e_4$

| $-e_5$

| $e_{10}$

| $e_{11}$

| $-e_8$

| $-e_9$

| $-e_{14}$

| $-e_{15}$

| $e_{12}$

| $e_{13}$

e_3

| $e_3$

| $e_2$

| $-e_1$

| $-e_0$

| $e_7$

| $-e_6$

| $e_5$

| $-e_4$

| $e_{11}$

| $-e_{10}$

| $e_9$

| $-e_8$

| $-e_{15}$

| $e_{14}$

| $-e_{13}$

| $e_{12}$

e_4

| $e_4$

| $-e_5$

| $-e_6$

| $-e_7$

| $-e_0$

| $e_1$

| $e_2$

| $e_3$

| $e_{12}$

| $e_{13}$

| $e_{14}$

| $e_{15}$

| $-e_8$

| $-e_9$

| $-e_{10}$

| $-e_{11}$

e_5

| $e_5$

| $e_4$

| $-e_7$

| $e_6$

| $-e_1$

| $-e_0$

| $-e_3$

| $e_2$

| $e_{13}$

| $-e_{12}$

| $e_{15}$

| $-e_{14}$

| $e_9$

| $-e_8$

| $e_{11}$

| $-e_{10}$

e_6

| $e_6$

| $e_7$

| $e_4$

| $-e_5$

| $-e_2$

| $e_3$

| $-e_0$

| $-e_1$

| $e_{14}$

| $-e_{15}$

| $-e_{12}$

| $e_{13}$

| $e_{10}$

| $-e_{11}$

| $-e_8$

| $e_9$

e_7

| $e_7$

| $-e_6$

| $e_5$

| $e_4$

| $-e_3$

| $-e_2$

| $e_1$

| $-e_0$

| $e_{15}$

| $e_{14}$

| $-e_{13}$

| $-e_{12}$

| $e_{11}$

| $e_{10}$

| $-e_9$

| $-e_8$

e_8

| $e_8$

| $-e_9$

| $-e_{10}$

| $-e_{11}$

| $-e_{12}$

| $-e_{13}$

| $-e_{14}$

| $-e_{15}$

| $-e_0$

| $e_1$

| $e_2$

| $e_3$

| $e_4$

| $e_5$

| $e_6$

| $e_7$

e_9

| $e_9$

| $e_8$

| $-e_{11}$

| $e_{10}$

| $-e_{13}$

| $e_{12}$

| $e_{15}$

| $-e_{14}$

| $-e_1$

| $-e_0$

| $-e_3$

| $e_2$

| $-e_5$

| $e_4$

| $e_7$

| $-e_6$

e_{10}

| $e_{10}$

| $e_{11}$

| $e_8$

| $-e_9$

| $-e_{14}$

| $-e_{15}$

| $e_{12}$

| $e_{13}$

| $-e_2$

| $e_3$

| $-e_0$

| $-e_1$

| $-e_6$

| $-e_7$

| $e_4$

| $e_5$

e_{11}

| $e_{11}$

| $-e_{10}$

| $e_9$

| $e_8$

| $-e_{15}$

| $e_{14}$

| $-e_{13}$

| $e_{12}$

| $-e_3$

| $-e_2$

| $e_1$

| $-e_0$

| $-e_7$

| $e_6$

| $-e_5$

| $e_4$

e_{12}

| $e_{12}$

| $e_{13}$

| $e_{14}$

| $e_{15}$

| $e_8$

| $-e_9$

| $-e_{10}$

| $-e_{11}$

| $-e_4$

| $e_5$

| $e_6$

| $e_7$

| $-e_0$

| $-e_1$

| $-e_2$

| $-e_3$

e_{13}

| $e_{13}$

| $-e_{12}$

| $e_{15}$

| $-e_{14}$

| $e_9$

| $e_8$

| $e_{11}$

| $-e_{10}$

| $-e_5$

| $-e_4$

| $e_7$

| $-e_6$

| $e_1$

| $-e_0$

| $e_3$

| $-e_2$

e_{14}

| $e_{14}$

| $-e_{15}$

| $-e_{12}$

| $e_{13}$

