Sedenion

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In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the reals obtained by applying the Cayley–Dickson construction to the octonions. Unlike the octonions, the sedenions are not an alternative algebra. The set of sedenions is denoted by ${\displaystyle \mathbb {S} }$.

Sedenions
Symbol ${\displaystyle \mathbb {S} }$
Type nonassociative algebra
Units e0...e15
Multiplicative identity e0
Main properties distributivity
Common Systems
• ${\displaystyle \mathbb {N} }$ Natural numbers
• ${\displaystyle \mathbb {Z} }$ Integers
• ${\displaystyle \mathbb {Q} }$ Rational numbers
• ${\displaystyle \mathbb {R} }$ Real numbers
• ${\displaystyle \mathbb {C} }$ Complex numbers
• ${\displaystyle \mathbb {H} }$ Quaternions
Less common systems

Octonions (${\displaystyle \mathbb {O} }$) Sedenions (${\displaystyle \mathbb {S} }$)

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 by 4 matrices over the reals, or that studied by Smith (1995).

ArithmeticEdit

A visualization of a 4D extension to the cubic Octonion,[1] showing the 35 triads as hyperplanes through the Real (${\displaystyle e_{0}}$ ) vertex of the sedenion example given.

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 ${\displaystyle \mathbb {S} }$ , the power ${\displaystyle x^{n}}$  is well defined. They are also flexible.

Every sedenion is a linear combination of the unit sedenions ${\displaystyle e_{0}}$ , ${\displaystyle e_{1}}$ , ${\displaystyle e_{2}}$ , ${\displaystyle e_{3}}$ , ...,${\displaystyle e_{15}}$ , which form a basis of the vector space of sedenions. Every sedenion can be represented in the form

${\displaystyle x=x_{0}e_{0}+x_{1}e_{1}+x_{2}e_{2}+\ldots +x_{14}e_{14}+x_{15}e_{15},\,}$ .

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 (${\displaystyle e_{0}}$  to ${\displaystyle e_{7}}$  in the table below), and therefore also the quaternions (${\displaystyle e_{0}}$  to ${\displaystyle e_{3}}$ ), complex numbers (${\displaystyle e_{0}}$  and ${\displaystyle e_{1}}$ ) and reals (${\displaystyle e_{0}}$ ).

The sedenions have a multiplicative identity element ${\displaystyle e_{0}}$  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 (${\displaystyle e_{3}}$  + ${\displaystyle e_{10}}$ )(${\displaystyle e_{6}}$ ${\displaystyle e_{15}}$ ). All hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction contain zero divisors.

