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SandBoxNonAssociativeAlgebra

last edited 18 years ago by Bill Page

Edit detail for SandBoxNonAssociativeAlgebra revision 14 of 25

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Editor: Bill Page Time: 2008/04/28 21:52:57 GMT-7
Note: properties

changed:
-
\begin{axiom}
V has FramedNonAssociativeAlgebra(Fraction Integer)
basisOfCenter()$AlgebraPackage(Fraction Integer,V)
basisOfCentroid()$AlgebraPackage(Fraction Integer,V)
basisOfNucleus()$AlgebraPackage(Fraction Integer,V)
basisOfLeftNucloid()$AlgebraPackage(Fraction Integer,V)
\end{axiom}

\begin{axiom}
G:=GenericNonAssociativeAlgebra(Fraction Integer, 4, [e,i,j,k],sc)
)show G
p1:=generic([x1,y1,z1,w1])$G
p2:=generic([x2,y2,z2,w2])$G
p3:=generic([x3,y3,z3,w3])$G
associator(p1,p2,p3)$G
leftRegularRepresentation(p1)
rightRegularRepresentation(p1)
\end{axiom}

Ref: http://arxiv.org/abs/0711.3220

Fourvector algebra

Author: Diego Saa (Submitted on 20 Nov 2007)

Abstract: The algebra of fourvectors is described. The fourvectors are more appropriate than the Hamilton quaternions for its use in Physics and the sciences in general. The fourvectors embrace the 3D vectors in a natural form. It is shown the excellent ability to perform rotations with the use of fourvectors, as well as their use in relativity for producing Lorentz boosts, which are understood as simple rotations.

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(1) -> _*_*(x,y)==concat(x(1) * y(1) + dot(x(2..), y(2..)), x(1) * y(2..) - x(2..) * y(1) + cross(x(2..), y(2..)))

Type: Void

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e:Vector INT:=[1,0,0,0]

$\label{eq1}\left[ 1, \: 0, \: 0, \: 0 \right]$

(1)

Type: Vector(Integer)

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i:Vector INT:=[0,1,0,0]

$\label{eq2}\left[ 0, \: 1, \: 0, \: 0 \right]$

(2)

Type: Vector(Integer)

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j:Vector INT:=[0,0,1,0]

$\label{eq3}\left[ 0, \: 0, \: 1, \: 0 \right]$

(3)

Type: Vector(Integer)

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k:Vector INT:=[0,0,0,1]

$\label{eq4}\left[ 0, \: 0, \: 0, \: 1 \right]$

(4)

Type: Vector(Integer)

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test(e**e=e)  and _
test(i**i=e)  and _
test(j**j=e)  and _
test(k**k=e)  and _
test(e**i=i)  and _
test(e**j=j)  and _
test(e**k=k)  and _
test(i**e=-i) and _
test(j**e=-j) and _
test(k**e=-k) and _
test(i**j=k)  and _
test(j**i=-k) and _
test(k**i=j)  and _
test(i**k=-j) and _
test(j**k=i)  and _
test(k**j=-i)

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Compiling function ** with type (Vector(Integer), Vector(Integer))
       -> Vector(Integer)

$\label{eq5} \mbox{\rm true}$

(5)

Type: Boolean

Axiom has a domain for NonAssociative? Algebra

This is documented in the article: Computations in Algebras of Fixed Rank by Johannes Grabmeir and Robert Wisbauer, from the book "Computational Algebra" By Klaus G. Fischer, Philippe Loustaunau, Jay Shapiro.

The algebra above can be given by structural constants.

