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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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_*_*(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)

$\label{eq6}\hbox{\axiomType{AlgebraGivenByStructuralConstants}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ }) , 4, [ e , i , j , k ] , [ [ [ 1, 0, 0, 0 ] , [ 0, 1, 0, 0 ] , [ 0, 0, 1, 0 ] , [ 0, 0, 0, 1 ] ] , [ [ 0, 1, 0, 0 ] , [ - 1, 0, 0, 0 ] , [ 0, 0, 0, 1 ] , [ 0, 0, - 1, 0 ] ] , [ [ 0, 0, 1, 0 ] , [ 0, 0, 0, - 1 ] , [ - 1, 0, 0, 0 ] , [ 0, 1, 0, 0 ] ] , [ [ 0, 0, 0, 1 ] , [ 0, 0, 1, 0 ] , [ 0, - 1, 0, 0 ] , [ - 1, 0, 0, 0 ] ] ])$

(6)

Type: Type

Multiplication

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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?

Commutator and Associator

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matrix([[commutator(a.x,a.y) for x in 1..4] for y in 1..4])$OutputForm

$\label{eq9}\left[ \begin{array}{cccc} 0 & -{2 \ i}& -{2 \ j}& -{2 \ k} \ {2 \ i}& 0 & -{2 \ k}&{2 \ j} \ {2 \ j}&{2 \ k}& 0 & -{2 \ i} \ {2 \ k}& -{2 \ j}&{2 \ i}& 0$

(9)

Type: OutputForm?

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[matrix([[associator(a.x,a.y,a.z) for x in 1..4] for y in 1..4])$OutputForm for z in 1..4]

$\label{eq10}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cccc} 0 &{2 \ i}&{2 \ j}&{2 \ k} \ 0 &{2 \ e}& 0 & 0 \ 0 & 0 &{2 \ e}& 0 \ 0 & 0 & 0 &{2 \ e}$

(10)

Type: List(OutputForm?)

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for x in 1..4 repeat
  for y in 1..4 repeat
    for z in 1..4 repeat
      if associator(a.x,a.y,a.z) ~= 0$V then
        output([[a.x,a.y,a.z],"=",associator(a.x,a.y,a.z)])

   [[i, e, e], "=", 2 i]
   [[i, e, i], "=", - 2 e]
   [[i, e, j], "=", - 2 k]
   [[i, e, k], "=", 2 j]
   [[i, i, e], "=", 2 e]
   [[i, i, i], "=", 2 i]
   [[i, i, j], "=", 2 j]
   [[i, i, k], "=", 2 k]
   [[j, e, e], "=", 2 j]
   [[j, e, i], "=", 2 k]
   [[j, e, j], "=", - 2 e]
   [[j, e, k], "=", - 2 i]
   [[j, j, e], "=", 2 e]
   [[j, j, i], "=", 2 i]
   [[j, j, j], "=", 2 j]
   [[j, j, k], "=", 2 k]
   [[k, e, e], "=", 2 k]
   [[k, e, i], "=", - 2 j]
   [[k, e, j], "=", 2 i]
   [[k, e, k], "=", - 2 e]
   [[k, k, e], "=", 2 e]
   [[k, k, i], "=", 2 i]
   [[k, k, j], "=", 2 j]
   [[k, k, k], "=", 2 k]

Type: Void

Volume form?

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a.2 * (a.3 * a.4) = (a.2 * a.3) * a.4

$\label{eq11}e = e$

(11)

Type: Equation(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]]]))

Check standard properties

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leftUnit()$V

$\label{eq12}e$

(12)

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

   this algebra has no right unit

$\label{eq13}\verb#"failed"#$

(13)

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

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alternative?()$V

   algebra is not left alternative

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

(14)

Type: Boolean

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leftAlternative?()$V

   algebra is not left alternative

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

(15)

Type: Boolean

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rightAlternative?()$V

   algebra is not right alternative

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

(16)

Type: Boolean

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associative?()$V

   algebra is not associative

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

(17)

Type: Boolean

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antiAssociative?()$V

   algebra is not anti-associative

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

(18)

Type: Boolean

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

   algebra is not commutative

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

(19)

Type: Boolean

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antiCommutative?()$V

   algebra is not anti-commutative

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

(20)

