Sedenion

Sedenions
Symbol	$\mathbb {S}$
Type	nonassociative algebra
Units	e0, ..., e15
Multiplicative identity	e0
Main properties	power associativity distributivity
Common systems
$\mathbb {N}$ Natural numbers $\mathbb {Z}$ Integers $\mathbb {Q}$ Rational numbers $\mathbb {R}$ Real numbers $\mathbb {C}$ Complex numbers $\mathbb {H}$ Quaternions Less common systems $\mathbb {O}$ Octonions $\mathbb {S}$ Sedenions $\mathbb {T}$ Trigintaduonions

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 S or blackboard bold $\mathbb {S}$. 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 Smith (1995).

Arithmetic

A visualization of a 4D extension to the cubic octonion, showing the 35 triads as hyperplanes through the real $(e_{0})$ vertex of the sedenion example given. Note that the only exception is that the triple $(e_{1})$, $(e_{2})$, $(e_{3})$ doesn't form a hyperplane with $(e_{0})$.

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

Every sedenion is a linear combination of the unit sedenions $e_{0}$, $e_{1}$, $e_{2}$, $e_{3}$, ..., $e_{15}$, 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 $e_{0}$).

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 nonzero sedenions can be multiplied to obtain zero: an example is $(e_{3}+e_{10})(e_{6}-e_{15})$. All hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.

A sedenion multiplication table is shown below:

$e_{i}e_{j}$		$e_{j}$
		$e_{0}$	$e_{1}$	$e_{2}$	$e_{3}$	$e_{4}$	$e_{5}$	$e_{6}$	$e_{7}$	$e_{8}$	$e_{9}$	$e_{10}$	$e_{11}$	$e_{12}$	$e_{13}$	$e_{14}$	$e_{15}$
$e_{i}$	$e_{0}$	$e_{0}$	$e_{1}$	$e_{2}$	$e_{3}$	$e_{4}$	$e_{5}$	$e_{6}$	$e_{7}$	$e_{8}$	$e_{9}$	$e_{10}$	$e_{11}$	$e_{12}$	$e_{13}$	$e_{14}$	$e_{15}$
	$e_{1}$	$e_{1}$	$-e_{0}$	$e_{3}$	$-e_{2}$	$e_{5}$	$-e_{4}$	$-e_{7}$	$e_{6}$	$e_{9}$	$-e_{8}$	$-e_{11}$	$e_{10}$	$-e_{13}$	$e_{12}$	$e_{15}$	$-e_{14}$
	$e_{2}$	$e_{2}$	$-e_{3}$	$-e_{0}$	$e_{1}$	$e_{6}$	$e_{7}$	$-e_{4}$	$-e_{5}$	$e_{10}$	$e_{11}$	$-e_{8}$	$-e_{9}$	$-e_{14}$	$-e_{15}$	$e_{12}$	$e_{13}$
	$e_{3}$	$e_{3}$	$e_{2}$	$-e_{1}$	$-e_{0}$	$e_{7}$	$-e_{6}$	$e_{5}$	$-e_{4}$	$e_{11}$	$-e_{10}$	$e_{9}$	$-e_{8}$	$-e_{15}$	$e_{14}$	$-e_{13}$	$e_{12}$
	$e_{4}$	$e_{4}$	$-e_{5}$	$-e_{6}$	$-e_{7}$	$-e_{0}$	$e_{1}$	$e_{2}$	$e_{3}$	$e_{12}$	$e_{13}$	$e_{14}$	$e_{15}$	$-e_{8}$	$-e_{9}$	$-e_{10}$	$-e_{11}$
	$e_{5}$	$e_{5}$	$e_{4}$	$-e_{7}$	$e_{6}$	$-e_{1}$	$-e_{0}$	$-e_{3}$	$e_{2}$	$e_{13}$	$-e_{12}$	$e_{15}$	$-e_{14}$	$e_{9}$	$-e_{8}$	$e_{11}$	$-e_{10}$
	$e_{6}$	$e_{6}$	$e_{7}$	$e_{4}$	$-e_{5}$	$-e_{2}$	$e_{3}$	$-e_{0}$	$-e_{1}$	$e_{14}$	$-e_{15}$	$-e_{12}$	$e_{13}$	$e_{10}$	$-e_{11}$	$-e_{8}$	$e_{9}$
	$e_{7}$	$e_{7}$	$-e_{6}$	$e_{5}$	$e_{4}$	$-e_{3}$	$-e_{2}$	$e_{1}$	$-e_{0}$	$e_{15}$	$e_{14}$	$-e_{13}$	$-e_{12}$	$e_{11}$	$e_{10}$	$-e_{9}$	$-e_{8}$
	$e_{8}$	$e_{8}$	$-e_{9}$	$-e_{10}$	$-e_{11}$	$-e_{12}$	$-e_{13}$	$-e_{14}$	$-e_{15}$	$-e_{0}$	$e_{1}$	$e_{2}$	$e_{3}$	$e_{4}$	$e_{5}$	$e_{6}$	$e_{7}$
	$e_{9}$	$e_{9}$	$e_{8}$	$-e_{11}$	$e_{10}$	$-e_{13}$	$e_{12}$	$e_{15}$	$-e_{14}$	$-e_{1}$	$-e_{0}$	$-e_{3}$	$e_{2}$	$-e_{5}$	$e_{4}$	$e_{7}$	$-e_{6}$
	$e_{10}$	$e_{10}$	$e_{11}$	$e_{8}$	$-e_{9}$	$-e_{14}$	$-e_{15}$	$e_{12}$	$e_{13}$	$-e_{2}$	$e_{3}$	$-e_{0}$	$-e_{1}$	$-e_{6}$	$-e_{7}$	$e_{4}$	$e_{5}$
	$e_{11}$	$e_{11}$	$-e_{10}$	$e_{9}$	$e_{8}$	$-e_{15}$	$e_{14}$	$-e_{13}$	$e_{12}$	$-e_{3}$	$-e_{2}$	$e_{1}$	$-e_{0}$	$-e_{7}$	$e_{6}$	$-e_{5}$	$e_{4}$
	$e_{12}$	$e_{12}$	$e_{13}$	$e_{14}$	$e_{15}$	$e_{8}$	$-e_{9}$	$-e_{10}$	$-e_{11}$	$-e_{4}$	$e_{5}$	$e_{6}$	$e_{7}$	$-e_{0}$	$-e_{1}$	$-e_{2}$	$-e_{3}$
	$e_{13}$	$e_{13}$	$-e_{12}$	$e_{15}$	$-e_{14}$	$e_{9}$	$e_{8}$	$e_{11}$	$-e_{10}$	$-e_{5}$	$-e_{4}$	$e_{7}$	$-e_{6}$	$e_{1}$	$-e_{0}$	$e_{3}$	$-e_{2}$
	$e_{14}$	$e_{14}$	$-e_{15}$	$-e_{12}$	$e_{13}$	$e_{10}$	$-e_{11}$	$e_{8}$	$e_{9}$	$-e_{6}$	$-e_{7}$	$-e_{4}$	$e_{5}$	$e_{2}$	$-e_{3}$	$-e_{0}$	$e_{1}$
	$e_{15}$	$e_{15}$	$e_{14}$	$-e_{13}$	$-e_{12}$	$e_{11}$	$e_{10}$	$-e_{9}$	$e_{8}$	$-e_{7}$	$e_{6}$	$-e_{5}$	$-e_{4}$	$e_{3}$	$e_{2}$	$-e_{1}$	$-e_{0}$

