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Edit detail for SandBoxNonAssociativeAlgebra revision 23 of 25

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Editor: Bill Page
Time: 2008/05/08 06:53:52 GMT-7
Note:

removed:
-)show V

added:
basisOfCommutingElements()$AlgebraPackage(Fraction Integer,V)

removed:
-)show G

added:

Look for Idempotents
\begin{axiom}
conditionsForIdempotents()$G
gb:=groebnerFactorize %
\end{axiom}

removed:
-conditionsForIdempotents()$G

added:
\end{axiom}

\begin{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.

axiom
_*_*(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
axiom
e:Vector INT:=[1,0,0,0]
LatexWiki Image(1)
Type: Vector Integer
axiom
i:Vector INT:=[0,1,0,0]
LatexWiki Image(2)
Type: Vector Integer
axiom
j:Vector INT:=[0,0,1,0]
LatexWiki Image(3)
Type: Vector Integer
axiom
k:Vector INT:=[0,0,0,1]
LatexWiki Image(4)
Type: Vector Integer
axiom
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)
axiom
Compiling function ** with type (Vector Integer,Vector Integer) -> 
      Vector Integer
LatexWiki Image(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.

axiom
)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
axiom
V:=AlgebraGivenByStructuralConstants(Fraction Integer, 4, [e,i,j,k],sc)
LatexWiki Image(6)
Type: Domain

axiom
a:=basis()$V
LatexWiki Image(7)
Type: Vector AlgebraGivenByStructuralConstants?(Fraction Integer,4,[e,i,j,k]?,[MATRIX,MATRIX,MATRIX,MATRIX]?)
axiom
matrix([[(a.i * a.j) for j in 1..4] for i in 1..4])$OutputForm
LatexWiki Image(8)
Type: OutputForm?
axiom
leftUnit()$V
LatexWiki Image(9)
Type: Union(AlgebraGivenByStructuralConstants?(Fraction Integer,4,[e,i,j,k]?,[MATRIX,MATRIX,MATRIX,MATRIX]?),...)
axiom
rightUnit()$V this algebra has no right unit
LatexWiki Image(10)
Type: Union("failed",...)
axiom
alternative?()$V algebra is not left alternative
LatexWiki Image(11)
Type: Boolean
axiom
leftAlternative?()$V algebra is not left alternative
LatexWiki Image(12)
Type: Boolean
axiom
rightAlternative?()$V algebra is not right alternative
LatexWiki Image(13)
Type: Boolean
axiom
associative?()$V algebra is not associative
LatexWiki Image(14)
Type: Boolean
axiom
antiAssociative?()$V algebra is not anti-associative
LatexWiki Image(15)
Type: Boolean
axiom
--powerAssociative?()$V commutative?()$V algebra is not commutative
LatexWiki Image(16)
Type: Boolean
axiom
jordanAlgebra?()$V algebra is not commutative this is not a Jordan algebra
LatexWiki Image(17)
Type: Boolean
axiom
jordanAdmissible?()$V algebra is not Jordan admissible
LatexWiki Image(18)
Type: Boolean
axiom
noncommutativeJordanAlgebra?()$V algebra is not flexible this is not a noncommutative Jordan algebra, as it is not flexible
LatexWiki Image(19)
Type: Boolean
axiom
lieAlgebra?()$V algebra is not anti-commutative this is not a Lie algebra
LatexWiki Image(20)
Type: Boolean
axiom
lieAdmissible?()$V algebra is not Lie admissible
LatexWiki Image(21)
Type: Boolean
axiom
jacobiIdentity?()$V Jacobi identity does not hold
LatexWiki Image(22)
Type: Boolean

axiom
V has FramedNonAssociativeAlgebra(Fraction Integer)
LatexWiki Image(23)
Type: Boolean
axiom
basisOfCommutingElements()$AlgebraPackage(Fraction Integer,V)
LatexWiki Image(24)
Type: List AlgebraGivenByStructuralConstants?(Fraction Integer,4,[e,i,j,k]?,[MATRIX,MATRIX,MATRIX,MATRIX]?)
axiom
basisOfCenter()$AlgebraPackage(Fraction Integer,V)
LatexWiki Image(25)
Type: List AlgebraGivenByStructuralConstants?(Fraction Integer,4,[e,i,j,k]?,[MATRIX,MATRIX,MATRIX,MATRIX]?)
axiom
basisOfCentroid()$AlgebraPackage(Fraction Integer,V)
LatexWiki Image(26)
Type: List Matrix Fraction Integer
axiom
basisOfNucleus()$AlgebraPackage(Fraction Integer,V)
LatexWiki Image(27)
Type: List AlgebraGivenByStructuralConstants?(Fraction Integer,4,[e,i,j,k]?,[MATRIX,MATRIX,MATRIX,MATRIX]?)
axiom
basisOfLeftNucloid()$AlgebraPackage(Fraction Integer,V)
LatexWiki Image(28)
Type: List Matrix Fraction Integer

axiom
G:=GenericNonAssociativeAlgebra(Fraction Integer, 4, [e,i,j,k],sc)
LatexWiki Image(29)
Type: Domain

