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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.

(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..)))

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

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

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

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

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)

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

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.

)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]]];

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

Multiplication

a:=basis()$V

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

Commutator and Associator

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

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

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]

Volume form?

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

Check standard properties

leftUnit()$V

rightUnit()$V

 this algebra has no right unit

alternative?()$V

 algebra is not left alternative

leftAlternative?()$V

 algebra is not left alternative

rightAlternative?()$V

 algebra is not right alternative

associative?()$V

 algebra is not associative

antiAssociative?()$V

 algebra is not anti-associative

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

 algebra is not commutative

antiCommutative?()$V

 algebra is not anti-commutative

jordanAlgebra?()$V

 algebra is not commutative
 this is not a Jordan algebra

jordanAdmissible?()$V

 algebra is not Jordan admissible

noncommutativeJordanAlgebra?()$V

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

lieAlgebra?()$V

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

lieAdmissible?()$V

 algebra is not Lie admissible

jacobiIdentity?()$V

 Jacobi identity does not hold

Commuting elements

V has FramedNonAssociativeAlgebra(Fraction Integer)

basisOfCommutingElements()$AlgebraPackage(Fraction Integer,V)

basisOfCenter()$AlgebraPackage(Fraction Integer,V)

basisOfCentroid()$AlgebraPackage(Fraction Integer,V)

basisOfNucleus()$AlgebraPackage(Fraction Integer,V)

basisOfLeftNucloid()$AlgebraPackage(Fraction Integer,V)

Symbolic computations

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

Look for Idempotents

conditionsForIdempotents()$G

gb:=groebnerFactorize %

associatorDependence()$G

q:=leftRankPolynomial()$G

map(factor,coefficients q)

rightUnit()$G

 this algebra has no right unit

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

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

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

leftRecip(p1)$G

rightRecip(p1)$G

 this algebra has no right unit

leftRegularRepresentation(p1)

rightRegularRepresentation(p1)

associator(p1,p2,p3)$G

associator(p1,p1,p2)

associator(p1,p2,p2)

p1*p1

p1*p2 + p2*p1

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