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Sedenion

From Wikipedia, the free encyclopedia
Hypercomplex number system
Sedenions
SymbolS{\displaystyle \mathbb {S} }
TypeHypercomplexalgebra
Unitse0, ..., e15
Multiplicative identitye0
Main properties

Inabstract algebra, thesedenions form a 16-dimensionalnoncommutative andnonassociativealgebra over thereal numbers, usually represented by the capital letter S, boldfaceS orblackboard bold⁠S{\displaystyle \mathbb {S} }⁠.

The sedenions are obtained by applying theCayley–Dickson construction to theoctonions, which can be mathematically expressed as⁠S=CD(O,1){\displaystyle \mathbb {S} ={\mathcal {CD}}(\mathbb {O} ,1)}⁠.[1] As such, the octonions areisomorphic to asubalgebra of the sedenions. Unlike the octonions, the sedenions are not analternative algebra. Applying the Cayley–Dickson construction to the sedenions yields a 32-dimensional algebra, called thetrigintaduonions or sometimes the 32-nions.[2]

The termsedenion is also used for other 16-dimensional algebraic structures, such as a tensor product of two copies of thebiquaternions, or the algebra of4 × 4matrices over the real numbers, or that studied bySmith (1995).

Arithmetic

[edit]
A visualization of a 4D extension to the cubicoctonion,[3] showing the 35 triads ashyperplanes through the real(e0){\displaystyle (e_{0})} vertex of the sedenion example given

Every sedenion is alinear combination of the unit sedenionse0{\displaystyle e_{0}},e1{\displaystyle e_{1}},e2{\displaystyle e_{2}},e3{\displaystyle e_{3}}, ...,e15{\displaystyle e_{15}},which form abasis of thevector space of sedenions. Every sedenion can be represented in the form

x=x0e0+x1e1+x2e2+⋯+x14e14+x15e15.{\displaystyle 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 isdistributive over addition.

Like other algebras based on theCayley–Dickson construction, the sedenions contain the algebra they were constructed from. So they contain the octonions (generated bye0{\displaystyle e_{0}} toe7{\displaystyle e_{7}} in the table below), and therefore also thequaternions (generated bye0{\displaystyle e_{0}} toe3{\displaystyle e_{3}}),complex numbers (generated bye0{\displaystyle e_{0}} ande1{\displaystyle e_{1}}) and real numbers (generated bye0{\displaystyle e_{0}}).

Multiplication

[edit]

Likeoctonions,multiplication of sedenions is neithercommutative norassociative. However, in contrast to the octonions, the sedenions do not even have the property of beingalternative. They do, however, have the property ofpower associativity, which can be stated as that, for any elementx{\displaystyle x} of⁠S{\displaystyle \mathbb {S} }⁠, the powerxn{\displaystyle x^{n}} is well defined. They are alsoflexible.

The sedenions have a multiplicativeidentity elemente0{\displaystyle e_{0}} and multiplicative inverses, but they are not adivision algebra because they havezero divisors: two nonzero sedenions can be multiplied to obtain zero, for example⁠(e3+e10)(e6−e15){\displaystyle (e_{3}+e_{10})(e_{6}-e_{15})}⁠. Allhypercomplex number systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.

The sedenion multiplication table is shown below:

