View source for Sedenion

{{Short description|Hypercomplex number system}}
{{Infobox number system
| official_name = Sedenions
| symbol = <math>\mathbb S</math>
| type = [[Hypercomplex number|Hypercomplex]] [[algebra over a field|algebra]]
| units = e<sub>0</sub>, ..., e<sub>15</sub>
| identity = e<sub>0</sub>
| properties = {{Plainlist|
*[[Power associativity]]
*[[Distributivity]]
}}
}}
In [[abstract algebra]], the '''sedenions''' form a 16-[[dimension of a vector space|dimensional]] [[commutative property|noncommutative]] and [[associative property|nonassociative]] [[algebra over a field|algebra]] over the [[real number]]s, usually represented by the capital letter S, boldface {{math|'''S'''}} or [[blackboard bold]] <math>\mathbb S</math>.

The sedenions are obtained by applying the [[Cayley–Dickson construction]] to the [[octonion]]s, which can be mathematically expressed as <math>\mathbb{S}=\mathcal{CD}(\mathbb{O},1)</math>.<ref name="Ensembles">{{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}}</ref> As such, the octonions are [[isomorphism|isomorphic]] to a [[Subalgebra#Subalgebras for algebras over a ring or field|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 [[trigintaduonion]]s or sometimes the 32-nions.<ref>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)"].</ref>

The term ''sedenion'' is also used for other 16-dimensional algebraic structures, such as a tensor product of two copies of the [[biquaternion]]s, or the algebra of 4 × 4 [[matrix (mathematics)|matrices]] over the real numbers, or that studied by {{harvtxt|Smith |1995}}.

== Arithmetic ==
[[File:Sedenion-Fano Tesseract.gif|thumb|A visualization of a 4D extension to the cubic [[Octonion#Fano plane mnemonic|octonion]],<ref>{{Harv|Baez|2002|loc=p. 6}}</ref> showing the 35 triads as [[hyperplane]]s through the real <math>(e_0)</math> vertex of the sedenion example given]]

Every sedenion is a [[linear combination]] of the unit sedenions <math>e_0</math>, <math>e_1</math>, <math>e_2</math>, <math>e_3</math>, ..., <math>e_{15}</math>,
which form a [[Basis (linear algebra)|basis]] of the [[vector space]] of sedenions. Every sedenion can be represented in the form

:<math>x = x_0 e_0 + x_1 e_1 + x_2 e_2 + \cdots + x_{14} e_{14} + x_{15} e_{15}.</math>

Addition and subtraction are defined by the addition and subtraction of corresponding coefficients and multiplication is [[distributive property|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 <math>e_0</math> to <math>e_7</math> in the table below), and therefore also the [[quaternion]]s (generated by <math>e_0</math> to <math>e_3</math>), [[complex number]]s (generated by <math>e_0</math> and <math>e_1</math>) and real numbers (generated by <math>e_0</math>).

=== Multiplication ===
Like [[octonion]]s, [[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 algebra|alternative]]. They do, however, have the property of [[power associativity]], which can be stated as that, for any element <math>x</math> of <math>\mathbb{S}</math>, the power <math>x^n</math> is well defined. They are also [[Flexible algebra|flexible]].