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)clear all

   All user variables and function definitions have been cleared.
sc:Vector Matrix Fraction Integer := [ _
[[ 1, 0, 0, 0], _
 [ 0, 1, 0, 0], _
 [ 0, 0, 1, 0], _
 [ 0, 0, 0, 1]], _
[[ 0, 1, 0, 0], _
 [-1, 0, 0, 0], _
 [ 0, 0, 0, 1], _
 [ 0, 0,-1, 0]], _
[[ 0, 0, 1, 0], _
 [ 0, 0, 0,-1], _
 [-1, 0, 0, 0], _
 [ 0, 1, 0, 0]], _
[[ 0, 0, 0, 1], _
 [ 0, 0, 1, 0], _
 [ 0,-1, 0, 0], _
 [-1, 0, 0, 0]]];

Type: Vector(Matrix(Fraction(Integer)))

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V:=AlgebraGivenByStructuralConstants(Fraction Integer, 4, [e,i,j,k],sc)

$\label{eq6}\hbox{\axiomType{AlgebraGivenByStructuralConstants}\ } \left({{\hbox{\axiomType{Fraction}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \: 4, \:{\left[ e , \: i , \: j , \: k \right]}, \:{\left[{\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(6)

Type: Type

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)show V

 AlgebraGivenByStructuralConstants(Fraction(Integer),4,[e,i,j,k],[[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],[[0,1,0,0],[-1,0,0,0],[0,0,0,1],[0,0,-1,0]],[[0,0,1,0],[0,0,0,-1],[-1,0,0,0],[0,1,0,0]],[[0,0,0,1],[0,0,1,0],[0,-1,0,0],[-1,0,0,0]]]) is a domain constructor.
 Abbreviation for AlgebraGivenByStructuralConstants is ALGSC 
 This constructor is exposed in this frame.
 89 names for 110 operations in this domain.
------------------------------- Operations --------------------------------