eiej{\displaystyle e_{i}e_{j}}ej{\displaystyle e_{j}}
e0{\displaystyle e_{0}}e1{\displaystyle e_{1}}e2{\displaystyle e_{2}}e3{\displaystyle e_{3}}e4{\displaystyle e_{4}}e5{\displaystyle e_{5}}e6{\displaystyle e_{6}}e7{\displaystyle e_{7}}e8{\displaystyle e_{8}}e9{\displaystyle e_{9}}e10{\displaystyle e_{10}}e11{\displaystyle e_{11}}e12{\displaystyle e_{12}}e13{\displaystyle e_{13}}e14{\displaystyle e_{14}}e15{\displaystyle e_{15}}
ei{\displaystyle e_{i}}e0{\displaystyle e_{0}}e0{\displaystyle e_{0}}e1{\displaystyle e_{1}}e2{\displaystyle e_{2}}e3{\displaystyle e_{3}}e4{\displaystyle e_{4}}e5{\displaystyle e_{5}}e6{\displaystyle e_{6}}e7{\displaystyle e_{7}}e8{\displaystyle e_{8}}e9{\displaystyle e_{9}}e10{\displaystyle e_{10}}e11{\displaystyle e_{11}}e12{\displaystyle e_{12}}e13{\displaystyle e_{13}}e14{\displaystyle e_{14}}e15{\displaystyle e_{15}}
e1{\displaystyle e_{1}}e1{\displaystyle e_{1}}−e0{\displaystyle -e_{0}}e3{\displaystyle e_{3}}−e2{\displaystyle -e_{2}}e5{\displaystyle e_{5}}−e4{\displaystyle -e_{4}}−e7{\displaystyle -e_{7}}e6{\displaystyle e_{6}}e9{\displaystyle e_{9}}−e8{\displaystyle -e_{8}}−e11{\displaystyle -e_{11}}e10{\displaystyle e_{10}}−e13{\displaystyle -e_{13}}e12{\displaystyle e_{12}}e15{\displaystyle e_{15}}−e14{\displaystyle -e_{14}}
e2{\displaystyle e_{2}}e2{\displaystyle e_{2}}−e3{\displaystyle -e_{3}}−e0{\displaystyle -e_{0}}e1{\displaystyle e_{1}}e6{\displaystyle e_{6}}e7{\displaystyle e_{7}}−e4{\displaystyle -e_{4}}−e5{\displaystyle -e_{5}}e10{\displaystyle e_{10}}e11{\displaystyle e_{11}}−e8{\displaystyle -e_{8}}−e9{\displaystyle -e_{9}}−e14{\displaystyle -e_{14}}−e15{\displaystyle -e_{15}}e12{\displaystyle e_{12}}e13{\displaystyle e_{13}}
e3{\displaystyle e_{3}}e3{\displaystyle e_{3}}e2{\displaystyle e_{2}}−e1{\displaystyle -e_{1}}−e0{\displaystyle -e_{0}}e7{\displaystyle e_{7}}−e6{\displaystyle -e_{6}}e5{\displaystyle e_{5}}−e4{\displaystyle -e_{4}}e11{\displaystyle e_{11}}−e10{\displaystyle -e_{10}}e9{\displaystyle e_{9}}−e8{\displaystyle -e_{8}}−e15{\displaystyle -e_{15}}e14{\displaystyle e_{14}}−e13{\displaystyle -e_{13}}e12{\displaystyle e_{12}}
e4{\displaystyle e_{4}}e4{\displaystyle e_{4}}−e5{\displaystyle -e_{5}}−e6{\displaystyle -e_{6}}−e7{\displaystyle -e_{7}}−e0{\displaystyle -e_{0}}e1{\displaystyle e_{1}}e2{\displaystyle e_{2}}e3{\displaystyle e_{3}}e12{\displaystyle e_{12}}e13{\displaystyle e_{13}}e14{\displaystyle e_{14}}e15{\displaystyle e_{15}}−e8{\displaystyle -e_{8}}−e9{\displaystyle -e_{9}}−e10{\displaystyle -e_{10}}−e11{\displaystyle -e_{11}}
e5{\displaystyle e_{5}}e5{\displaystyle e_{5}}e4{\displaystyle e_{4}}−e7{\displaystyle -e_{7}}e6{\displaystyle e_{6}}−e1{\displaystyle -e_{1}}−e0{\displaystyle -e_{0}}−e3{\displaystyle -e_{3}}e2{\displaystyle e_{2}}e13{\displaystyle e_{13}}−e12{\displaystyle -e_{12}}e15{\displaystyle e_{15}}−e14{\displaystyle -e_{14}}e9{\displaystyle e_{9}}−e8{\displaystyle -e_{8}}e11{\displaystyle e_{11}}−e10{\displaystyle -e_{10}}
e6{\displaystyle e_{6}}e6{\displaystyle e_{6}}e7{\displaystyle e_{7}}e4{\displaystyle e_{4}}−e5{\displaystyle -e_{5}}−e2{\displaystyle -e_{2}}e3{\displaystyle e_{3}}−e0{\displaystyle -e_{0}}−e1{\displaystyle -e_{1}}e14{\displaystyle e_{14}}−e15{\displaystyle -e_{15}}−e12{\displaystyle -e_{12}}e13{\displaystyle e_{13}}e10{\displaystyle e_{10}}−e11{\displaystyle -e_{11}}−e8{\displaystyle -e_{8}}e9{\displaystyle e_{9}}
