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SandBoxNonAssociativeAlgebra

last edited 16 years ago by Bill Page

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]

(1)

Type: Vector Integer

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

(2)

Type: Vector Integer

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

(3)

Type: Vector Integer

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

(4)

Type: Vector Integer

axiomtest(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

(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)

(6)

Type: Domain

axiom
a:=basis()$V

(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

(8)

Type: OutputForm?

axiom
leftUnit()$V

(9)

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

axiomrightUnit()$V
   this algebra has no right unit

(10)

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

axiomalternative?()$V
   algebra is not left alternative

(11)

Type: Boolean

axiomleftAlternative?()$V
   algebra is not left alternative

(12)

Type: Boolean

axiomrightAlternative?()$V
   algebra is not right alternative

(13)

Type: Boolean

axiomassociative?()$V
   algebra is not associative

(14)

Type: Boolean

axiomantiAssociative?()$V
   algebra is not anti-associative

(15)

Type: Boolean

axiom--powerAssociative?()$V
commutative?()$V
   algebra is not commutative

(16)

Type: Boolean

axiomjordanAlgebra?()$V
   algebra is not commutative
   this is not a Jordan algebra

(17)

Type: Boolean

axiomjordanAdmissible?()$V
   algebra is not Jordan admissible

(18)

Type: Boolean

axiomnoncommutativeJordanAlgebra?()$V
   algebra is not flexible
   this is not a noncommutative Jordan algebra, as it is not flexible

(19)

Type: Boolean

axiomlieAlgebra?()$V
   algebra is not anti-commutative
   this is not a Lie algebra

(20)

Type: Boolean

axiomlieAdmissible?()$V
   algebra is not Lie admissible

(21)

Type: Boolean

axiomjacobiIdentity?()$V
   Jacobi identity does not hold

(22)

Type: Boolean

axiom
V has FramedNonAssociativeAlgebra(Fraction Integer)

(23)

Type: Boolean

axiom
basisOfCommutingElements()$AlgebraPackage(Fraction Integer,V)

(24)

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

axiom
basisOfCenter()$AlgebraPackage(Fraction Integer,V)

(25)

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

axiom
basisOfCentroid()$AlgebraPackage(Fraction Integer,V)

(26)

Type: List Matrix Fraction Integer

axiom
basisOfNucleus()$AlgebraPackage(Fraction Integer,V)

(27)

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

axiom
basisOfLeftNucloid()$AlgebraPackage(Fraction Integer,V)

(28)

Type: List Matrix Fraction Integer

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

(29)

Type: Domain

Look for Idempotents

axiom
conditionsForIdempotents()$G

(30)

Type: List Polynomial Fraction Integer

axiom
gb:=groebnerFactorize %

(31)

Type: List List Polynomial Fraction Integer

axiom
associatorDependence()$G

(32)

Type: List Vector Fraction Polynomial Fraction Integer

axiom
leftRankPolynomial()$G

(33)

Type: SparseUnivariatePolynomial? Fraction Polynomial Fraction Integer

axiomrightUnit()$G
   this algebra has no right unit

(34)

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

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

(35)

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

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

(36)

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

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

(37)

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

axiom
leftRecip(p1)$G

(38)

Type: Union(GenericNonAssociativeAlgebra?(Fraction Integer,4,[e,i,j,k]?,[MATRIX,MATRIX,MATRIX,MATRIX]?),...)

axiomrightRecip(p1)$G
   this algebra has no right unit

(39)

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

axiom
leftRegularRepresentation(p1)

(40)

Type: Matrix Fraction Polynomial Fraction Integer

axiom
rightRegularRepresentation(p1)

(41)

Type: Matrix Fraction Polynomial Fraction Integer

axiom
associator(p1,p2,p3)$G

(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

(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]

(44)

Type: List OutputForm?

axiomfor 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

(45)

Type: Equation AlgebraGivenByStructuralConstants?(Fraction Integer,4,[e,i,j,k]?,[MATRIX,MATRIX,MATRIX,MATRIX]?)