 0 : () -> %                           ?*? : (%, %) -> %
 ?*? : (Integer, %) -> %               ?*? : (PositiveInteger, %) -> %
 ?+? : (%, %) -> %                     -? : % -> %
 ?-? : (%, %) -> %                     ?=? : (%, %) -> Boolean
 ?^? : (%, PositiveInteger) -> %       alternative? : () -> Boolean
 antiAssociative? : () -> Boolean      antiCommutative? : () -> Boolean
 antiCommutator : (%, %) -> %          associative? : () -> Boolean
 associator : (%, %, %) -> %           basis : () -> Vector(%)
 coerce : % -> OutputForm              commutative? : () -> Boolean
 commutator : (%, %) -> %              flexible? : () -> Boolean
 hash : % -> SingleInteger             jacobiIdentity? : () -> Boolean
 jordanAdmissible? : () -> Boolean     jordanAlgebra? : () -> Boolean
 latex : % -> String                   leadingMonomial : % -> %
 leftAlternative? : () -> Boolean      leftNorm : % -> Fraction(Integer)
 lieAdmissible? : () -> Boolean        lieAlgebra? : () -> Boolean
 monomial? : % -> Boolean              monomials : % -> List(%)
 opposite? : (%, %) -> Boolean         powerAssociative? : () -> Boolean
 rank : () -> PositiveInteger          recip : % -> Union(%,"failed")
 reductum : % -> %                     rightAlternative? : () -> Boolean
 sample : () -> %                      smaller? : (%, %) -> Boolean
 someBasis : () -> Vector(%)           unit : () -> Union(%,"failed")
 zero? : % -> Boolean                  ?~=? : (%, %) -> Boolean
 ?*? : (%, Fraction(Integer)) -> %
 ?*? : (Fraction(Integer), %) -> %
 ?*? : (NonNegativeInteger, %) -> %
 ?*? : (SquareMatrix(4,Fraction(Integer)), %) -> %
 apply : (Matrix(Fraction(Integer)), %) -> %
 associatorDependence : () -> List(Vector(Fraction(Integer)))
 coefficient : (%, OrderedVariableList([e,i,j,k])) -> Fraction(Integer)
 coefficients : % -> List(Fraction(Integer))
 coerce : Vector(Fraction(Integer)) -> %
 conditionsForIdempotents : () -> List(Polynomial(Fraction(Integer)))
 conditionsForIdempotents : Vector(%) -> List(Polynomial(Fraction(Integer)))
 construct : List(Record(k: OrderedVariableList([e,i,j,k]),c: Fraction(Integer))) -> %
 constructOrdered : List(Record(k: OrderedVariableList([e,i,j,k]),c: Fraction(Integer))) -> %
 convert : Vector(Fraction(Integer)) -> %
 convert : % -> Vector(Fraction(Integer))
 coordinates : Vector(%) -> Matrix(Fraction(Integer))
 coordinates : (Vector(%), Vector(%)) -> Matrix(Fraction(Integer))
 coordinates : % -> Vector(Fraction(Integer))
 coordinates : (%, Vector(%)) -> Vector(Fraction(Integer))
 elt : (%, Integer) -> Fraction(Integer)
 hashUpdate! : (HashState, %) -> HashState
 leadingCoefficient : % -> Fraction(Integer)
 leadingSupport : % -> OrderedVariableList([e,i,j,k])
 leadingTerm : % -> Record(k: OrderedVariableList([e,i,j,k]),c: Fraction(Integer))
 leftCharacteristicPolynomial : % -> SparseUnivariatePolynomial(Fraction(Integer))
 leftDiscriminant : () -> Fraction(Integer)
 leftDiscriminant : Vector(%) -> Fraction(Integer)
 leftMinimalPolynomial : % -> SparseUnivariatePolynomial(Fraction(Integer))
 leftPower : (%, PositiveInteger) -> %
 leftRankPolynomial : () -> SparseUnivariatePolynomial(Polynomial(Fraction(Integer)))
 leftRecip : % -> Union(%,"failed")
 leftRegularRepresentation : % -> Matrix(Fraction(Integer))
 leftRegularRepresentation : (%, Vector(%)) -> Matrix(Fraction(Integer))
 leftTrace : % -> Fraction(Integer)
 leftTraceMatrix : () -> Matrix(Fraction(Integer))
 leftTraceMatrix : Vector(%) -> Matrix(Fraction(Integer))
 leftUnit : () -> Union(%,"failed")
 leftUnits : () -> Union(Record(particular: %,basis: List(%)),"failed")
 linearExtend : ((OrderedVariableList([e,i,j,k]) -> Fraction(Integer)), %) -> Fraction(Integer)
 listOfTerms : % -> List(Record(k: OrderedVariableList([e,i,j,k]),c: Fraction(Integer)))
 map : ((Fraction(Integer) -> Fraction(Integer)), %) -> %
 monomial : (Fraction(Integer), OrderedVariableList([e,i,j,k])) -> %
 noncommutativeJordanAlgebra? : () -> Boolean
 numberOfMonomials : % -> NonNegativeInteger
 plenaryPower : (%, PositiveInteger) -> %
 represents : Vector(Fraction(Integer)) -> %
 represents : (Vector(Fraction(Integer)), Vector(%)) -> %
 rightCharacteristicPolynomial : % -> SparseUnivariatePolynomial(Fraction(Integer))
 rightDiscriminant : () -> Fraction(Integer)
 rightDiscriminant : Vector(%) -> Fraction(Integer)
 rightMinimalPolynomial : % -> SparseUnivariatePolynomial(Fraction(Integer))
 rightNorm : % -> Fraction(Integer)
 rightPower : (%, PositiveInteger) -> %
 rightRankPolynomial : () -> SparseUnivariatePolynomial(Polynomial(Fraction(Integer)))
 rightRecip : % -> Union(%,"failed")
 rightRegularRepresentation : % -> Matrix(Fraction(Integer))
 rightRegularRepresentation : (%, Vector(%)) -> Matrix(Fraction(Integer))
 rightTrace : % -> Fraction(Integer)
 rightTraceMatrix : () -> Matrix(Fraction(Integer))
 rightTraceMatrix : Vector(%) -> Matrix(Fraction(Integer))
 rightUnit : () -> Union(%,"failed")
 rightUnits : () -> Union(Record(particular: %,basis: List(%)),"failed")
 structuralConstants : () -> Vector(Matrix(Fraction(Integer)))
 structuralConstants : Vector(%) -> Vector(Matrix(Fraction(Integer)))
 subtractIfCan : (%, %) -> Union(%,"failed")
 support : % -> List(OrderedVariableList([e,i,j,k]))