Look for Idempotents

axiom
conditionsForIdempotents()$G
LatexWiki Image(30)
Type: List Polynomial Fraction Integer
axiom
gb:=groebnerFactorize %
LatexWiki Image(31)
Type: List List Polynomial Fraction Integer

axiom
associatorDependence()$G
LatexWiki Image(32)
Type: List Vector Fraction Polynomial Fraction Integer
axiom
leftRankPolynomial()$G
LatexWiki Image(33)
Type: SparseUnivariatePolynomial? Fraction Polynomial Fraction Integer
axiom
rightUnit()$G this algebra has no right unit
LatexWiki Image(34)
Type: Union("failed",...)

axiom
p1:=generic([x1,y1,z1,w1])$G
LatexWiki Image(35)
Type: GenericNonAssociativeAlgebra?(Fraction Integer,4,[e,i,j,k]?,[MATRIX,MATRIX,MATRIX,MATRIX]?)
axiom
p2:=generic([x2,y2,z2,w2])$G
LatexWiki Image(36)
Type: GenericNonAssociativeAlgebra?(Fraction Integer,4,[e,i,j,k]?,[MATRIX,MATRIX,MATRIX,MATRIX]?)
axiom
p3:=generic([x3,y3,z3,w3])$G
LatexWiki Image(37)
Type: GenericNonAssociativeAlgebra?(Fraction Integer,4,[e,i,j,k]?,[MATRIX,MATRIX,MATRIX,MATRIX]?)
axiom
leftRecip(p1)$G
LatexWiki Image(38)
Type: Union(GenericNonAssociativeAlgebra?(Fraction Integer,4,[e,i,j,k]?,[MATRIX,MATRIX,MATRIX,MATRIX]?),...)
axiom
rightRecip(p1)$G this algebra has no right unit
LatexWiki Image(39)
Type: Union("failed",...)
axiom
leftRegularRepresentation(p1)
LatexWiki Image(40)
Type: Matrix Fraction Polynomial Fraction Integer
axiom
rightRegularRepresentation(p1)
LatexWiki Image(41)
Type: Matrix Fraction Polynomial Fraction Integer
axiom
associator(p1,p2,p3)$G
LatexWiki Image(42)
Type: GenericNonAssociativeAlgebra?(Fraction Integer,4,[e,i,j,k]?,[MATRIX,MATRIX,MATRIX,MATRIX]?)

commutator and associator --Bill Page, Fri, 02 May 2008 08:38:36 -0700 reply
axiom
matrix([[commutator(a.x,a.y) for x in 1..4] for y in 1..4])$OutputForm
LatexWiki Image(43)
Type: OutputForm?

axiom
[matrix([[associator(a.x,a.y,a.z) for x in 1..4] for y in 1..4])$OutputForm for z in 1..4]
LatexWiki Image(44)
Type: List OutputForm?
axiom
for x in 1..4 repeat for y in 1..4 repeat for z in 1..4 repeat if associator(a.x,a.y,a.z) ~= 0$V then output([[a.x,a.y,a.z],"=",associator(a.x,a.y,a.z)]) [[i,e,e],"=",2i] [[i,e,i],"=",- 2e] [[i,e,j],"=",- 2k] [[i,e,k],"=",2j] [[i,i,e],"=",2e] [[i,i,i],"=",2i] [[i,i,j],"=",2j] [[i,i,k],"=",2k] [[j,e,e],"=",2j] [[j,e,i],"=",2k] [[j,e,j],"=",- 2e] [[j,e,k],"=",- 2i] [[j,j,e],"=",2e] [[j,j,i],"=",2i] [[j,j,j],"=",2j] [[j,j,k],"=",2k] [[k,e,e],"=",2k] [[k,e,i],"=",- 2j] [[k,e,j],"=",2i] [[k,e,k],"=",- 2e] [[k,k,e],"=",2e] [[k,k,i],"=",2i] [[k,k,j],"=",2j] [[k,k,k],"=",2k]
Type: Void

Volume form?

axiom
a.2 * (a.3 * a.4) = (a.2 * a.3) * a.4
LatexWiki Image(45)
Type: Equation AlgebraGivenByStructuralConstants?(Fraction Integer,4,[e,i,j,k]?,[MATRIX,MATRIX,MATRIX,MATRIX]?)