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SandBoxNonAssociativeAlgebra

last edited 18 years ago by Bill Page

Edit detail for SandBoxNonAssociativeAlgebra revision 12 of 25

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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.

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)

fricas

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.

fricas

)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)))

fricas

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

fricas

)show V

 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]]]) is a domain constructor.
 Abbreviation for AlgebraGivenByStructuralConstants is ALGSC 
 This constructor is exposed in this frame.
 89 names for 110 operations in this domain.
------------------------------- Operations --------------------------------

 0 : () -> %                           ?*? : (%, %) -> %
 ?*? : (Integer, %) -> %               ?*? : (PositiveInteger, %) -> %
 ?+? : (%, %) -> %                     -? : % -> %
 ?-? : (%, %) -> %                     ?=? : (%, %) -> Boolean
 ?^? : (%, PositiveInteger) -> %       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                   leadingMonomial : % -> %
 leftAlternative? : () -> Boolean      leftNorm : % -> Fraction(Integer)
 lieAdmissible? : () -> Boolean        lieAlgebra? : () -> Boolean
 monomial? : % -> Boolean              monomials : % -> List(%)
 opposite? : (%, %) -> Boolean         powerAssociative? : () -> Boolean
 rank : () -> PositiveInteger          recip : % -> Union(%,"failed")
 reductum : % -> %                     rightAlternative? : () -> Boolean
 sample : () -> %                      smaller? : (%, %) -> Boolean
 someBasis : () -> Vector(%)           unit : () -> Union(%,"failed")
 zero? : % -> Boolean                  ?~=? : (%, %) -> Boolean
 ?*? : (%, Fraction(Integer)) -> %
 ?*? : (Fraction(Integer), %) -> %
 ?*? : (NonNegativeInteger, %) -> %
 ?*? : (SquareMatrix(4,Fraction(Integer)), %) -> %
 apply : (Matrix(Fraction(Integer)), %) -> %
 associatorDependence : () -> List(Vector(Fraction(Integer)))
 coefficient : (%, OrderedVariableList([e,i,j,k])) -> Fraction(Integer)
 coefficients : % -> List(Fraction(Integer))
 coerce : Vector(Fraction(Integer)) -> %
 conditionsForIdempotents : () -> List(Polynomial(Fraction(Integer)))
 conditionsForIdempotents : Vector(%) -> List(Polynomial(Fraction(Integer)))
 construct : List(Record(k: OrderedVariableList([e,i,j,k]),c: Fraction(Integer))) -> %
 constructOrdered : List(Record(k: OrderedVariableList([e,i,j,k]),c: Fraction(Integer))) -> %
 convert : Vector(Fraction(Integer)) -> %
 convert : % -> Vector(Fraction(Integer))
 coordinates : Vector(%) -> Matrix(Fraction(Integer))
 coordinates : (Vector(%), Vector(%)) -> Matrix(Fraction(Integer))
 coordinates : % -> Vector(Fraction(Integer))
 coordinates : (%, Vector(%)) -> Vector(Fraction(Integer))
 elt : (%, Integer) -> Fraction(Integer)
 hashUpdate! : (HashState, %) -> HashState
 leadingCoefficient : % -> Fraction(Integer)
 leadingSupport : % -> OrderedVariableList([e,i,j,k])
 leadingTerm : % -> Record(k: OrderedVariableList([e,i,j,k]),c: Fraction(Integer))
 leftCharacteristicPolynomial : % -> SparseUnivariatePolynomial(Fraction(Integer))
 leftDiscriminant : () -> Fraction(Integer)
 leftDiscriminant : Vector(%) -> Fraction(Integer)
 leftMinimalPolynomial : % -> SparseUnivariatePolynomial(Fraction(Integer))
 leftPower : (%, PositiveInteger) -> %
 leftRankPolynomial : () -> SparseUnivariatePolynomial(Polynomial(Fraction(Integer)))
 leftRecip : % -> Union(%,"failed")