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a:=basis()$V

$\label{eq7}\left[ e , \: i , \: j , \: k \right]$

(7)

Type: Vector(AlgebraGivenByStructuralConstants?(Fraction(Integer),4,[e, i, j, k],[[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], [[0, 1, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 1], [0, 0, -1, 0]], [[0, 0, 1, 0], [0, 0, 0, -1], [-1, 0, 0, 0], [0, 1, 0, 0]], [[0, 0, 0, 1], [0, 0, 1, 0], [0, -1, 0, 0], [-1, 0, 0, 0]]]))

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matrix([[(a.i * a.j) for j in 1..4] for i in 1..4])$OutputForm

$\label{eq8}\left[ \begin{array}{cccc} e & i & j & k \ - i & e & k & - j \ - j & - k & e & i \ - k & j & - i & e$

(8)

Type: OutputForm?

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leftUnit()$V

$\label{eq9}e$

(9)

Type: Union(AlgebraGivenByStructuralConstants?(Fraction(Integer),4,[e, i, j, k],[[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], [[0, 1, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 1], [0, 0, -1, 0]], [[0, 0, 1, 0], [0, 0, 0, -1], [-1, 0, 0, 0], [0, 1, 0, 0]], [[0, 0, 0, 1], [0, 0, 1, 0], [0, -1, 0, 0], [-1, 0, 0, 0]]]),...)

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rightUnit()$V

$\label{eq10}\verb#"failed"#$

(10)

Type: Union("failed",...)

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alternative?()$V

$\label{eq11} \mbox{\rm false}$

(11)

Type: Boolean

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leftAlternative?()$V

$\label{eq12} \mbox{\rm false}$

(12)

Type: Boolean

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rightAlternative?()$V

$\label{eq13} \mbox{\rm false}$

(13)

Type: Boolean

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associative?()$V

$\label{eq14} \mbox{\rm false}$

(14)

Type: Boolean

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antiAssociative?()$V

$\label{eq15} \mbox{\rm false}$

(15)

Type: Boolean

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--powerAssociative?()$V
commutative?()$V

$\label{eq16} \mbox{\rm false}$

(16)

Type: Boolean

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jordanAlgebra?()$V

$\label{eq17} \mbox{\rm false}$

(17)

Type: Boolean

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jordanAdmissible?()$V

$\label{eq18} \mbox{\rm false}$

(18)

Type: Boolean

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noncommutativeJordanAlgebra?()$V

$\label{eq19} \mbox{\rm false}$

(19)

Type: Boolean

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lieAlgebra?()$V

$\label{eq20} \mbox{\rm false}$

(20)

Type: Boolean

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lieAdmissible?()$V

$\label{eq21} \mbox{\rm false}$

(21)

Type: Boolean

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jacobiIdentity?()$V

$\label{eq22} \mbox{\rm false}$

(22)

Type: Boolean

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V has FramedNonAssociativeAlgebra(Fraction Integer)

$\label{eq23} \mbox{\rm true}$

(23)

Type: Boolean

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basisOfCenter()$AlgebraPackage(Fraction Integer,V)

$\label{eq24}\left[ \right]$

(24)

Type: List(AlgebraGivenByStructuralConstants?(Fraction(Integer),4,[e, i, j, k],[[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], [[0, 1, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 1], [0, 0, -1, 0]], [[0, 0, 1, 0], [0, 0, 0, -1], [-1, 0, 0, 0], [0, 1, 0, 0]], [[0, 0, 0, 1], [0, 0, 1, 0], [0, -1, 0, 0], [-1, 0, 0, 0]]]))

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basisOfCentroid()$AlgebraPackage(Fraction Integer,V)