The sedenions have a multiplicative [[identity element]] <math>e_0</math> 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 <math>(e_3 + e_{10})(e_6 - e_{15})</math>. 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"| <math>e_ie_j</math>
  !colspan="16" |<math>e_j</math>
|-
  ! <math>e_0</math>
  ! <math>e_1</math>
  ! <math>e_2</math>
  ! <math>e_3</math>
  ! <math>e_4</math>
  ! <math>e_5</math>
  ! <math>e_6</math>
  ! <math>e_7</math>
  ! <math>e_8</math>
  ! <math>e_9</math>
  ! <math>e_{10}</math>
  ! <math>e_{11}</math>
  ! <math>e_{12}</math>
  ! <math>e_{13}</math>
  ! <math>e_{14}</math>
  ! <math>e_{15}</math>
|- <!-- Would color coding the cells be nice or annoying? There are too many similar greenish shades though... -->
  ! rowspan="16" | <math>e_i</math>
  ! width="30pt" | <math>e_0</math><!-- color: white; when minus sign -->
  | width="30pt" <!-- style="color: black; background-color: hsla(  0,   0%, 50%, 1);" --> | <math>e_0</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(  0, 100%, 50%, 1);" --> | <math>e_1</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 16, 100%, 50%, 1);" --> | <math>e_2</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 32, 100%, 50%, 1);" --> | <math>e_3</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 48, 100%, 50%, 1);" --> | <math>e_4</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 64, 100%, 50%, 1);" --> | <math>e_5</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 80, 100%, 50%, 1);" --> | <math>e_6</math>
  | width="30pt" <!-- style="color: black; background-color: hsla( 96, 100%, 50%, 1);" --> | <math>e_7</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(112, 100%, 50%, 1);" --> | <math>e_8</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(128, 100%, 50%, 1);" --> | <math>e_9</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(144, 100%, 50%, 1);" --> | <math>e_{10}</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(160, 100%, 50%, 1);" --> | <math>e_{11}</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(176, 100%, 50%, 1);" --> | <math>e_{12}</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(192, 100%, 50%, 1);" --> | <math>e_{13}</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(208, 100%, 50%, 1);" --> | <math>e_{14}</math>
  | width="30pt" <!-- style="color: black; background-color: hsla(224, 100%, 50%, 1);" --> | <math>e_{15}</math>
|-
  ! <math>e_1</math>
  | <math>e_1</math>
  | <math>-e_0</math>
  | <math>e_3</math>
  | <math>-e_2</math>
  | <math>e_5</math>
  | <math>-e_4</math>
  | <math>-e_7</math>
  | <math>e_6</math>
  | <math>e_9</math>
  | <math>-e_8</math>
  | <math>-e_{11}</math>
  | <math>e_{10}</math>
  | <math>-e_{13}</math>
  | <math>e_{12}</math>
  | <math>e_{15}</math>
  | <math>-e_{14}</math>
|-
  ! <math>e_2</math>
  | <math>e_2</math>
  | <math>-e_3</math>
  | <math>-e_0</math>
  | <math>e_1</math>
  | <math>e_6</math>
  | <math>e_7</math>
  | <math>-e_4</math>
  | <math>-e_5</math>
  | <math>e_{10}</math>
  | <math>e_{11}</math>
  | <math>-e_8</math>
  | <math>-e_9</math>
  | <math>-e_{14}</math>
  | <math>-e_{15}</math>
  | <math>e_{12}</math>
  | <math>e_{13}</math>
|-
  ! <math>e_3</math>
  | <math>e_3</math>
  | <math>e_2</math>
  | <math>-e_1</math>
  | <math>-e_0</math>
  | <math>e_7</math>
  | <math>-e_6</math>
  | <math>e_5</math>
  | <math>-e_4</math>
  | <math>e_{11}</math>
  | <math>-e_{10}</math>
  | <math>e_9</math>
  | <math>-e_8</math>
  | <math>-e_{15}</math>
  | <math>e_{14}</math>
  | <math>-e_{13}</math>
  | <math>e_{12}</math>
|-
  ! <math>e_4</math>
  | <math>e_4</math>
  | <math>-e_5</math>
  | <math>-e_6</math>
  | <math>-e_7</math>
  | <math>-e_0</math>
  | <math>e_1</math>
  | <math>e_2</math>
  | <math>e_3</math>
  | <math>e_{12}</math>
  | <math>e_{13}</math>
  | <math>e_{14}</math>
  | <math>e_{15}</math>
  | <math>-e_8</math>
  | <math>-e_9</math>
  | <math>-e_{10}</math>
  | <math>-e_{11}</math>
|-
  ! <math>e_5</math>
  | <math>e_5</math>
  | <math>e_4</math>
  | <math>-e_7</math>
  | <math>e_6</math>
  | <math>-e_1</math>
  | <math>-e_0</math>