 leftRegularRepresentation : % -> Matrix(Fraction(Integer))
 leftRegularRepresentation : (%, Vector(%)) -> Matrix(Fraction(Integer))
 leftTrace : % -> Fraction(Integer)
 leftTraceMatrix : () -> Matrix(Fraction(Integer))
 leftTraceMatrix : Vector(%) -> Matrix(Fraction(Integer))
 leftUnit : () -> Union(%,"failed")
 leftUnits : () -> Union(Record(particular: %,basis: List(%)),"failed")
 linearExtend : ((OrderedVariableList([e,i,j,k]) -> Fraction(Integer)), %) -> Fraction(Integer)
 listOfTerms : % -> List(Record(k: OrderedVariableList([e,i,j,k]),c: Fraction(Integer)))
 map : ((Fraction(Integer) -> Fraction(Integer)), %) -> %
 monomial : (Fraction(Integer), OrderedVariableList([e,i,j,k])) -> %
 noncommutativeJordanAlgebra? : () -> Boolean
 numberOfMonomials : % -> NonNegativeInteger
 plenaryPower : (%, PositiveInteger) -> %
 represents : Vector(Fraction(Integer)) -> %
 represents : (Vector(Fraction(Integer)), Vector(%)) -> %
 rightCharacteristicPolynomial : % -> SparseUnivariatePolynomial(Fraction(Integer))
 rightDiscriminant : () -> Fraction(Integer)
 rightDiscriminant : Vector(%) -> Fraction(Integer)
 rightMinimalPolynomial : % -> SparseUnivariatePolynomial(Fraction(Integer))
 rightNorm : % -> Fraction(Integer)
 rightPower : (%, PositiveInteger) -> %
 rightRankPolynomial : () -> SparseUnivariatePolynomial(Polynomial(Fraction(Integer)))
 rightRecip : % -> Union(%,"failed")
 rightRegularRepresentation : % -> Matrix(Fraction(Integer))
 rightRegularRepresentation : (%, Vector(%)) -> Matrix(Fraction(Integer))
 rightTrace : % -> Fraction(Integer)
 rightTraceMatrix : () -> Matrix(Fraction(Integer))
 rightTraceMatrix : Vector(%) -> Matrix(Fraction(Integer))
 rightUnit : () -> Union(%,"failed")
 rightUnits : () -> Union(Record(particular: %,basis: List(%)),"failed")
 structuralConstants : () -> Vector(Matrix(Fraction(Integer)))
 structuralConstants : Vector(%) -> Vector(Matrix(Fraction(Integer)))
 subtractIfCan : (%, %) -> Union(%,"failed")
 support : % -> List(OrderedVariableList([e,i,j,k]))

fricas

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]]]))

fricas

matrix([[(a.i * a.j)::OutputForm 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?

fricas

leftUnit()$V

$\label{eq9}e$

(9)

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]]]),...)

fricas

rightUnit()$V

$\label{eq10}\verb#"failed"#$

(10)

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

fricas

alternative?()$V

$\label{eq11} \mbox{\rm false}$

(11)

Type: Boolean

fricas

leftAlternative?()$V

$\label{eq12} \mbox{\rm false}$

(12)

Type: Boolean

fricas

rightAlternative?()$V

$\label{eq13} \mbox{\rm false}$

(13)

Type: Boolean

fricas

associative?()$V

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

(14)

Type: Boolean

fricas

antiAssociative?()$V

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

(15)

Type: Boolean

fricas

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

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

(16)

Type: Boolean

fricas

jordanAlgebra?()$V

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

(17)

Type: Boolean

fricas

jordanAdmissible?()$V

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

(18)

Type: Boolean

fricas

noncommutativeJordanAlgebra?()$V

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

(19)

Type: Boolean

fricas

lieAlgebra?()$V

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

(20)

Type: Boolean

fricas

lieAdmissible?()$V

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

(21)

Type: Boolean

fricas

jacobiIdentity?()$V

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

(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