e7{\displaystyle e_{7}}e7{\displaystyle e_{7}}−e6{\displaystyle -e_{6}}e5{\displaystyle e_{5}}e4{\displaystyle e_{4}}−e3{\displaystyle -e_{3}}−e2{\displaystyle -e_{2}}e1{\displaystyle e_{1}}−e0{\displaystyle -e_{0}}e15{\displaystyle e_{15}}e14{\displaystyle e_{14}}−e13{\displaystyle -e_{13}}−e12{\displaystyle -e_{12}}e11{\displaystyle e_{11}}e10{\displaystyle e_{10}}−e9{\displaystyle -e_{9}}−e8{\displaystyle -e_{8}}
e8{\displaystyle e_{8}}e8{\displaystyle e_{8}}−e9{\displaystyle -e_{9}}−e10{\displaystyle -e_{10}}−e11{\displaystyle -e_{11}}−e12{\displaystyle -e_{12}}−e13{\displaystyle -e_{13}}−e14{\displaystyle -e_{14}}−e15{\displaystyle -e_{15}}−e0{\displaystyle -e_{0}}e1{\displaystyle e_{1}}e2{\displaystyle e_{2}}e3{\displaystyle e_{3}}e4{\displaystyle e_{4}}e5{\displaystyle e_{5}}e6{\displaystyle e_{6}}e7{\displaystyle e_{7}}
e9{\displaystyle e_{9}}e9{\displaystyle e_{9}}e8{\displaystyle e_{8}}−e11{\displaystyle -e_{11}}e10{\displaystyle e_{10}}−e13{\displaystyle -e_{13}}e12{\displaystyle e_{12}}e15{\displaystyle e_{15}}−e14{\displaystyle -e_{14}}−e1{\displaystyle -e_{1}}−e0{\displaystyle -e_{0}}−e3{\displaystyle -e_{3}}e2{\displaystyle e_{2}}−e5{\displaystyle -e_{5}}e4{\displaystyle e_{4}}e7{\displaystyle e_{7}}−e6{\displaystyle -e_{6}}
e10{\displaystyle e_{10}}e10{\displaystyle e_{10}}e11{\displaystyle e_{11}}e8{\displaystyle e_{8}}−e9{\displaystyle -e_{9}}−e14{\displaystyle -e_{14}}−e15{\displaystyle -e_{15}}e12{\displaystyle e_{12}}e13{\displaystyle e_{13}}−e2{\displaystyle -e_{2}}e3{\displaystyle e_{3}}−e0{\displaystyle -e_{0}}−e1{\displaystyle -e_{1}}−e6{\displaystyle -e_{6}}−e7{\displaystyle -e_{7}}e4{\displaystyle e_{4}}e5{\displaystyle e_{5}}
e11{\displaystyle e_{11}}e11{\displaystyle e_{11}}−e10{\displaystyle -e_{10}}e9{\displaystyle e_{9}}e8{\displaystyle e_{8}}−e15{\displaystyle -e_{15}}e14{\displaystyle e_{14}}−e13{\displaystyle -e_{13}}e12{\displaystyle e_{12}}−e3{\displaystyle -e_{3}}−e2{\displaystyle -e_{2}}e1{\displaystyle e_{1}}−e0{\displaystyle -e_{0}}−e7{\displaystyle -e_{7}}e6{\displaystyle e_{6}}−e5{\displaystyle -e_{5}}e4{\displaystyle e_{4}}
e12{\displaystyle e_{12}}e12{\displaystyle e_{12}}e13{\displaystyle e_{13}}e14{\displaystyle e_{14}}e15{\displaystyle e_{15}}e8{\displaystyle e_{8}}−e9{\displaystyle -e_{9}}−e10{\displaystyle -e_{10}}−e11{\displaystyle -e_{11}}−e4{\displaystyle -e_{4}}e5{\displaystyle e_{5}}e6{\displaystyle e_{6}}e7{\displaystyle e_{7}}−e0{\displaystyle -e_{0}}−e1{\displaystyle -e_{1}}−e2{\displaystyle -e_{2}}−e3{\displaystyle -e_{3}}
e13{\displaystyle e_{13}}e13{\displaystyle e_{13}}−e12{\displaystyle -e_{12}}e15{\displaystyle e_{15}}−e14{\displaystyle -e_{14}}e9{\displaystyle e_{9}}e8{\displaystyle e_{8}}e11{\displaystyle e_{11}}−e10{\displaystyle -e_{10}}−e5{\displaystyle -e_{5}}−e4{\displaystyle -e_{4}}e7{\displaystyle e_{7}}−e6{\displaystyle -e_{6}}e1{\displaystyle e_{1}}−e0{\displaystyle -e_{0}}e3{\displaystyle e_{3}}−e2{\displaystyle -e_{2}}
e14{\displaystyle e_{14}}e14{\displaystyle e_{14}}−e15{\displaystyle -e_{15}}−e12{\displaystyle -e_{12}}e13{\displaystyle e_{13}}e10{\displaystyle e_{10}}−e11{\displaystyle -e_{11}}e8{\displaystyle e_{8}}e9{\displaystyle e_{9}}−e6{\displaystyle -e_{6}}−e7{\displaystyle -e_{7}}−e4{\displaystyle -e_{4}}e5{\displaystyle e_{5}}e2{\displaystyle e_{2}}−e3{\displaystyle -e_{3}}−e0{\displaystyle -e_{0}}e1{\displaystyle e_{1}}
e15{\displaystyle e_{15}}e15{\displaystyle e_{15}}e14{\displaystyle e_{14}}−e13{\displaystyle -e_{13}}−e12{\displaystyle -e_{12}}e11{\displaystyle e_{11}}e10{\displaystyle e_{10}}−e9{\displaystyle -e_{9}}e8{\displaystyle e_{8}}−e7{\displaystyle -e_{7}}e6{\displaystyle e_{6}}−e5{\displaystyle -e_{5}}−e4{\displaystyle -e_{4}}e3{\displaystyle e_{3}}e2{\displaystyle e_{2}}−e1{\displaystyle -e_{1}}−e0{\displaystyle -e_{0}}

