Sedenion Explained

Official Name:	Sedenions
Symbol:	S
Type:	Hypercomplex algebra
Units:	e₀, ..., e₁₅
Identity:	e₀

The sedenions are obtained by applying the Cayley–Dickson construction to the octonions, which can be mathematically expressed as

S=l{CD}(O,1)

.^[1] 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.^[2] 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

Every sedenion is a linear combination of the unit sedenions

e₀

e₁

e₂

e₃

, ...,

e₁₅

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

x=x₀e₀+x₁e₁+x₂e₂+ … +x₁₄e₁₄+x₁₅e₁₅.

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

e₀

e₇

in the table below), and therefore also the quaternions (generated by

e₀

e₃

), complex numbers (generated by

e₀

and

e₁

) and real numbers (generated by

e₀

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

, the power

xⁿ

is well defined. They are also flexible.

e₀

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

(e₃+e₁₀)(e₆-e₁₅)

. 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:

e_ie_j		e_j
e_ie_j		e₀ ! e₁	e₂ ! e₃	e₄ ! e₅	e₆ ! e₇	e₈ ! e₉	e₁₀ ! e₁₁	e₁₂ ! e₁₃	e₁₄ ! e₁₅
e_i	e₀ <	-- color: white; when minus sign -->	e₀	e₁	e₂	e₃	e₄	e₅	e₆	e₇	e₈	e₉	e₁₀	e₁₁	e₁₂	e₁₃	e₁₄	e₁₅
	e₁	e₁	-e₀	e₃	-e₂	e₅	-e₄	-e₇	e₆	e₉	-e₈	-e₁₁	e₁₀	-e₁₃	e₁₂	e₁₅	-e₁₄
	e₂	e₂	-e₃	-e₀	e₁	e₆	e₇	-e₄	-e₅	e₁₀	e₁₁	-e₈	-e₉	-e₁₄	-e₁₅	e₁₂	e₁₃
	e₃	e₃	e₂	-e₁	-e₀	e₇	-e₆	e₅	-e₄	e₁₁	-e₁₀	e₉	-e₈	-e₁₅	e₁₄	-e₁₃	e₁₂
	e₄	e₄	-e₅	-e₆	-e₇	-e₀	e₁	e₂	e₃	e₁₂	e₁₃	e₁₄	e₁₅	-e₈	-e₉	-e₁₀	-e₁₁
	e₅	e₅	e₄	-e₇	e₆	-e₁	-e₀	-e₃	e₂	e₁₃	-e₁₂	e₁₅	-e₁₄	e₉	-e₈	e₁₁	-e₁₀
	e₆	e₆	e₇	e₄	-e₅	-e₂	e₃	-e₀	-e₁	e₁₄	-e₁₅	-e₁₂	e₁₃	e₁₀	-e₁₁	-e₈	e₉
	e₇	e₇	-e₆	e₅	e₄	-e₃	-e₂	e₁	-e₀	e₁₅	e₁₄	-e₁₃	-e₁₂	e₁₁	e₁₀	-e₉	-e₈
	e₈	e₈	-e₉	-e₁₀	-e₁₁	-e₁₂	-e₁₃	-e₁₄	-e₁₅	-e₀	e₁	e₂	e₃	e₄	e₅	e₆	e₇
	e₉	e₉	e₈	-e₁₁	e₁₀	-e₁₃	e₁₂	e₁₅	-e₁₄	-e₁	-e₀	-e₃	e₂	-e₅	e₄	e₇	-e₆
	e₁₀	e₁₀	e₁₁	e₈	-e₉	-e₁₄	-e₁₅	e₁₂	e₁₃	-e₂	e₃	-e₀	-e₁	-e₆	-e₇	e₄	e₅
	e₁₁	e₁₁	-e₁₀	e₉	e₈	-e₁₅	e₁₄	-e₁₃	e₁₂	-e₃	-e₂	e₁	-e₀	-e₇	e₆	-e₅	e₄
	e₁₂	e₁₂	e₁₃	e₁₄	e₁₅	e₈	-e₉	-e₁₀	-e₁₁	-e₄	e₅	e₆	e₇	-e₀	-e₁	-e₂	-e₃
	e₁₃	e₁₃	-e₁₂	e₁₅	-e₁₄	e₉	e₈	e₁₁	-e₁₀	-e₅	-e₄	e₇	-e₆	e₁	-e₀	e₃	-e₂
	e₁₄	e₁₄	-e₁₅	-e₁₂	e₁₃	e₁₀	-e₁₁	e₈	e₉	-e₆	-e₇	-e₄	e₅	e₂	-e₃	-e₀	e₁
	e₁₅	e₁₅	e₁₄	-e₁₃	-e₁₂	e₁₁	e₁₀	-e₉	e₈	-e₇	e₆	-e₅	-e₄	e₃	e₂	-e₁	-e₀

Sedenion properties

From the above table, we can see that:

e_0e_i=e_ie₀=e_iforalli,

e_ie_i=-e₀fori ≠ 0,

and

e_ie_j=-e_je_ifori ≠ jwithi,j ≠ 0.

Anti-associative

The sedenions are not fully anti-associative. Choose any four generators,

i,j,k

and

. 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,e₄

and

e₈

the last expression associates.