$\label{eq25}\left[ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(25)

Type: List(Matrix(Fraction(Integer)))

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basisOfNucleus()$AlgebraPackage(Fraction Integer,V)

$\label{eq26}\left[ \right]$

(26)

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basisOfLeftNucloid()$AlgebraPackage(Fraction Integer,V)

$\label{eq27}\left[ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(27)

Type: List(Matrix(Fraction(Integer)))

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G:=GenericNonAssociativeAlgebra(Fraction Integer, 4, [e,i,j,k],sc)

$\label{eq28}\hbox{\axiomType{GenericNonAssociativeAlgebra}\ } \left({{\hbox{\axiomType{Fraction}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \: 4, \:{\left[ e , \: i , \: j , \: k \right]}, \:{\left[{\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(28)

Type: Type

fricas

)show G

 GenericNonAssociativeAlgebra(Fraction(Integer),4,[e,i,j,k],[[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]],[[0,1,0,0],[-1,0,0,0],[0,0,0,1],[0,0,-1,0]],[[0,0,1,0],[0,0,0,-1],[-1,0,0,0],[0,1,0,0]],[[0,0,0,1],[0,0,1,0],[0,-1,0,0],[-1,0,0,0]]]) is a domain constructor.
 Abbreviation for GenericNonAssociativeAlgebra is GCNAALG 
 This constructor is not exposed in this frame.
 82 names for 112 operations in this domain.
------------------------------- Operations --------------------------------

