UOR Division Algebras

IRI
https://uor.foundation/division/
Prefix
division:
Space
kernel
Comment
The four normed division algebras R, C, H, O and the Cayley-Dickson construction.

Imports

  • https://uor.foundation/op/
  • https://uor.foundation/convergence/

Classes

NameIRISubclass OfDisjoint WithComment
NormedDivisionAlgebrahttps://uor.foundation/division/NormedDivisionAlgebrahttp://www.w3.org/2002/07/owl#ThingAn algebra over R that is a division ring with multiplicative norm. Exactly four exist (Hurwitz theorem): R, C, H, O.
CayleyDicksonConstructionhttps://uor.foundation/division/CayleyDicksonConstructionhttp://www.w3.org/2002/07/owl#ThingThe doubling construction that builds each division algebra from the previous: R → C → H → O.
MultiplicationTablehttps://uor.foundation/division/MultiplicationTablehttp://www.w3.org/2002/07/owl#ThingThe explicit product rules for a division algebra’s basis elements.
AlgebraCommutatorhttps://uor.foundation/division/AlgebraCommutatorhttp://www.w3.org/2002/07/owl#ThingThe commutator [a,b] = ab − ba. Zero for R and C; non-zero for H and O.
AlgebraAssociatorhttps://uor.foundation/division/AlgebraAssociatorhttp://www.w3.org/2002/07/owl#ThingThe associator [a,b,c] = (ab)c − a(bc). Zero for R, C, H; non-zero for O.

Properties

NameKindFunctionalDomainRangeComment
algebraDimensionDatatypetruehttps://uor.foundation/division/NormedDivisionAlgebrahttp://www.w3.org/2001/XMLSchema#nonNegativeIntegerThe dimension of this division algebra (1, 2, 4, or 8).
isCommutativeDatatypetruehttps://uor.foundation/division/NormedDivisionAlgebrahttp://www.w3.org/2001/XMLSchema#booleanWhether multiplication in this algebra is commutative.
isAssociativeDatatypetruehttps://uor.foundation/division/NormedDivisionAlgebrahttp://www.w3.org/2001/XMLSchema#booleanWhether multiplication in this algebra is associative.
basisElementsDatatypetruehttps://uor.foundation/division/NormedDivisionAlgebrahttp://www.w3.org/2001/XMLSchema#stringThe basis elements of this division algebra.
algebraMultiplicationTableObjecttruehttps://uor.foundation/division/NormedDivisionAlgebrahttps://uor.foundation/division/MultiplicationTableThe multiplication table for this algebra.
cayleyDicksonSourceObjecttruehttps://uor.foundation/division/CayleyDicksonConstructionhttps://uor.foundation/division/NormedDivisionAlgebraThe source algebra of the Cayley-Dickson doubling.
cayleyDicksonTargetObjecttruehttps://uor.foundation/division/CayleyDicksonConstructionhttps://uor.foundation/division/NormedDivisionAlgebraThe target algebra of the Cayley-Dickson doubling.
adjoinedElementDatatypetruehttps://uor.foundation/division/CayleyDicksonConstructionhttp://www.w3.org/2001/XMLSchema#stringThe new basis element adjoined by this doubling step.
conjugationRuleDatatypetruehttps://uor.foundation/division/CayleyDicksonConstructionhttp://www.w3.org/2001/XMLSchema#stringThe conjugation and multiplication rule for the adjoined element.

Named Individuals

NameTypePropertiesComment
RealAlgebrahttps://uor.foundation/division/NormedDivisionAlgebra
  • algebraDimension: 1
  • isCommutative: true
  • isAssociative: true
  • basisElements: {1}
The real numbers R: dimension 1, commutative, associative.
ComplexAlgebrahttps://uor.foundation/division/NormedDivisionAlgebra
  • algebraDimension: 2
  • isCommutative: true
  • isAssociative: true
  • basisElements: {1, i}
The complex numbers C: dimension 2, commutative, associative.
QuaternionAlgebrahttps://uor.foundation/division/NormedDivisionAlgebra
  • algebraDimension: 4
  • isCommutative: false
  • isAssociative: true
  • basisElements: {1, i, j, k}
The quaternions H: dimension 4, non-commutative, associative.
OctonionAlgebrahttps://uor.foundation/division/NormedDivisionAlgebra
  • algebraDimension: 8
  • isCommutative: false
  • isAssociative: false
  • basisElements: {1, i, j, k, l, il, jl, kl}
The octonions O: dimension 8, non-commutative, non-associative.
cayleyDickson_R_to_Chttps://uor.foundation/division/CayleyDicksonConstruction
  • cayleyDicksonSource: https://uor.foundation/division/RealAlgebra
  • cayleyDicksonTarget: https://uor.foundation/division/ComplexAlgebra
  • adjoinedElement: i
  • conjugationRule: i² = −1
Cayley-Dickson doubling R → C: adjoin i with i² = −1.
cayleyDickson_C_to_Hhttps://uor.foundation/division/CayleyDicksonConstruction
  • cayleyDicksonSource: https://uor.foundation/division/ComplexAlgebra
  • cayleyDicksonTarget: https://uor.foundation/division/QuaternionAlgebra
  • adjoinedElement: j
  • conjugationRule: ij = k, ji = −k
Cayley-Dickson doubling C → H: adjoin j with j² = −1, ij = k, ji = −k.
cayleyDickson_H_to_Ohttps://uor.foundation/division/CayleyDicksonConstruction
  • cayleyDicksonSource: https://uor.foundation/division/QuaternionAlgebra
  • cayleyDicksonTarget: https://uor.foundation/division/OctonionAlgebra
  • adjoinedElement: l
  • conjugationRule: non-associative Fano plane products
Cayley-Dickson doubling H → O: adjoin l, non-associative Fano plane products.