  | <math>-e_3</math>
  | <math>e_2</math>
  | <math>e_{13}</math>
  | <math>-e_{12}</math>
  | <math>e_{15}</math>
  | <math>-e_{14}</math>
  | <math>e_9</math>
  | <math>-e_8</math>
  | <math>e_{11}</math>
  | <math>-e_{10}</math>
|-
  ! <math>e_6</math>
  | <math>e_6</math>
  | <math>e_7</math>
  | <math>e_4</math>
  | <math>-e_5</math>
  | <math>-e_2</math>
  | <math>e_3</math>
  | <math>-e_0</math>
  | <math>-e_1</math>
  | <math>e_{14}</math>
  | <math>-e_{15}</math>
  | <math>-e_{12}</math>
  | <math>e_{13}</math>
  | <math>e_{10}</math>
  | <math>-e_{11}</math>
  | <math>-e_8</math>
  | <math>e_9</math>
|-
  ! <math>e_7</math>
  | <math>e_7</math>
  | <math>-e_6</math>
  | <math>e_5</math>
  | <math>e_4</math>
  | <math>-e_3</math>
  | <math>-e_2</math>
  | <math>e_1</math>
  | <math>-e_0</math>
  | <math>e_{15}</math>
  | <math>e_{14}</math>
  | <math>-e_{13}</math>
  | <math>-e_{12}</math>
  | <math>e_{11}</math>
  | <math>e_{10}</math>
  | <math>-e_9</math>
  | <math>-e_8</math>
|-
  ! <math>e_8</math>
  | <math>e_8</math>
  | <math>-e_9</math>
  | <math>-e_{10}</math>
  | <math>-e_{11}</math>
  | <math>-e_{12}</math>
  | <math>-e_{13}</math>
  | <math>-e_{14}</math>
  | <math>-e_{15}</math>
  | <math>-e_0</math>
  | <math>e_1</math>
  | <math>e_2</math>
  | <math>e_3</math>
  | <math>e_4</math>
  | <math>e_5</math>
  | <math>e_6</math>
  | <math>e_7</math>
|-
  ! <math>e_9</math>
  | <math>e_9</math>
  | <math>e_8</math>
  | <math>-e_{11}</math>
  | <math>e_{10}</math>
  | <math>-e_{13}</math>
  | <math>e_{12}</math>
  | <math>e_{15}</math>
  | <math>-e_{14}</math>
  | <math>-e_1</math>
  | <math>-e_0</math>
  | <math>-e_3</math>
  | <math>e_2</math>
  | <math>-e_5</math>
  | <math>e_4</math>
  | <math>e_7</math>
  | <math>-e_6</math>
|-
  ! <math>e_{10}</math>
  | <math>e_{10}</math>
  | <math>e_{11}</math>
  | <math>e_8</math>
  | <math>-e_9</math>
  | <math>-e_{14}</math>
  | <math>-e_{15}</math>
  | <math>e_{12}</math>
  | <math>e_{13}</math>
  | <math>-e_2</math>
  | <math>e_3</math>
  | <math>-e_0</math>
  | <math>-e_1</math>
  | <math>-e_6</math>
  | <math>-e_7</math>
  | <math>e_4</math>
  | <math>e_5</math>
|-
  ! <math>e_{11}</math>
  | <math>e_{11}</math>
  | <math>-e_{10}</math>
  | <math>e_9</math>
  | <math>e_8</math>
  | <math>-e_{15}</math>
  | <math>e_{14}</math>
  | <math>-e_{13}</math>
  | <math>e_{12}</math>
  | <math>-e_3</math>
  | <math>-e_2</math>
  | <math>e_1</math>
  | <math>-e_0</math>
  | <math>-e_7</math>
  | <math>e_6</math>
  | <math>-e_5</math>
  | <math>e_4</math>
|-
  ! <math>e_{12}</math>
  | <math>e_{12}</math>
  | <math>e_{13}</math>
  | <math>e_{14}</math>
  | <math>e_{15}</math>
  | <math>e_8</math>
  | <math>-e_9</math>
  | <math>-e_{10}</math>
  | <math>-e_{11}</math>
  | <math>-e_4</math>
  | <math>e_5</math>
  | <math>e_6</math>
  | <math>e_7</math>
  | <math>-e_0</math>
  | <math>-e_1</math>
  | <math>-e_2</math>
  | <math>-e_3</math>
|-
  ! <math>e_{13}</math>
  | <math>e_{13}</math>
  | <math>-e_{12}</math>
  | <math>e_{15}</math>
  | <math>-e_{14}</math>
  | <math>e_9</math>
  | <math>e_8</math>
  | <math>e_{11}</math>
  | <math>-e_{10}</math>
  | <math>-e_5</math>
  | <math>-e_4</math>
  | <math>e_7</math>
  | <math>-e_6</math>
  | <math>e_1</math>
  | <math>-e_0</math>
  | <math>e_3</math>
  | <math>-e_2</math>
|-
  ! <math>e_{14}</math>
  | <math>e_{14}</math>
  | <math>-e_{15}</math>
  | <math>-e_{12}</math>
  | <math>e_{13}</math>
  | <math>e_{10}</math>
  | <math>-e_{11}</math>
  | <math>e_8</math>
  | <math>e_9</math>
  | <math>-e_6</math>
  | <math>-e_7</math>
  | <math>-e_4</math>
  | <math>e_5</math>
  | <math>e_2</math>
  | <math>-e_3</math>
  | <math>-e_0</math>
  | <math>e_1</math>
|-
  ! <math>e_{15}</math>
  | <math>e_{15}</math>
  | <math>e_{14}</math>
  | <math>-e_{13}</math>
  | <math>-e_{12}</math>
  | <math>e_{11}</math>
  | <math>e_{10}</math>
  | <math>-e_9</math>
  | <math>e_8</math>
  | <math>-e_7</math>
  | <math>e_6</math>
  | <math>-e_5</math>
  | <math>-e_4</math>
  | <math>e_3</math>
  | <math>e_2</math>
  | <math>-e_1</math>
  | <math>-e_0</math>
|}