| $e_{10}$

| $-e_{11}$

| $e_8$

| $e_9$

| $-e_6$

| $-e_7$

| $-e_4$

| $e_5$

| $e_2$

| $-e_3$

| $-e_0$

| $e_1$

e_{15}

| $e_{15}$

| $e_{14}$

| $-e_{13}$

| $-e_{12}$

| $e_{11}$

| $e_{10}$

| $-e_9$

| $e_8$

| $-e_7$

| $e_6$

| $-e_5$

| $-e_4$

| $e_3$

| $e_2$

| $-e_1$

| $-e_0$

= Sedenion properties =

File:PG(3,2) g005.png that provides the multiplication law for sedenions, as shown by {{harvtxt|Saniga|Holweck|Pracna|2015}}. Any three points (representing three sedenion imaginary units) lying on the same line are such that the product of two of them yields the third one, sign disregarded.]]

From the above table, we can see that:

: $e_0e_i = e_ie_0 = e_i \, \text{for all} \, i,$

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

: $e_ie_j = -e_je_i \,\, \text{for}\,\, i \neq j \,\,\text{with}\,\, i,j \neq 0.$

== 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.

$(e_1e_2)e_{12} = e_1(e_2e_{12}) = -e_{15}$

= Quaternionic subalgebras =

The particular sedenion multiplication table shown above is represented by 35 triads. The table and its triads have been constructed from an octonion represented by the bolded set of 7 triads using Cayley–Dickson construction. It is one of 480 possible sets of 7 triads (one of two shown in the octonion article) and is the one based on the Cayley–Dickson construction of quaternions from two possible quaternion constructions from the complex numbers. The binary representations of the indices of these triples bitwise XOR to 0. These 35 triads are:

{ {1, 2, 3}, {1, 4, 5}, {1, 7, 6}, {1, 8, 9}, {1, 11, 10}, {1, 13, 12}, {1, 14, 15},

{2, 4, 6}, {2, 5, 7}, {2, 8, 10}, {2, 9, 11}, {2, 14, 12}, {2, 15, 13}, {3, 4, 7},

{3, 6, 5}, {3, 8, 11}, {3, 10, 9}, {3, 13, 14}, {3, 15, 12}, {4, 8, 12}, {4, 9, 13},

{4, 10, 14}, {4, 11, 15}, {5, 8, 13}, {5, 10, 15}, {5, 12, 9}, {5, 14, 11}, {6, 8, 14},

{6, 11, 13}, {6, 12, 10}, {6, 15, 9}, {7, 8, 15}, {7, 9, 14}, {7, 12, 11}, {7, 13, 10} }

= Zero divisors =

The list of 84 sets of zero divisors $\{e_a, e_b, e_c, e_d\}$ , where

$(e_a + e_b) \circ (e_c + e_d) = 0$ :