Sedenion multiplication tableEdit

${\displaystyle e_{i}e_{j}}$  ${\displaystyle e_{0}}$  ${\displaystyle e_{1}}$  ${\displaystyle e_{2}}$  ${\displaystyle e_{3}}$  ${\displaystyle e_{4}}$  ${\displaystyle e_{5}}$  ${\displaystyle e_{6}}$  ${\displaystyle e_{7}}$  ${\displaystyle e_{8}}$  ${\displaystyle e_{9}}$  ${\displaystyle e_{10}}$  ${\displaystyle e_{11}}$  ${\displaystyle e_{12}}$  ${\displaystyle e_{13}}$  ${\displaystyle e_{14}}$  ${\displaystyle e_{15}}$ multiplier ${\displaystyle e_{j}}$ multiplicand ${\displaystyle e_{i}}$ ${\displaystyle e_{0}}$ ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$ ${\displaystyle e_{3}}$ ${\displaystyle e_{4}}$ ${\displaystyle e_{5}}$ ${\displaystyle e_{6}}$ ${\displaystyle e_{7}}$ ${\displaystyle e_{8}}$ ${\displaystyle e_{9}}$ ${\displaystyle e_{10}}$ ${\displaystyle e_{11}}$ ${\displaystyle e_{12}}$ ${\displaystyle e_{13}}$ ${\displaystyle e_{14}}$ ${\displaystyle e_{15}}$ ${\displaystyle e_{1}}$ −${\displaystyle e_{0}}$ ${\displaystyle e_{3}}$ −${\displaystyle e_{2}}$ ${\displaystyle e_{5}}$ −${\displaystyle e_{4}}$ −${\displaystyle e_{7}}$ ${\displaystyle e_{6}}$ ${\displaystyle e_{9}}$ −${\displaystyle e_{8}}$ −${\displaystyle e_{11}}$ ${\displaystyle e_{10}}$ −${\displaystyle e_{13}}$ ${\displaystyle e_{12}}$ ${\displaystyle e_{15}}$ −${\displaystyle e_{14}}$ ${\displaystyle e_{2}}$ −${\displaystyle e_{3}}$ −${\displaystyle e_{0}}$ ${\displaystyle e_{1}}$ ${\displaystyle e_{6}}$ ${\displaystyle e_{7}}$ −${\displaystyle e_{4}}$ −${\displaystyle e_{5}}$ ${\displaystyle e_{10}}$ ${\displaystyle e_{11}}$ −${\displaystyle e_{8}}$ −${\displaystyle e_{9}}$ −${\displaystyle e_{14}}$ −${\displaystyle e_{15}}$ ${\displaystyle e_{12}}$ ${\displaystyle e_{13}}$ ${\displaystyle e_{3}}$ ${\displaystyle e_{2}}$ −${\displaystyle e_{1}}$ −${\displaystyle e_{0}}$ ${\displaystyle e_{7}}$ −${\displaystyle e_{6}}$ ${\displaystyle e_{5}}$ −${\displaystyle e_{4}}$ ${\displaystyle e_{11}}$ −${\displaystyle e_{10}}$ ${\displaystyle e_{9}}$ −${\displaystyle e_{8}}$ −${\displaystyle e_{15}}$ ${\displaystyle e_{14}}$ −${\displaystyle e_{13}}$ ${\displaystyle e_{12}}$ ${\displaystyle e_{4}}$ −${\displaystyle e_{5}}$ −${\displaystyle e_{6}}$ −${\displaystyle e_{7}}$ −${\displaystyle e_{0}}$ ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$ ${\displaystyle e_{3}}$ ${\displaystyle e_{12}}$ ${\displaystyle e_{13}}$ ${\displaystyle e_{14}}$ ${\displaystyle e_{15}}$ −${\displaystyle e_{8}}$ −${\displaystyle e_{9}}$ −${\displaystyle e_{10}}$ −${\displaystyle e_{11}}$ ${\displaystyle e_{5}}$ ${\displaystyle e_{4}}$ −${\displaystyle e_{7}}$ ${\displaystyle e_{6}}$ −${\displaystyle e_{1}}$ −${\displaystyle e_{0}}$ −${\displaystyle e_{3}}$ ${\displaystyle e_{2}}$ ${\displaystyle e_{13}}$ −${\displaystyle e_{12}}$ ${\displaystyle e_{15}}$ −${\displaystyle e_{14}}$ ${\displaystyle e_{9}}$ −${\displaystyle e_{8}}$ ${\displaystyle e_{11}}$ −${\displaystyle e_{10}}$ ${\displaystyle e_{6}}$ ${\displaystyle e_{7}}$ ${\displaystyle e_{4}}$ −${\displaystyle e_{5}}$ −${\displaystyle e_{2}}$ ${\displaystyle e_{3}}$ −${\displaystyle e_{0}}$ −${\displaystyle e_{1}}$ ${\displaystyle e_{14}}$ −${\displaystyle e_{15}}$ −${\displaystyle e_{12}}$ ${\displaystyle e_{13}}$ ${\displaystyle e_{10}}$ −${\displaystyle e_{11}}$ −${\displaystyle e_{8}}$ ${\displaystyle e_{9}}$ ${\displaystyle e_{7}}$ −${\displaystyle e_{6}}$ ${\displaystyle e_{5}}$ ${\displaystyle e_{4}}$ −${\displaystyle e_{3}}$ −${\displaystyle e_{2}}$ ${\displaystyle e_{1}}$ −${\displaystyle e_{0}}$ ${\displaystyle e_{15}}$ ${\displaystyle e_{14}}$ −${\displaystyle e_{13}}$ −${\displaystyle e_{12}}$ ${\displaystyle e_{11}}$ ${\displaystyle e_{10}}$ −${\displaystyle e_{9}}$ −${\displaystyle e_{8}}$ ${\displaystyle e_{8}}$ −${\displaystyle e_{9}}$ −${\displaystyle e_{10}}$ −${\displaystyle e_{11}}$ −${\displaystyle e_{12}}$ −${\displaystyle e_{13}}$ −${\displaystyle e_{14}}$ −${\displaystyle e_{15}}$ −${\displaystyle e_{0}}$ ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$ ${\displaystyle e_{3}}$ ${\displaystyle e_{4}}$ ${\displaystyle e_{5}}$ ${\displaystyle e_{6}}$ ${\displaystyle e_{7}}$ ${\displaystyle e_{9}}$ ${\displaystyle e_{8}}$ −${\displaystyle e_{11}}$ ${\displaystyle e_{10}}$ −${\displaystyle e_{13}}$ ${\displaystyle e_{12}}$ ${\displaystyle e_{15}}$ −${\displaystyle e_{14}}$ −${\displaystyle e_{1}}$ −${\displaystyle e_{0}}$ −${\displaystyle e_{3}}$ ${\displaystyle e_{2}}$ −${\displaystyle e_{5}}$ ${\displaystyle e_{4}}$ ${\displaystyle e_{7}}$ −${\displaystyle e_{6}}$ ${\displaystyle e_{10}}$ ${\displaystyle e_{11}}$ ${\displaystyle e_{8}}$ −${\displaystyle e_{9}}$ −${\displaystyle e_{14}}$ −${\displaystyle e_{15}}$ ${\displaystyle e_{12}}$ ${\displaystyle e_{13}}$ −${\displaystyle e_{2}}$ ${\displaystyle e_{3}}$ −${\displaystyle e_{0}}$ −${\displaystyle e_{1}}$ −${\displaystyle e_{6}}$ −${\displaystyle e_{7}}$ ${\displaystyle e_{4}}$ ${\displaystyle e_{5}}$ ${\displaystyle e_{11}}$ −${\displaystyle e_{10}}$ ${\displaystyle e_{9}}$ ${\displaystyle e_{8}}$ −${\displaystyle e_{15}}$ ${\displaystyle e_{14}}$ −${\displaystyle e_{13}}$ ${\displaystyle e_{12}}$ −${\displaystyle e_{3}}$ −${\displaystyle e_{2}}$ ${\displaystyle e_{1}}$ −${\displaystyle e_{0}}$ −${\displaystyle e_{7}}$ ${\displaystyle e_{6}}$ −${\displaystyle e_{5}}$ ${\displaystyle e_{4}}$ ${\displaystyle e_{12}}$ ${\displaystyle e_{13}}$ ${\displaystyle e_{14}}$ ${\displaystyle e_{15}}$ ${\displaystyle e_{8}}$ −${\displaystyle e_{9}}$ −${\displaystyle e_{10}}$ −${\displaystyle e_{11}}$ −${\displaystyle e_{4}}$ ${\displaystyle e_{5}}$ ${\displaystyle e_{6}}$ ${\displaystyle e_{7}}$ −${\displaystyle e_{0}}$ −${\displaystyle e_{1}}$ −${\displaystyle e_{2}}$ −${\displaystyle e_{3}}$ ${\displaystyle e_{13}}$ −${\displaystyle e_{12}}$ ${\displaystyle e_{15}}$ −${\displaystyle e_{14}}$ ${\displaystyle e_{9}}$ ${\displaystyle e_{8}}$ ${\displaystyle e_{11}}$ −${\displaystyle e_{10}}$ −${\displaystyle e_{5}}$ −${\displaystyle e_{4}}$ ${\displaystyle e_{7}}$ −${\displaystyle e_{6}}$ ${\displaystyle e_{1}}$ −${\displaystyle e_{0}}$ ${\displaystyle e_{3}}$ −${\displaystyle e_{2}}$ ${\displaystyle e_{14}}$ −${\displaystyle e_{15}}$ −${\displaystyle e_{12}}$ ${\displaystyle e_{13}}$ ${\displaystyle e_{10}}$ −${\displaystyle e_{11}}$ ${\displaystyle e_{8}}$ ${\displaystyle e_{9}}$ −${\displaystyle e_{6}}$ −${\displaystyle e_{7}}$ −${\displaystyle e_{4}}$ ${\displaystyle e_{5}}$ ${\displaystyle e_{2}}$ −${\displaystyle e_{3}}$ −${\displaystyle e_{0}}$ ${\displaystyle e_{1}}$ ${\displaystyle e_{15}}$ ${\displaystyle e_{14}}$ −${\displaystyle e_{13}}$ −${\displaystyle e_{12}}$ ${\displaystyle e_{11}}$ ${\displaystyle e_{10}}$ −${\displaystyle e_{9}}$ ${\displaystyle e_{8}}$ −${\displaystyle e_{7}}$ ${\displaystyle e_{6}}$ −${\displaystyle e_{5}}$ −${\displaystyle e_{4}}$ ${\displaystyle e_{3}}$ ${\displaystyle e_{2}}$ −${\displaystyle e_{1}}$ −${\displaystyle e_{0}}$