Sedenion properties

[edit]
An illustration of the structure ofPG(3,2) that provides the multiplication law for sedenions, as shown bySaniga, 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:

e0ei=eie0=eifor alli,{\displaystyle e_{0}e_{i}=e_{i}e_{0}=e_{i}\,{\text{for all}}\,i,}
eiei=−e0fori≠0,{\displaystyle e_{i}e_{i}=-e_{0}\,\,{\text{for}}\,\,i\neq 0,} and
eiej=−ejeifori≠jwithi,j≠0.{\displaystyle e_{i}e_{j}=-e_{j}e_{i}\,\,{\text{for}}\,\,i\neq j\,\,{\text{with}}\,\,i,j\neq 0.}

Anti-associative

[edit]

The sedenions are not fully anti-associative. Choose any four generators,i,j,k{\displaystyle i,j,k} and⁠l{\displaystyle 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){\displaystyle (ij)(kl)=-((ij)k)l=(i(jk))l=-i((jk)l)=i(j(kl))=-(ij)(kl)}

In particular, in the table above, usinge1,e2,e4{\displaystyle e_{1},e_{2},e_{4}} ande8{\displaystyle e_{8}} the last expression associates.(e1e2)e12=e1(e2e12)=−e15{\displaystyle (e_{1}e_{2})e_{12}=e_{1}(e_{2}e_{12})=-e_{15}}

