Editing Sedenion (section)

== 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>