Sedenion propertiesEdit

From the above table, we can see that:

${\displaystyle e_{0}e_{i}=e_{i}e_{0}=e_{i},}$
${\displaystyle e_{i}e_{i}=-e_{0}\,\,{\text{for}}\,\,i\neq 0,}$
${\displaystyle e_{i}e_{j}=-e_{j}e_{i}\,\,{\text{for}}\,\,i\neq j\,\,{\text{and}}\,\,i,j\neq 0.}$

Anti-associativeEdit

The sedenions are NOT fully anti-associative. Choose any four generators, ${\displaystyle i,j,k}$  and ${\displaystyle l}$ . The following 5-cycle shows that at least one of these relations must associate.

${\displaystyle (ij)(kl)=-((ij)k)l=(i(jk))l=-i((jk)l)=i(j(kl))=-(ij)(kl)=0}$

In particular, in the table above, using ${\displaystyle e_{1},e_{2},e_{4}}$  and ${\displaystyle e_{8}}$  the last expression associates. ${\displaystyle (e_{1}e_{2})e_{12}=e_{1}(e_{2}e_{12})=-e_{15}}$

Quaternionic subalgebrasEdit

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

The binary representations of the indicies of these triples xor to 0.

{​{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}​}

The list of 84 sets of zero divisors {${\displaystyle e_{a}}$ , ${\displaystyle e_{b}}$ , ${\displaystyle e_{c}}$ , ${\displaystyle e_{d}}$ }, where (${\displaystyle e_{a}}$  + ${\displaystyle e_{b}}$ )${\displaystyle \circ }$ (${\displaystyle e_{c}}$  + ${\displaystyle e_{d}}$ )=0:

ApplicationsEdit

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.)