 0 : () -> %                           ?*? : (%, %) -> %
 ?*? : (Integer, %) -> %               ?*? : (PositiveInteger, %) -> %
 ?+? : (%, %) -> %                     -? : % -> %
 ?-? : (%, %) -> %                     ?=? : (%, %) -> Boolean
 ?^? : (%, PositiveInteger) -> %       alternative? : () -> Boolean
 antiAssociative? : () -> Boolean      antiCommutative? : () -> Boolean
 antiCommutator : (%, %) -> %          associative? : () -> Boolean
 associator : (%, %, %) -> %           basis : () -> Vector(%)
 coerce : % -> OutputForm              commutative? : () -> Boolean
 commutator : (%, %) -> %              flexible? : () -> Boolean
 generic : () -> %                     generic : Symbol -> %
 generic : Vector(%) -> %              generic : Vector(Symbol) -> %
 hash : % -> SingleInteger             jacobiIdentity? : () -> Boolean
 jordanAdmissible? : () -> Boolean     jordanAlgebra? : () -> Boolean
 latex : % -> String                   leftAlternative? : () -> Boolean
 lieAdmissible? : () -> Boolean        lieAlgebra? : () -> Boolean
 opposite? : (%, %) -> Boolean         powerAssociative? : () -> Boolean
 rank : () -> PositiveInteger          recip : % -> Union(%,"failed")
 rightAlternative? : () -> Boolean     sample : () -> %
 someBasis : () -> Vector(%)           unit : () -> Union(%,"failed")
 zero? : % -> Boolean                  ?~=? : (%, %) -> Boolean
 ?*? : (%, Fraction(Polynomial(Fraction(Integer)))) -> %
 ?*? : (Fraction(Polynomial(Fraction(Integer))), %) -> %
 ?*? : (NonNegativeInteger, %) -> %
 ?*? : (SquareMatrix(4,Fraction(Polynomial(Fraction(Integer)))), %) -> %
 apply : (Matrix(Fraction(Polynomial(Fraction(Integer)))), %) -> %
 associatorDependence : () -> List(Vector(Fraction(Polynomial(Fraction(Integer)))))
 coerce : Vector(Fraction(Polynomial(Fraction(Integer)))) -> %
 conditionsForIdempotents : () -> List(Polynomial(Fraction(Integer)))
 conditionsForIdempotents : Vector(%) -> List(Polynomial(Fraction(Integer)))
 conditionsForIdempotents : () -> List(Polynomial(Fraction(Polynomial(Fraction(Integer)))))
 conditionsForIdempotents : Vector(%) -> List(Polynomial(Fraction(Polynomial(Fraction(Integer)))))
 convert : Vector(Fraction(Polynomial(Fraction(Integer)))) -> %
 convert : % -> Vector(Fraction(Polynomial(Fraction(Integer))))
 coordinates : Vector(%) -> Matrix(Fraction(Polynomial(Fraction(Integer))))
 coordinates : (Vector(%), Vector(%)) -> Matrix(Fraction(Polynomial(Fraction(Integer))))
 coordinates : % -> Vector(Fraction(Polynomial(Fraction(Integer))))
 coordinates : (%, Vector(%)) -> Vector(Fraction(Polynomial(Fraction(Integer))))
 elt : (%, Integer) -> Fraction(Polynomial(Fraction(Integer)))
 generic : (Symbol, Vector(%)) -> %
 generic : (Vector(Symbol), Vector(%)) -> %
 genericLeftDiscriminant : () -> Fraction(Polynomial(Fraction(Integer)))
 genericLeftMinimalPolynomial : % -> SparseUnivariatePolynomial(Fraction(Polynomial(Fraction(Integer))))
 genericLeftNorm : % -> Fraction(Polynomial(Fraction(Integer)))
 genericLeftTrace : % -> Fraction(Polynomial(Fraction(Integer)))
 genericLeftTraceForm : (%, %) -> Fraction(Polynomial(Fraction(Integer)))
 genericRightDiscriminant : () -> Fraction(Polynomial(Fraction(Integer)))
 genericRightMinimalPolynomial : % -> SparseUnivariatePolynomial(Fraction(Polynomial(Fraction(Integer))))
 genericRightNorm : % -> Fraction(Polynomial(Fraction(Integer)))
 genericRightTrace : % -> Fraction(Polynomial(Fraction(Integer)))
 genericRightTraceForm : (%, %) -> Fraction(Polynomial(Fraction(Integer)))
 hashUpdate! : (HashState, %) -> HashState
 leftCharacteristicPolynomial : % -> SparseUnivariatePolynomial(Fraction(Polynomial(Fraction(Integer))))
 leftDiscriminant : () -> Fraction(Polynomial(Fraction(Integer)))
 leftDiscriminant : Vector(%) -> Fraction(Polynomial(Fraction(Integer)))
 leftMinimalPolynomial : % -> SparseUnivariatePolynomial(Fraction(Polynomial(Fraction(Integer))))
 leftNorm : % -> Fraction(Polynomial(Fraction(Integer)))
 leftPower : (%, PositiveInteger) -> %
 leftRankPolynomial : () -> SparseUnivariatePolynomial(Fraction(Polynomial(Fraction(Integer))))
 leftRankPolynomial : () -> SparseUnivariatePolynomial(Polynomial(Fraction(Polynomial(Fraction(Integer)))))
 leftRecip : % -> Union(%,"failed")
 leftRegularRepresentation : % -> Matrix(Fraction(Polynomial(Fraction(Integer))))
 leftRegularRepresentation : (%, Vector(%)) -> Matrix(Fraction(Polynomial(Fraction(Integer))))
 leftTrace : % -> Fraction(Polynomial(Fraction(Integer)))
 leftTraceMatrix : () -> Matrix(Fraction(Polynomial(Fraction(Integer))))
 leftTraceMatrix : Vector(%) -> Matrix(Fraction(Polynomial(Fraction(Integer))))
 leftUnit : () -> Union(%,"failed")
 leftUnits : () -> Union(Record(particular: %,basis: List(%)),"failed")
 noncommutativeJordanAlgebra? : () -> Boolean
 plenaryPower : (%, PositiveInteger) -> %
 represents : Vector(Fraction(Polynomial(Fraction(Integer)))) -> %
 represents : (Vector(Fraction(Polynomial(Fraction(Integer)))), Vector(%)) -> %
 rightCharacteristicPolynomial : % -> SparseUnivariatePolynomial(Fraction(Polynomial(Fraction(Integer))))
 rightDiscriminant : () -> Fraction(Polynomial(Fraction(Integer)))
 rightDiscriminant : Vector(%) -> Fraction(Polynomial(Fraction(Integer)))
 rightMinimalPolynomial : % -> SparseUnivariatePolynomial(Fraction(Polynomial(Fraction(Integer))))
 rightNorm : % -> Fraction(Polynomial(Fraction(Integer)))
 rightPower : (%, PositiveInteger) -> %
 rightRankPolynomial : () -> SparseUnivariatePolynomial(Fraction(Polynomial(Fraction(Integer))))
 rightRankPolynomial : () -> SparseUnivariatePolynomial(Polynomial(Fraction(Polynomial(Fraction(Integer)))))
 rightRecip : % -> Union(%,"failed")
 rightRegularRepresentation : % -> Matrix(Fraction(Polynomial(Fraction(Integer))))
 rightRegularRepresentation : (%, Vector(%)) -> Matrix(Fraction(Polynomial(Fraction(Integer))))
 rightTrace : % -> Fraction(Polynomial(Fraction(Integer)))
 rightTraceMatrix : () -> Matrix(Fraction(Polynomial(Fraction(Integer))))
 rightTraceMatrix : Vector(%) -> Matrix(Fraction(Polynomial(Fraction(Integer))))
 rightUnit : () -> Union(%,"failed")
 rightUnits : () -> Union(Record(particular: %,basis: List(%)),"failed")
 structuralConstants : () -> Vector(Matrix(Fraction(Polynomial(Fraction(Integer)))))
 structuralConstants : Vector(%) -> Vector(Matrix(Fraction(Polynomial(Fraction(Integer)))))
 subtractIfCan : (%, %) -> Union(%,"failed")