Quaternionic subalgebras

[edit]

The particular sedenion multiplication table shown above is represented by 35 triads. The table and its triads have been constructed from anoctonion represented by the bolded set of 7 triads usingCayley–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 ofquaternions from two possible quaternion constructions from thecomplex numbers. The binary representations of the indices of these triplesbitwise 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

[edit]

The list of 84 sets of zero divisors⁠{ea,eb,ec,ed}{\displaystyle \{e_{a},e_{b},e_{c},e_{d}\}}⁠, where⁠(ea+eb)(ec+ed)=0{\displaystyle (e_{a}+e_{b})(e_{c}+e_{d})=0}⁠:Sedenion zero divisors{ea,eb,ec,ed}where (ea+eb)(ec+ed)=01≤a≤6,c>a,9≤b≤15{9≤d≤15}{−9≥d≥−15}{9≤d≤15}{−9≥d≥−15}{e1,e10,e5,e14}{e1,e10,e4,−e15}{e1,e10,e7,e12}{e1,e10,e6,−e13}{e1,e11,e4,e14}{e1,e11,e6,−e12}{e1,e11,e5,e15}{e1,e11,e7,−e13}{e1,e12,e2,e15}{e1,e12,e3,−e14}{e1,e12,e6,e11}{e1,e12,e7,−e10}{e1,e13,e6,e10}{e1,e13,e2,−e14}{e1,e13,e7,e11}{e1,e13,e3,−e15}{e1,e14,e2,e13}{e1,e14,e4,−e11}{e1,e14,e3,e12}{e1,e14,e5,−e10}{e1,e15,e3,e13}{e1,e15,e2,−e12}{e1,e15,e4,e10}{e1,e15,e5,−e11}{e2,e9,e4,e15}{e2,e9,e5,−e14}{e2,e9,e6,e13}{e2,e9,e7,−e12}{e2,e11,e5,e12}{e2,e11,e4,−e13}{e2,e11,e6,e15}{e2,e11,e7,−e14}{e2,e12,e3,e13}{e2,e12,e5,−e11}{e2,e12,e7,e9}{e2,e13,e3,−e12}{e2,e13,e4,e11}{e2,e13,e6,−e9}{e2,e14,e5,e9}{e2,e14,e3,−e15}{e2,e14,e7,e11}{e2,e15,e4,−e9}{e2,e15,e3,e14}{e2,e15,e6,−e11}{e3,e9,e6,e12}{e3,e9,e4,−e14}{e3,e9,e7,e13}{e3,e9,e5,−e15}{e3,e10,e4,e13}{e3,e10,e5,−e12}{e3,e10,e7,e14}{e3,e10,e6,−e15}{e3,e12,e5,e10}{e3,e12,e6,−e9}{e3,e14,e4,e9}{e3,e13,e4,−e10}{e3,e15,e5,e9}{e3,e13,e7,−e9}{e3,e15,e6,e10}{e3,e14,e7,−e10}{e4,e9,e7,e10}{e4,e9,e6,−e11}{e4,e10,e5,e11}{e4,e10,e7,−e9}{e4,e11,e6,e9}{e4,e11,e5,−e10}{e4,e13,e6,e15}{e4,e13,e7,−e14}{e4,e14,e7,e13}{e4,e14,e5,−e15}{e4,e15,e5,e14}{e4,e15,e6,−e13}{e5,e10,e6,e9}{e5,e9,e6,−e10}{e5,e11,e7,e9}{e5,e9,e7,−e11}{e5,e12,e7,e14}{e5,e12,e6,−e15}{e5,e15,e6,e12}{e5,e14,e7,−e12}{e6,e11,e7,e10}{e6,e10,e7,−e11}{e6,e13,e7,e12}{e6,e12,e7,−e13}{\displaystyle {\begin{array}{c}{\text{Sedenion zero divisors}}\quad \{e_{a},e_{b},e_{c},e_{d}\}\\{\text{where}}~(e_{a}+e_{b})(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}}}

Space of Zero Divisors

[edit]

It has been shown that the pairs of zero divisors in the unit sedonions form a manifold isomorphic to the Lie groupG2 in the space⁠S2{\displaystyle \mathbb {S} ^{2}}⁠.[4]

Applications

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Moreno (1998) showed that the space of pairs of norm-one sedenions that multiply to zero ishomeomorphic to the compact form of the exceptionalLie groupG2. (Note that in his paper, a "zero divisor" means apair of elements that multiply to zero.)

