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SandBoxNonAssociativeAlgebra

last edited 17 years ago by Bill Page

Edit detail for SandBoxNonAssociativeAlgebra revision 25 of 25

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Editor: Bill Page Time: 2008/05/08 08:42:07 GMT-7
Note:

added:
antiCommutative?()$V

added:
associator(p1,p1,p2)
associator(p1,p2,p2)
p1*p1
p1*p2 + p2*p1
p1*(p1*p1)+(p1*p1)*p1

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.

fricas

(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

fricas

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

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

(1)

Type: Vector(Integer)

fricas

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)

fricas

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)

fricas

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

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?

fricas

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

fricas

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

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

(13)

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

fricas

alternative?()$V

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

(14)

Type: Boolean

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leftAlternative?()$V

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

(15)

Type: Boolean

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rightAlternative?()$V

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

(16)

Type: Boolean

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associative?()$V

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

(17)

Type: Boolean

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antiAssociative?()$V

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

(18)

Type: Boolean

fricas

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

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

(19)

Type: Boolean

fricas

antiCommutative?()$V

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

(20)

Type: Boolean

fricas

jordanAlgebra?()$V

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

(21)

Type: Boolean

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jordanAdmissible?()$V

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

(22)

Type: Boolean

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noncommutativeJordanAlgebra?()$V

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

(23)

Type: Boolean

fricas

lieAlgebra?()$V

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

(24)

Type: Boolean

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lieAdmissible?()$V

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

(25)

Type: Boolean

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jacobiIdentity?()$V

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

(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)

fricas

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

fricas

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

$\label{eq33}\hbox{\axiomType{GenericNonAssociativeAlgebra}\ } \left({{\hbox{\axiomType{Fraction}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \: 4, \:{\left[ e , \: i , \: j , \: k \right]}, \:{\left[{\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(33)

Type: Type

Look for Idempotents

fricas

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)))

fricas

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))))

fricas

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

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

(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]]])

fricas

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)

fricas

leftRecip(p1)$G

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

(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]]]),...)

fricas

rightRecip(p1)$G

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

(44)

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

fricas

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))))

fricas

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))))

fricas

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)

fricas

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)

fricas

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)

fricas

p1*p1

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

(50)

fricas

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)

fricas

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)