$\begin{array}{c}
\text{Sedenion Zero Divisors} \quad \{ e_a, e_b, e_c, e_d \} \\
\text{where} ~ (e_a + e_b) \circ (e_c + e_d) = 0 \\
\begin{array}{ccc}
1 \leq a \leq 6, & c > a, & 9 \leq b \leq 15 \\
\end{array} \\
\\
\begin{array}{lccr}
\{ 9 \leq d \leq 15 \} & \{ -9 \geq d \geq -15 \} &
\{ 9 \leq d \leq 15 \} & \{ -9 \geq d \geq -15 \}\\
\end{array} \\
\\
\begin{array}{lccr}
\{e_1, e_{10}, e_5, e_{14}\} & \{e_1, e_{10}, e_4, -e_{15}\} &
\{e_1, e_{10}, e_7, e_{12}\} & \{e_1, e_{10}, e_6, -e_{13}\} \\
\{e_1, e_{11}, e_4, e_{14}\} & \{e_1, e_{11}, e_6, -e_{12}\} &
\{e_1, e_{11}, e_5, e_{15}\} & \{e_1, e_{11}, e_7, -e_{13}\} \\
\{e_1, e_{12}, e_2, e_{15}\} & \{e_1, e_{12}, e_3, -e_{14}\} &
\{e_1, e_{12}, e_6, e_{11}\} & \{e_1, e_{12}, e_7, -e_{10}\} \\
\{e_1, e_{13}, e_6, e_{10}\} & \{e_1, e_{13}, e_2, -e_{14}\} &
\{e_1, e_{13}, e_7, e_{11}\} & \{e_1, e_{13}, e_3, -e_{15}\} \\
\{e_1, e_{14}, e_2, e_{13}\} & \{e_1, e_{14}, e_4, -e_{11}\} &
\{e_1, e_{14}, e_3, e_{12}\} & \{e_1, e_{14}, e_5, -e_{10}\} \\
\{e_1, e_{15}, e_3, e_{13}\} & \{e_1, e_{15}, e_2, -e_{12}\} &
\{e_1, e_{15}, e_4, e_{10}\} & \{e_1, e_{15}, e_5, -e_{11}\} \\
\\
\{e_2, e_9, e_4, e_{15}\} & \{e_2, e_9, e_5, -e_{14}\} &
\{e_2, e_9, e_6, e_{13}\} & \{e_2, e_9, e_7, -e_{12}\} \\
\{e_2, e_{11}, e_5, e_{12}\} & \{e_2, e_{11}, e_4, -e_{13}\} &
\{e_2, e_{11}, e_6, e_{15}\} & \{e_2, e_{11}, e_7, -e_{14}\} \\
\{e_2, e_{12}, e_3, e_{13}\} & \{e_2, e_{12}, e_5, -e_{11}\} &
\{e_2, e_{12}, e_7, e_9 \} & \{e_2, e_{13}, e_3, -e_{12}\} \\
\{e_2, e_{13}, e_4, e_{11}\} & \{e_2, e_{13}, e_6, -e_9 \} &
\{e_2, e_{14}, e_5, e_9 \} & \{e_2, e_{14}, e_3, -e_{15}\} \\
\{e_2, e_{14}, e_7, e_{11}\} & \{e_2, e_{15}, e_4, -e_9 \} &
\{e_2, e_{15}, e_3, e_{14}\} & \{e_2, e_{15}, e_6, -e_{11}\} \\
\\
\{e_3, e_9, e_6, e_{12}\} & \{e_3, e_9, e_4, -e_{14}\} &
\{e_3, e_9, e_7, e_{13}\} & \{e_3, e_9, e_5, -e_{15}\} \\
\{e_3, e_{10}, e_4, e_{13}\} & \{e_3, e_{10}, e_5, -e_{12}\} &
\{e_3, e_{10}, e_7, e_{14}\} & \{e_3, e_{10}, e_6, -e_{15}\} \\
\{e_3, e_{12}, e_5, e_{10}\} & \{e_3, e_{12}, e_6, -e_9 \} &
\{e_3, e_{14}, e_4, e_9 \} & \{e_3, e_{13}, e_4, -e_{10}\} \\
\{e_3, e_{15}, e_5, e_9 \} & \{e_3, e_{13}, e_7, -e_9 \} &
\{e_3, e_{15}, e_6, e_{10}\} & \{e_3, e_{14}, e_7, -e_{10}\} \\
\\
\{e_4, e_9, e_7, e_{10}\} & \{e_4, e_9, e_6, -e_{11}\} &
\{e_4, e_{10}, e_5, e_{11}\} & \{e_4, e_{10}, e_7, -e_9 \} \\
\{e_4, e_{11}, e_6, e_9 \} & \{e_4, e_{11}, e_5, -e_{10}\} &
\{e_4, e_{13}, e_6, e_{15}\} & \{e_4, e_{13}, e_7, -e_{14}\} \\
\{e_4, e_{14}, e_7, e_{13}\} & \{e_4, e_{14}, e_5, -e_{15}\} &
\{e_4, e_{15}, e_5, e_{14}\} & \{e_4, e_{15}, e_6, -e_{13}\} \\
\\
\{e_5, e_{10}, e_6, e_9 \} & \{e_5, e_9, e_6, -e_{10}\} &
\{e_5, e_{11}, e_7, e_9 \} & \{e_5, e_9, e_7, -e_{11}\} \\
\{e_5, e_{12}, e_7, e_{14}\} & \{e_5, e_{12}, e_6, -e_{15}\} &
\{e_5, e_{15}, e_6, e_{12}\} & \{e_5, e_{14}, e_7, -e_{12}\} \\
\\
\{e_6, e_{11}, e_7, e_{10}\} & \{e_6, e_{10}, e_7, -e_{11}\} &
\{e_6, e_{13}, e_7, e_{12}\} & \{e_6, e_{12}, e_7, -e_{13}\}
\end{array}
\end{array}$