Guillard & Gresnigt (2019) demonstrated that the three generations ofleptons andquarks that are associated with unbrokengauge symmetrySU(3)c×U(1)em{\displaystyle \mathrm {SU(3)_{c}\times U(1)_{em}} } can be represented using the algebra of the complexified sedenions⁠C⊗S{\displaystyle \mathbb {C\otimes S} }⁠. Their reasoning follows that a primitiveidempotentprojectorρ+=1/2(1+ie15){\displaystyle \rho _{+}=1/2(1+ie_{15})} – wheree15{\displaystyle e_{15}} is chosen as animaginary unit akin toe7{\displaystyle e_{7}} forO{\displaystyle \mathbb {O} } in theFano plane – thatacts on thestandard basis of the sedenions uniquely divides the algebra into three sets ofsplit basis elements for⁠C⊗O{\displaystyle \mathbb {C\otimes O} }⁠, whose adjointleft actionson themselves generate three copies of theClifford algebraCl(6){\displaystyle \mathrm {Cl} (6)} which in turn containminimal left ideals that describe a single generation offermions with unbrokenSU(3)c×U(1)em{\displaystyle \mathrm {SU(3)_{c}\times U(1)_{em}} } gauge symmetry. In particular, they note thattensor products between normed division algebras generate zero divisors akin to those inside⁠S{\displaystyle \mathbb {S} }⁠, where forC⊗O{\displaystyle \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, andisomorphic to a Clifford algebra. Altogether, this permits three copies of(C⊗O)L≅Cl(6){\displaystyle (\mathbb {C\otimes O} )_{L}\cong \mathrm {Cl(6)} } to exist inside⁠(C⊗S)L{\displaystyle \mathbb {(C\otimes S)} _{L}}⁠. Furthermore, these three complexified octonion subalgebras are not independent; they share a commonCl(2){\displaystyle \mathrm {Cl} (2)} subalgebra, which the authors note could form a theoretical basis forCKM andPMNS matrices that, respectively, describequark mixing andneutrino oscillations.

Sedenion neural networks provide[further explanation needed] a means of efficient and compact expression in machine learning applications and have been used in solving multiple time-series and traffic forecasting problems[5][6] as well ascomputer chess.[7]

See also

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Notes

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  1. ^"Ensembles de nombre"(PDF) (in French). Forum Futura-Science. 6 September 2011. Retrieved11 October 2024.
  2. ^Raoul E. Cawagas, et al. (2009)."THE BASIC SUBALGEBRA STRUCTURE OF THE CAYLEY-DICKSON ALGEBRA OF DIMENSION 32 (TRIGINTADUONIONS)".
  3. ^Baez 2002, p. 6
  4. ^ The geometry of sedenion zero divisorsSilvio Reggianihttps://arxiv.org/pdf/2411.18881
  5. ^Saoud, Lyes Saad; Al-Marzouqi, Hasan (2020)."Metacognitive Sedenion-Valued Neural Network and its Learning Algorithm".IEEE Access.8:144823–144838.doi:10.1109/ACCESS.2020.3014690.ISSN 2169-3536.
  6. ^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. PMLR:325–343.
  7. ^"pchavez2029/zdtp-chess". December 8, 2025 – via GitHub.

References

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Number systems
Sets ofdefinable numbers
Composition algebras
Split
types
Otherhypercomplex
Infinities andinfinitesimals
Other types
Dimensional spaces
Animated tesseract
Other dimensions
Polytopes andshapes
Number systems
Dimensions by number
See also
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