Type: Boolean

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jordanAlgebra?()$V

   algebra is not commutative
   this is not a Jordan algebra

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

(21)

Type: Boolean

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jordanAdmissible?()$V

   algebra is not Jordan admissible

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

(22)

Type: Boolean

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noncommutativeJordanAlgebra?()$V

   algebra is not flexible
   this is not a noncommutative Jordan algebra, as it is not flexible

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

(23)

Type: Boolean

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lieAlgebra?()$V

   algebra is not anti-commutative
   this is not a Lie algebra

$\label{eq24} \mbox{\rm false}$

(24)

Type: Boolean

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lieAdmissible?()$V

   algebra is not Lie admissible

$\label{eq25} \mbox{\rm false}$

(25)

Type: Boolean

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jacobiIdentity?()$V

   Jacobi identity does not hold

$\label{eq26} \mbox{\rm false}$

(26)

Type: Boolean

Commuting elements

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V has FramedNonAssociativeAlgebra(Fraction Integer)

$\label{eq27} \mbox{\rm true}$

(27)

Type: Boolean

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basisOfCommutingElements()$AlgebraPackage(Fraction Integer,V)

$\label{eq28}\left[ \right]$

(28)

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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basisOfCenter()$AlgebraPackage(Fraction Integer,V)

$\label{eq29}\left[ \right]$

(29)

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basisOfCentroid()$AlgebraPackage(Fraction Integer,V)

$\label{eq30}\left[ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(30)

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

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basisOfNucleus()$AlgebraPackage(Fraction Integer,V)

$\label{eq31}\left[ \right]$

(31)

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basisOfLeftNucloid()$AlgebraPackage(Fraction Integer,V)

$\label{eq32}\left[ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(32)

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

Symbolic computations

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

$\label{eq33}\hbox{\axiomType{GenericNonAssociativeAlgebra}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ }) , 4, [ e , i , j , k ] , [ [ [ 1, 0, 0, 0 ] , [ 0, 1, 0, 0 ] , [ 0, 0, 1, 0 ] , [ 0, 0, 0, 1 ] ] , [ [ 0, 1, 0, 0 ] , [ - 1, 0, 0, 0 ] , [ 0, 0, 0, 1 ] , [ 0, 0, - 1, 0 ] ] , [ [ 0, 0, 1, 0 ] , [ 0, 0, 0, - 1 ] , [ - 1, 0, 0, 0 ] , [ 0, 1, 0, 0 ] ] , [ [ 0, 0, 0, 1 ] , [ 0, 0, 1, 0 ] , [ 0, - 1, 0, 0 ] , [ - 1, 0, 0, 0 ] ] ])$

(33)

Type: Type

Look for Idempotents

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conditionsForIdempotents()$G

$\label{eq34}\left[{{{\%x 4}^{2}}+{{\%x 3}^{2}}+{{\%x 2}^{2}}+{{\%x 1}^{2}}- \%x 1}, \: - \%x 2, \: - \%x 3, \: - \%x 4 \right]$

(34)

Type: List(Polynomial(Fraction(Integer)))

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gb:=groebnerFactorize %

$\label{eq35}\left[{\left[ \%x 4, \: \%x 3, \: \%x 2, \: \%x 1 \right]}, \:{\left[ \%x 4, \: \%x 3, \: \%x 2, \:{\%x 1 - 1}\right]}\right]$

(35)

Type: List(List(Polynomial(Fraction(Integer))))

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associatorDependence()$G

$\label{eq36}\left[ \right]$

(36)

Type: List(Vector(Fraction(Polynomial(Fraction(Integer)))))

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q:=leftRankPolynomial()$G

$\label{eq37}{{?}^{3}}-{2 \ \%x 1 \ {{?}^{2}}}+{{\left({{\%x 4}^{2}}+{{\%x 3}^{2}}+{{\%x 2}^{2}}+{{\%x 1}^{2}}\right)}\ ?}$

(37)

Type: SparseUnivariatePolynomial?(Fraction(Polynomial(Fraction(Integer))))

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map(factor,coefficients q)

$\label{eq38}\left[ 1, \: -{2 \ \%x 1}, \:{{{\%x 4}^{2}}+{{\%x 3}^{2}}+{{\%x 2}^{2}}+{{\%x 1}^{2}}}\right]$

(38)

Type: List(Factored(Fraction(Polynomial(Fraction(Integer)))))