=== Sedenion properties ===
[[File:PG(3,2) g005.png|thumb|right|An illustration of the structure of [[PG(3,2)]] 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:

:<math>e_0e_i = e_ie_0 = e_i \, \text{for all} \, i,</math>

:<math>e_ie_i = -e_0 \,\, \text{for}\,\, i \neq 0,</math> and

:<math>e_ie_j = -e_je_i \,\, \text{for}\,\, i \neq j \,\,\text{with}\,\, i,j \neq 0.</math>

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

<math display="block">(ij)(kl) = -((ij)k)l = (i(jk))l = -i((jk)l) = i(j(kl)) = -(ij)(kl) = 0</math>

In particular, in the table above, using <math>e_1,e_2,e_4</math> and <math>e_8</math> the last expression associates.
<math>(e_1e_2)e_{12} = e_1(e_2e_{12}) = -e_{15}</math>

=== 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}, <br />
'''{2, 4, 6}''', '''{2, 5, 7}''', {2, 8, 10}, {2, 9, 11}, {2, 14, 12}, {2, 15, 13}, '''{3, 4, 7}''', <br />
'''{3, 6, 5}''', {3, 8, 11}, {3, 10, 9}, {3, 13, 14}, {3, 15, 12}, {4, 8, 12}, {4, 9, 13},  <br />
{4, 10, 14}, {4, 11, 15}, {5, 8, 13}, {5, 10, 15}, {5, 12, 9}, {5, 14, 11}, {6, 8, 14},  <br />
{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 <math>\{e_a, e_b, e_c, e_d\}</math>, where
<math>(e_a + e_b) \circ (e_c + e_d) = 0</math>:

<math display="block">\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}</math>

== Applications ==
{{harvtxt|Moreno|1998}} showed that the space of pairs of norm-one sedenions that multiply to zero is [[homeomorphism|homeomorphic]] to the compact form of the exceptional [[Lie group]] [[G2 (mathematics)|G<sub>2</sub>]].  (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 [[lepton]]s and [[quark]]s that are associated with unbroken [[gauge symmetry]] <math>\mathrm {SU(3)_{c} \times U(1)_{em}}</math> can be represented using the algebra of the complexified sedenions <math>\mathbb {C \otimes S}</math>. Their reasoning follows that a primitive [[idempotent]] [[Projection (linear algebra)|projector]] <math>\rho_{+} = 1/2(1+ie_{15})</math> — where <math>e_{15}</math> is chosen as an [[imaginary unit]] akin to <math>e_{7}</math> for <math>\mathbb {O}</math> in the [[Fano plane]] — that [[Group action|acts]] on the [[standard basis]] of the sedenions uniquely divides the algebra into three sets of [[Split-complex number|split basis]] elements for <math>\mathbb {C \otimes O}</math>, whose adjoint [[Group action#Left group action|left actions]] ''on themselves'' generate three copies of the [[Clifford algebra]] <math>\mathrm Cl(6)</math> which in-turn contain [[Ideal (ring theory)#Types of ideals|minimal left ideals]] that describe a single generation of [[fermion]]s with unbroken <math>\mathrm {SU(3)_{c} \times U(1)_{em}}</math> gauge symmetry. In particular, they note that [[tensor product]]s between normed division algebras generate zero divisors akin to those inside <math>\mathbb {S}</math>, where for <math>\mathbb {C \otimes O}</math> 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 <math>(\mathbb {C \otimes O})_{L} \cong \mathrm {Cl(6)}</math> to exist inside <math>\mathbb {(C \otimes S)}_{L}</math>. Furthermore, these three complexified octonion subalgebras are not independent; they share a common <math>\mathrm Cl(2)</math> subalgebra, which the authors note could form a theoretical basis for [[Cabibbo–Kobayashi–Maskawa matrix|CKM]] and [[Pontecorvo–Maki–Nakagawa–Sakata matrix|PMNS]] matrices that, respectively, describe [[quark mixing]] and [[neutrino oscillation]]s.

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.<ref>{{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}}</ref><ref>{{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}}</ref>

== See also ==
* [[Pfister's sixteen-square identity]]
* [[Split-complex number]]
* [[PG(3,2)]]

== Notes ==
{{Reflist}}

== 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 Science+Business Media|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.&nbsp;144823-144838, 2020, [[doi:10.1109/ACCESS.2020.3014690]].

{{Number systems}}
{{Dimension topics}}
{{Authority control}}

[[Category:Sedenions| ]]