Applications

{{harvtxt|Moreno|1998}} 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.)

{{harvtxt|Guillard|Gresnigt|2019}} demonstrated that the three generations of leptons and quarks that are associated with unbroken gauge symmetry $\mathrm {SU(3)_{c} \times U(1)_{em}}$ can be represented using the algebra of the complexified sedenions $\mathbb {C \otimes S}$ . Their reasoning follows that a primitive idempotent projector $\rho_{+} = 1/2(1+ie_{15})$ — where $e_{15}$ is chosen as an imaginary unit akin to $e_{7}$ for $\mathbb {O}$ in the Fano plane — that acts on the standard basis of the sedenions uniquely divides the algebra into three sets of split basis elements for $\mathbb {C \otimes O}$ , whose adjoint left actions on themselves generate three copies of the Clifford algebra $\mathrm Cl(6)$ which in-turn contain minimal left ideals that describe a single generation of fermions with unbroken $\mathrm {SU(3)_{c} \times U(1)_{em}}$ gauge symmetry. In particular, they note that tensor products between normed division algebras generate zero divisors akin to those inside $\mathbb {S}$ , where for $\mathbb {C \otimes O}$ the lack of alternativity and associativity does not affect the construction of minimal left ideals since their underlying split basis requires only two basis elements to be multiplied together, in-which associativity or alternativity are uninvolved. Still, these ideals constructed from an adjoint algebra of left actions of the algebra on itself remain associative, alternative, and isomorphic to a Clifford algebra. Altogether, this permits three copies of $(\mathbb {C \otimes O})_{L} \cong \mathrm {Cl(6)}$ to exist inside $\mathbb {(C \otimes S)}_{L}$ . Furthermore, these three complexified octonion subalgebras are not independent; they share a common $\mathrm Cl(2)$ subalgebra, which the authors note could form a theoretical basis for CKM and PMNS matrices that, respectively, describe quark mixing and neutrino oscillations.

Sedenion neural networks provide{{Explain|date=August 2022}} a means of efficient and compact expression in machine learning applications and have been used in solving multiple time-series and traffic forecasting problems.{{Cite journal|last1=Saoud|first1=Lyes Saad|last2=Al-Marzouqi|first2=Hasan|date=2020|title=Metacognitive Sedenion-Valued Neural Network and its Learning Algorithm|journal=IEEE Access|volume=8|pages=144823–144838|doi=10.1109/ACCESS.2020.3014690|issn=2169-3536|doi-access=free}}{{Cite journal |last1=Kopp |first1=Michael |last2=Kreil |first2=David |last3=Neun |first3=Moritz |last4=Jonietz |first4=David |last5=Martin |first5=Henry |last6=Herruzo |first6=Pedro |last7=Gruca |first7=Aleksandra |last8=Soleymani |first8=Ali |last9=Wu |first9=Fanyou |last10=Liu |first10=Yang |last11=Xu |first11=Jingwei |date=2021-08-07 |title=Traffic4cast at NeurIPS 2020 – yet more on the unreasonable effectiveness of gridded geo-spatial processes |url=https://proceedings.mlr.press/v133/kopp21a.html |journal=NeurIPS 2020 Competition and Demonstration Track |language=en |publisher=PMLR |pages=325–343}}