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rightUnit()$G

   this algebra has no right unit

$\label{eq39}\verb#"failed"#$

(39)

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

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

$\label{eq40}{w 1 \ k}+{z 1 \ j}+{y 1 \ i}+{x 1 \ e}$

(40)

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]]])

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

$\label{eq41}{w 2 \ k}+{z 2 \ j}+{y 2 \ i}+{x 2 \ e}$

(41)

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

$\label{eq42}{w 3 \ k}+{z 3 \ j}+{y 3 \ i}+{x 3 \ e}$

(42)

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leftRecip(p1)$G

$\label{eq43}\begin{array}{@{}l} \displaystyle {{w 1 \over{{{z 1}^{2}}+{{y 1}^{2}}+{{x 1}^{2}}+{{w 1}^{2}}}}\ k}+{{z 1 \over{{{z 1}^{2}}+{{y 1}^{2}}+{{x 1}^{2}}+{{w 1}^{2}}}}\ j}+ \ \ \displaystyle {{y 1 \over{{{z 1}^{2}}+{{y 1}^{2}}+{{x 1}^{2}}+{{w 1}^{2}}}}\ i}+{{x 1 \over{{{z 1}^{2}}+{{y 1}^{2}}+{{x 1}^{2}}+{{w 1}^{2}}}}\ e}$

(43)

Type: Union(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]]]),...)

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rightRecip(p1)$G

   this algebra has no right unit

$\label{eq44}\verb#"failed"#$

(44)

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

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leftRegularRepresentation(p1)

$\label{eq45}\left[ \begin{array}{cccc} x 1 & y 1 & z 1 & w 1 \ - y 1 & x 1 & - w 1 & z 1 \ - z 1 & w 1 & x 1 & - y 1 \ - w 1 & - z 1 & y 1 & x 1$

(45)

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

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rightRegularRepresentation(p1)

$\label{eq46}\left[ \begin{array}{cccc} x 1 & y 1 & z 1 & w 1 \ y 1 & - x 1 & w 1 & - z 1 \ z 1 & - w 1 & - x 1 & y 1 \ w 1 & z 1 & - y 1 & - x 1$

(46)

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

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associator(p1,p2,p3)$G

$\label{eq47}\begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{2 \ x 2 \ y 1 \ z 3}+{2 \ w 3 \ z 1 \ z 2}+{2 \ x 2 \ y 3 \ z 1}+ \ \ \displaystyle {2 \ w 3 \ y 1 \ y 2}+{2 \ w 1 \ x 2 \ x 3}+{2 \ w 1 \ w 2 \ w 3}$

(47)

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associator(p1,p1,p2)

$\label{eq48}\begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{2 \ x 1 \ y 1 \ z 2}+{2 \ w 2 \ {{z 1}^{2}}}+{2 \ x 1 \ y 2 \ z 1}+{2 \ w 2 \ {{y 1}^{2}}}+ \ \ \displaystyle {2 \ w 1 \ x 1 \ x 2}+{2 \ {{w 1}^{2}}\ w 2}$

(48)

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associator(p1,p2,p2)

$\label{eq49}\begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({2 \ w 2 \ z 1}-{2 \ x 2 \ y 1}\right)}\ z 2}+{2 \ x 2 \ y 2 \ z 1}+{2 \ w 2 \ y 1 \ y 2}+ \ \ \displaystyle {2 \ w 1 \ {{x 2}^{2}}}+{2 \ w 1 \ {{w 2}^{2}}}$

(49)

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p1*p1

$\label{eq50}{\left({{z 1}^{2}}+{{y 1}^{2}}+{{x 1}^{2}}+{{w 1}^{2}}\right)}\ e$

(50)

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p1*p2 + p2*p1

$\label{eq51}{\left({2 \ z 1 \ z 2}+{2 \ y 1 \ y 2}+{2 \ x 1 \ x 2}+{2 \ w 1 \ w 2}\right)}\ e$

(51)

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p1*(p1*p1)+(p1*p1)*p1

$\label{eq52}{\left({2 \ x 1 \ {{z 1}^{2}}}+{2 \ x 1 \ {{y 1}^{2}}}+{2 \ {{x 1}^{3}}}+{2 \ {{w 1}^{2}}\ x 1}\right)}\ e$

(52)