p1:=generic([x1,y1,z1,w1])$G

$\label{eq29}{w 1 \ k}+{z 1 \ j}+{y 1 \ i}+{x 1 \ e}$

(29)

Type: GenericNonAssociativeAlgebra?(Fraction(Integer),4,[e, i, j, k],[[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], [[0, 1, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 1], [0, 0, -1, 0]], [[0, 0, 1, 0], [0, 0, 0, -1], [-1, 0, 0, 0], [0, 1, 0, 0]], [[0, 0, 0, 1], [0, 0, 1, 0], [0, -1, 0, 0], [-1, 0, 0, 0]]])

fricas

p2:=generic([x2,y2,z2,w2])$G

$\label{eq30}{w 2 \ k}+{z 2 \ j}+{y 2 \ i}+{x 2 \ e}$

(30)

fricas

p3:=generic([x3,y3,z3,w3])$G

$\label{eq31}{w 3 \ k}+{z 3 \ j}+{y 3 \ i}+{x 3 \ e}$

(31)

fricas

associator(p1,p2,p3)$G

$\label{eq32}\begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{2 \ x 2 \ y 1 \ z 3}+{2 \ w 3 \ z 1 \ z 2}+{2 \ x 2 \ y 3 \ z 1}+ \ \ \displaystyle {2 \ w 3 \ y 1 \ y 2}+{2 \ w 1 \ x 2 \ x 3}+{2 \ w 1 \ w 2 \ w 3}$

(32)

fricas

leftRegularRepresentation(p1)

$\label{eq33}\left[ \begin{array}{cccc} x 1 & y 1 & z 1 & w 1 \ - y 1 & x 1 & - w 1 & z 1 \ - z 1 & w 1 & x 1 & - y 1 \ - w 1 & - z 1 & y 1 & x 1$

(33)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))

fricas

rightRegularRepresentation(p1)

$\label{eq34}\left[ \begin{array}{cccc} x 1 & y 1 & z 1 & w 1 \ y 1 & - x 1 & w 1 & - z 1 \ z 1 & - w 1 & - x 1 & y 1 \ w 1 & z 1 & - y 1 & - x 1$

(34)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))