Notes

References

{{Cite journal | last1=Imaeda | first1=K. | last2=Imaeda | first2=M. | title=Sedenions: algebra and analysis | doi=10.1016/S0096-3003(99)00140-X | mr=1786945 | year=2000 | journal=Applied Mathematics and Computation | volume=115 | issue=2 | pages=77–88}}
{{Cite journal | last1 = Baez | first1 = John C. | author-link = John Baez| title = The Octonions | journal = Bulletin of the American Mathematical Society |series=New Series | volume = 39 | issue = 2 | pages = 145–205 | year = 2002 | url = http://math.ucr.edu/home/baez/octonions/ | doi = 10.1090/S0273-0979-01-00934-X | arxiv = math/0105155| mr = 1886087 | s2cid = 586512 }}
{{cite journal |last1=Biss |first1=Daniel K. |last2=Christensen |first2=J. Daniel |last3=Dugger |first3=Daniel |last4=Isaksen |first4=Daniel C.|date=2007 |title=Large annihilators in Cayley-Dickson algebras II |journal=Boletin de la Sociedad Matematica Mexicana |volume=3 |pages=269–292 |arxiv=math/0702075|bibcode=2007math......2075B }}
{{cite journal |last1=Guillard |first=Adam B. |last2=Gresnigt |first2=Niels G. |title=Three fermion generations with two unbroken gauge symmetries from the complex sedenions |url=https://link.springer.com/article/10.1140/epjc/s10052-019-6967-1 |journal=The European Physical Journal C |publisher=Springer |volume=79 |issue=5 |year=2019 |pages= 1–11 (446) |doi=10.1140/epjc/s10052-019-6967-1 |bibcode=2019EPJC...79..446G |s2cid=102351250 |arxiv=1904.03186 }}
{{cite journal |first1=M.K. |last1=Kinyon |last2=Phillips |first2=J.D. |last3=Vojtěchovský |first3=P. |title=C-loops: Extensions and constructions |journal=Journal of Algebra and Its Applications |volume=6 |issue=1 |pages=1–20 |year=2007 |doi=10.1142/S0219498807001990 |arxiv=math/0412390|citeseerx=10.1.1.240.6208 |s2cid=48162304 }}
{{cite journal |first1=Benard M. |last1=Kivunge |last2=Smith |first2=Jonathan D. H |title=Subloops of sedenions |journal=Comment. Math. Univ. Carolinae |volume=45 |issue=2 |pages=295–302 |year=2004 |url=http://www.emis.de/journals/CMUC/pdf/cmuc0402/kivunge.pdf }}
{{Cite journal | last1=Moreno | first1=Guillermo | title=The zero divisors of the Cayley–Dickson algebras over the real numbers | arxiv=q-alg/9710013 | mr=1625585 | year=1998 | journal= Bol. Soc. Mat. Mexicana |series=Series 3| volume=4 | issue=1 | pages=13–28| bibcode=1997q.alg....10013G }}
{{cite journal | last=Saniga | first=Metod | last2=Holweck | first2=Frédéric | last3=Pracna | first3=Petr | title=From Cayley-Dickson Algebras to Combinatorial Grassmannians | journal=Mathematics | publisher=MDPI AG | volume=3 | issue=4 | date=2015 | issn=2227-7390 | arxiv=1405.6888 | doi=10.3390/math3041192 | doi-access=free | pages=1192–1221}} {{Creative Commons text attribution notice|cc=by4|from this source=yes}}
{{Cite journal | last1=Smith | first1=Jonathan D. H. | title=A left loop on the 15-sphere | doi=10.1006/jabr.1995.1237 | mr=1345298 | year=1995 | journal=Journal of Algebra | volume=176 | issue=1 | pages=128–138| doi-access=free }}
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.

sedenion

Arithmetic

= Multiplication =

= Sedenion properties =

== Anti-associative ==

= Quaternionic subalgebras =

= Zero divisors =

Applications

See also

Notes

References