(e_1e_2)e₁₂=e_1(e_2e₁₂)=-e₁₅

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:

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

$\begin\text \quad \ \\\text ~ (e_a + e_b) \circ (e_c + e_d) = 0 \\\begin1 \leq a \leq 6, & c > a, & 9 \leq b \leq 15 \\\end \\\\ \begin\ & \ &\ & \\\\end \\ \\\begin\ & \ &\ & \ \\\ & \ &\ & \ \\\ & \ &\ & \ \\\ & \ &\ & \ \\\ & \ &\ & \ \\\ & \ &\ & \ \\\\ \ & \ &\ & \ \\\ & \ &\ & \ \\\ & \ &\ & \ \\\ & \ &\ & \ \\\ & \ &\ & \ \\\\ \ & \ &\ & \ \\\ & \ &\ & \ \\\ & \ &\ & \ \\\ & \ &\ & \ \\\\ \ & \ &\ & \ \\\ & \ &\ & \ \\\ & \ &\ & \ \\\\ \ & \ &\ & \ \\\ & \ &\ & \ \\\\ \ & \ &\ & \\end\end$

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.)

SU(3)_c x U(1)_em

can be represented using the algebra of the complexified sedenions

C ⊗ S

. Their reasoning follows that a primitive idempotent projector

\rho₊=1/2(1+ie₁₅)

— where

e₁₅

is chosen as an imaginary unit akin to

e₇

for

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

C ⊗ O

, whose adjoint left actions on themselves generate three copies of the Clifford algebra

Cl(6)

which in-turn contain minimal left ideals that describe a single generation of fermions with unbroken

SU(3)_c x U(1)_em

gauge symmetry. In particular, they note that tensor products between normed division algebras generate zero divisors akin to those inside

, where for

C ⊗ 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

(C ⊗ O)_L\congCl(6)

to exist inside

(C ⊗ S)_L

. Furthermore, these three complexified octonion subalgebras are not independent; they share a common

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 a means of efficient and compact expression in machine learning applications and have been used in solving multiple time-series and traffic forecasting problems.^[3] ^[4]

References

Imaeda . K. . Imaeda . M. . Sedenions: algebra and analysis . 10.1016/S0096-3003(99)00140-X . 1786945 . 2000 . Applied Mathematics and Computation . 115 . 2 . 77–88.
Baez . John C. . John Baez. The Octonions . Bulletin of the American Mathematical Society . New Series . 39 . 2 . 145–205 . 2002 . 10.1090/S0273-0979-01-00934-X . math/0105155. 1886087 . 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 . math/0702075. 2007math......2075B .
Guillard . Adam B. . Gresnigt . Niels G. . Three fermion generations with two unbroken gauge symmetries from the complex sedenions . . . 79 . 5 . 2019 . 1–11 (446) . 10.1140/epjc/s10052-019-6967-1 . 2019EPJC...79..446G . 102351250 . 1904.03186 .
M.K. . Kinyon . Phillips . J.D. . Vojtěchovský . P. . C-loops: Extensions and constructions . Journal of Algebra and Its Applications . 6 . 1 . 1–20 . 2007 . 10.1142/S0219498807001990 . math/0412390. 10.1.1.240.6208 . 48162304 .
Benard M. . Kivunge . Smith . Jonathan D. H . Subloops of sedenions . Comment. Math. Univ. Carolinae . 45 . 2 . 295–302 . 2004 .
Moreno . Guillermo . The zero divisors of the Cayley–Dickson algebras over the real numbers . q-alg/9710013 . 1625585 . 1998 . Bol. Soc. Mat. Mexicana . Series 3. 4 . 1 . 13–28. 1997q.alg....10013G .
Saniga . Metod . Holweck . Frédéric . Pracna . Petr . From Cayley-Dickson Algebras to Combinatorial Grassmannians . Mathematics . MDPI AG . 3 . 4 . 2015 . 2227-7390 . 1405.6888 . 10.3390/math3041192 . free . 1192–1221.
Smith . Jonathan D. H. . A left loop on the 15-sphere . 10.1006/jabr.1995.1237 . 1345298 . 1995 . . 176 . 1 . 128–138. 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, .

Notes and References

Web site: Ensembles de nombre. 6 September 2011. Forum Futura-Science. 11 October 2024. fr.
Raoul E. Cawagas, et al. (2009). "THE BASIC SUBALGEBRA STRUCTURE OF THE CAYLEY-DICKSON ALGEBRA OF DIMENSION 32 (TRIGINTADUONIONS)".
Saoud. Lyes Saad. Al-Marzouqi. Hasan. 2020. Metacognitive Sedenion-Valued Neural Network and its Learning Algorithm. IEEE Access. 8. 144823–144838. 10.1109/ACCESS.2020.3014690. 2169-3536. free.
Kopp . Michael . Kreil . David . Neun . Moritz . Jonietz . David . Martin . Henry . Herruzo . Pedro . Gruca . Aleksandra . Soleymani . Ali . Wu . Fanyou . Liu . Yang . Xu . Jingwei . 2021-08-07 . Traffic4cast at NeurIPS 2020 – yet more on the unreasonable effectiveness of gridded geo-spatial processes . NeurIPS 2020 Competition and Demonstration Track . en . PMLR . 325–343.