Sedenion properties

From the above table, we can see that:

$e_{0}e_{i}=e_{i}e_{0}=e_{i}\,{\text{for all}}\,i,$

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

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

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. $(e_{1}e_{2})e_{12}=e_{1}(e_{2}e_{12})=-e_{15}$

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.

{ {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 $\{e_{a},e_{b},e_{c},e_{d}\}$, 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\\9\leq c\leq 15&&-9\geq d\geq -15\end{array}}\\{\begin{array}{ll}\{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_{7},-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_{3},e_{14}\}&\{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_{10},e_{7},-e_{13}\}\end{array}}\end{array}}$$

Applications

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.)

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 $\mathrm {C} l(6)$ 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 $\mathbb {C\otimes S}$ 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 $\mathrm {C} l(2)$ 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 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

• Pfister's sixteen-square identity
• Split-complex number

References

• Imaeda, K.; Imaeda, M. (2000), "Sedenions: algebra and analysis", Applied Mathematics and Computation, 115 (2): 77–88, doi:10.1016/S0096-3003(99)00140-X, MR 1786945
• Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society. New Series. 39 (2): 145–205. arXiv:math/0105155. doi:10.1090/S0273-0979-01-00934-X. MR 1886087. S2CID 586512.
• Biss, Daniel K.; Christensen, J. Daniel; Dugger, Daniel; Isaksen, Daniel C. (2007). "Large annihilators in Cayley-Dickson algebras II". Boletin de la Sociedad Matematica Mexicana. 3: 269–292. arXiv:math/0702075. Bibcode:2007math......2075B.
• Guillard, Adam B.; Gresnigt, Niels G. (2019). "Three fermion generations with two unbroken gauge symmetries from the complex sedenions". The European Physical Journal C. 79 (5): 1–11 (446). Bibcode:2019EPJC...79..446G. doi:10.1140/epjc/s10052-019-6967-1. S2CID 102351250.
• Kinyon, M.K.; Phillips, J.D.; Vojtěchovský, P. (2007). "C-loops: Extensions and constructions". Journal of Algebra and Its Applications. 6 (1): 1–20. arXiv:math/0412390. CiteSeerX 10.1.1.240.6208. doi:10.1142/S0219498807001990. S2CID 48162304.
• Kivunge, Benard M.; Smith, Jonathan D. H (2004). "Subloops of sedenions" (PDF). Comment. Math. Univ. Carolinae. 45 (2): 295–302.
• Moreno, Guillermo (1998), "The zero divisors of the Cayley–Dickson algebras over the real numbers", Bol. Soc. Mat. Mexicana, Series 3, 4 (1): 13–28, arXiv:q-alg/9710013, Bibcode:1997q.alg....10013G, MR 1625585
• Smith, Jonathan D. H. (1995), "A left loop on the 15-sphere", Journal of Algebra, 176 (1): 128–138, doi:10.1006/jabr.1995.1237, MR 1345298
• 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.