login  home  contents  what's new  discussion  bug reports     help  links  subscribe  changes  refresh  edit

Edit detail for SandBoxNonAssociativeAlgebra revision 12 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Editor: Bill Page
Time: 2008/04/24 18:53:09 GMT-7
Note: structural constants

changed:
-test(e**e=e)
-test(i**i=e)
-test(j**j=e)
-test(k**k=e)
-test(e**i=i)
-test(e**j=j)
-test(e**k=k)
-test(i**e=-i)
-test(j**e=-j)
-test(k**e=-k)
-test(i**j=k)
-test(j**i=-k)
-test(k**i=j)
-test(i**k=-j)
-test(j**k=i)
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 _

changed:
- [-1, 0, 0, 0]]]
 [-1, 0, 0, 0]]];

changed:
-[[a.i * a.j for i in 1..4] for j in 1..4]
-noncommutativeJordanAlgebra?()$V
matrix([[(a.i * a.j)::OutputForm for j in 1..4] for i in 1..4])$OutputForm
leftUnit()$V
rightUnit()$V
alternative?()$V
leftAlternative?()$V
rightAlternative?()$V
associative?()$V

changed:
-alternative?()$V
-associative?()$V
--powerAssociative?()$V

removed:
-jacobiIdentity?()$V

added:
noncommutativeJordanAlgebra?()$V

added:
jacobiIdentity?()$V

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
)show V AlgebraGivenByStructuralConstants(Fraction Integer,4,[e,i,j,k],[MATRIX,MATRIX,MATRIX,MATRIX]) is a domain constructor. Abbreviation for AlgebraGivenByStructuralConstants is ALGSC This constructor is exposed in this frame. Issue )edit /usr/local/lib/axiom/target/x86_64-unknown-linux/../../src/algebra/ALGSC.spad to see algebra source code for ALGSC ------------------------------- Operations -------------------------------- ?*? : (Fraction Integer,%) -> % ?*? : (Integer,%) -> % ?*? : (PositiveInteger,%) -> % ?*? : (%,Fraction Integer) -> % ?*? : (%,%) -> % ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % ?-? : (%,%) -> % -? : % -> % ?=? : (%,%) -> Boolean 0 : () -> % alternative? : () -> Boolean antiAssociative? : () -> Boolean antiCommutative? : () -> Boolean antiCommutator : (%,%) -> % associative? : () -> Boolean associator : (%,%,%) -> % basis : () -> Vector % coerce : % -> OutputForm commutative? : () -> Boolean commutator : (%,%) -> % flexible? : () -> Boolean hash : % -> SingleInteger jacobiIdentity? : () -> Boolean jordanAdmissible? : () -> Boolean jordanAlgebra? : () -> Boolean latex : % -> String leftAlternative? : () -> Boolean leftNorm : % -> Fraction Integer leftTrace : % -> Fraction Integer lieAdmissible? : () -> Boolean lieAlgebra? : () -> Boolean powerAssociative? : () -> Boolean rank : () -> PositiveInteger recip : % -> Union(%,"failed") rightAlternative? : () -> Boolean rightNorm : % -> Fraction Integer sample : () -> % someBasis : () -> Vector % unit : () -> Union(%,"failed") zero? : % -> Boolean ?~=? : (%,%) -> Boolean ?*? : (NonNegativeInteger,%) -> % ?*? : (SquareMatrix(4,Fraction Integer),%) -> % apply : (Matrix Fraction Integer,%) -> % associatorDependence : () -> List Vector Fraction Integer coerce : Vector Fraction Integer -> % conditionsForIdempotents : Vector % -> List Polynomial Fraction Integer conditionsForIdempotents : () -> List Polynomial Fraction Integer convert : % -> Vector Fraction Integer convert : Vector Fraction Integer -> % coordinates : (Vector %,Vector %) -> Matrix Fraction Integer coordinates : Vector % -> Matrix Fraction Integer coordinates : (%,Vector %) -> Vector Fraction Integer coordinates : % -> Vector Fraction Integer ?.? : (%,Integer) -> Fraction Integer leftCharacteristicPolynomial : % -> SparseUnivariatePolynomial Fraction Integer leftDiscriminant : Vector % -> Fraction Integer leftDiscriminant : () -> Fraction Integer leftMinimalPolynomial : % -> SparseUnivariatePolynomial Fraction Integer leftPower : (%,PositiveInteger) -> % leftRankPolynomial : () -> SparseUnivariatePolynomial Polynomial Fraction Integer leftRecip : % -> Union(%,"failed") leftRegularRepresentation : (%,Vector %) -> Matrix Fraction Integer leftRegularRepresentation : % -> Matrix Fraction Integer leftTraceMatrix : Vector % -> Matrix Fraction Integer leftTraceMatrix : () -> Matrix Fraction Integer leftUnit : () -> Union(%,"failed") leftUnits : () -> Union(Record(particular: %,basis: List %),"failed") noncommutativeJordanAlgebra? : () -> Boolean plenaryPower : (%,PositiveInteger) -> % represents : (Vector Fraction Integer,Vector %) -> % represents : Vector Fraction Integer -> % rightCharacteristicPolynomial : % -> SparseUnivariatePolynomial Fraction Integer rightDiscriminant : Vector % -> Fraction Integer rightDiscriminant : () -> Fraction Integer rightMinimalPolynomial : % -> SparseUnivariatePolynomial Fraction Integer rightPower : (%,PositiveInteger) -> % rightRankPolynomial : () -> SparseUnivariatePolynomial Polynomial Fraction Integer rightRecip : % -> Union(%,"failed") rightRegularRepresentation : (%,Vector %) -> Matrix Fraction Integer rightRegularRepresentation : % -> Matrix Fraction Integer rightTrace : % -> Fraction Integer rightTraceMatrix : Vector % -> Matrix Fraction Integer rightTraceMatrix : () -> Matrix Fraction Integer rightUnit : () -> Union(%,"failed") rightUnits : () -> Union(Record(particular: %,basis: List %),"failed") structuralConstants : Vector % -> Vector Matrix Fraction Integer structuralConstants : () -> Vector Matrix Fraction Integer subtractIfCan : (%,%) -> Union(%,"failed")

axiom
a:=basis()$V
LatexWiki Image(7)
Type: Vector AlgebraGivenByStructuralConstants?(Fraction Integer,4,[e,i,j,k]?,[MATRIX,MATRIX,MATRIX,MATRIX]?)
axiom
matrix([[(a.i * a.j)::OutputForm for j in 1..4] for i in 1..4])$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

(ee=e)$V (ii=e)$V (jj=e)$V (kk=e)$V (ei=i)$V (ej=j)$V (ek=k)$V (ie=-i)$V (je=-j)$V (ke=-k)$V (ij=k)$V (ji=-k)$V (ki=j)$V (ik=-j)$V (jk=i)$V (kj=-i)$V