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SandBoxNonAssociativeAlgebra

last edited 16 years ago by Bill Page

Edit detail for SandBoxNonAssociativeAlgebra revision 18 of 25

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Editor: Bill Page Time: 2008/05/02 08:39:54 GMT-7
Note: commutator and associator

changed:
-for a in basis()$V repeat
-  for b in basis()$V repeat
-    output([a,b,"=",a**b])
for x in basis()$V repeat
  for y in basis()$V repeat
    output(["commuator",x,y,"=",commutator(x,y)])

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)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

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

(24)

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

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

(25)

Type: List Matrix Fraction Integer

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

(26)

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

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

(27)

Type: List Matrix Fraction Integer

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

(28)

Type: Domain

axiom)show G
 GenericNonAssociativeAlgebra(Fraction Integer,4,[e,i,j,k],[MATRIX,MATRIX,MATRIX,MATRIX]) is a domain constructor.
 Abbreviation for GenericNonAssociativeAlgebra is GCNAALG 
 This constructor is not exposed in this frame.
 Issue )edit /usr/local/lib/axiom/target/x86_64-unknown-linux/../../src/algebra/GCNAALG.spad to see algebra source code for GCNAALG 
------------------------------- Operations --------------------------------
 ?*? : (Integer,%) -> %                ?*? : (PositiveInteger,%) -> %
 ?*? : (%,%) -> %                      ?**? : (%,PositiveInteger) -> %
 ?+? : (%,%) -> %                      ?-? : (%,%) -> %
 -? : % -> %                           ?=? : (%,%) -> Boolean
 0 : () -> %                           alternative? : () -> Boolean
 antiAssociative? : () -> Boolean      antiCommutative? : () -> Boolean
 antiCommutator : (%,%) -> %           associative? : () -> Boolean
 associator : (%,%,%) -> %             basis : () -> Vector %
 coerce : % -> OutputForm              commutative? : () -> Boolean
 commutator : (%,%) -> %               flexible? : () -> Boolean
 generic : (Symbol,Vector %) -> %      generic : Symbol -> %
 generic : Vector Symbol -> %          generic : Vector % -> %
 generic : () -> %                     hash : % -> SingleInteger
 jacobiIdentity? : () -> Boolean       jordanAdmissible? : () -> Boolean
 jordanAlgebra? : () -> Boolean        latex : % -> String
 leftAlternative? : () -> Boolean      lieAdmissible? : () -> Boolean
 lieAlgebra? : () -> Boolean           powerAssociative? : () -> Boolean
 rank : () -> PositiveInteger          recip : % -> Union(%,"failed")
 rightAlternative? : () -> Boolean     sample : () -> %
 someBasis : () -> Vector %            unit : () -> Union(%,"failed")
 zero? : % -> Boolean                  ?~=? : (%,%) -> Boolean
 ?*? : (Fraction Polynomial Fraction Integer,%) -> %
 ?*? : (NonNegativeInteger,%) -> %
 ?*? : (SquareMatrix(4,Fraction Polynomial Fraction Integer),%) -> %
 ?*? : (%,Fraction Polynomial Fraction Integer) -> %
 apply : (Matrix Fraction Polynomial Fraction Integer,%) -> %
 associatorDependence : () -> List Vector Fraction Polynomial Fraction Integer
 coerce : Vector Fraction Polynomial Fraction Integer -> %
 conditionsForIdempotents : Vector % -> List Polynomial Fraction Integer
 conditionsForIdempotents : () -> List Polynomial Fraction Integer
 conditionsForIdempotents : Vector % -> List Polynomial Fraction Polynomial Fraction Integer
 conditionsForIdempotents : () -> List Polynomial Fraction Polynomial Fraction Integer
 convert : % -> Vector Fraction Polynomial Fraction Integer
 convert : Vector Fraction Polynomial Fraction Integer -> %
 coordinates : (Vector %,Vector %) -> Matrix Fraction Polynomial Fraction Integer
 coordinates : Vector % -> Matrix Fraction Polynomial Fraction Integer
 coordinates : (%,Vector %) -> Vector Fraction Polynomial Fraction Integer
 coordinates : % -> Vector Fraction Polynomial Fraction Integer
 ?.? : (%,Integer) -> Fraction Polynomial Fraction Integer
 generic : (Vector Symbol,Vector %) -> %
 genericLeftDiscriminant : () -> Fraction Polynomial Fraction Integer
 genericLeftMinimalPolynomial : % -> SparseUnivariatePolynomial Fraction Polynomial Fraction Integer
 genericLeftNorm : % -> Fraction Polynomial Fraction Integer
 genericLeftTrace : % -> Fraction Polynomial Fraction Integer
 genericLeftTraceForm : (%,%) -> Fraction Polynomial Fraction Integer
 genericRightDiscriminant : () -> Fraction Polynomial Fraction Integer
 genericRightMinimalPolynomial : % -> SparseUnivariatePolynomial Fraction Polynomial Fraction Integer
 genericRightNorm : % -> Fraction Polynomial Fraction Integer
 genericRightTrace : % -> Fraction Polynomial Fraction Integer
 genericRightTraceForm : (%,%) -> Fraction Polynomial Fraction Integer
 leftCharacteristicPolynomial : % -> SparseUnivariatePolynomial Fraction Polynomial Fraction Integer
 leftDiscriminant : Vector % -> Fraction Polynomial Fraction Integer
 leftDiscriminant : () -> Fraction Polynomial Fraction Integer
 leftMinimalPolynomial : % -> SparseUnivariatePolynomial Fraction Polynomial Fraction Integer
 leftNorm : % -> Fraction Polynomial Fraction Integer
 leftPower : (%,PositiveInteger) -> %
 leftRankPolynomial : () -> SparseUnivariatePolynomial Fraction Polynomial Fraction Integer
 leftRankPolynomial : () -> SparseUnivariatePolynomial Polynomial Fraction Polynomial Fraction Integer
 leftRecip : % -> Union(%,"failed")
 leftRegularRepresentation : (%,Vector %) -> Matrix Fraction Polynomial Fraction Integer
 leftRegularRepresentation : % -> Matrix Fraction Polynomial Fraction Integer
 leftTrace : % -> Fraction Polynomial Fraction Integer
 leftTraceMatrix : Vector % -> Matrix Fraction Polynomial Fraction Integer
 leftTraceMatrix : () -> Matrix Fraction Polynomial Fraction Integer
 leftUnit : () -> Union(%,"failed")
 leftUnits : () -> Union(Record(particular: %,basis: List %),"failed")
 noncommutativeJordanAlgebra? : () -> Boolean
 plenaryPower : (%,PositiveInteger) -> %
 represents : (Vector Fraction Polynomial Fraction Integer,Vector %) -> %
 represents : Vector Fraction Polynomial Fraction Integer -> %
 rightCharacteristicPolynomial : % -> SparseUnivariatePolynomial Fraction Polynomial Fraction Integer
 rightDiscriminant : Vector % -> Fraction Polynomial Fraction Integer
 rightDiscriminant : () -> Fraction Polynomial Fraction Integer
 rightMinimalPolynomial : % -> SparseUnivariatePolynomial Fraction Polynomial Fraction Integer
 rightNorm : % -> Fraction Polynomial Fraction Integer
 rightPower : (%,PositiveInteger) -> %
 rightRankPolynomial : () -> SparseUnivariatePolynomial Fraction Polynomial Fraction Integer
 rightRankPolynomial : () -> SparseUnivariatePolynomial Polynomial Fraction Polynomial Fraction Integer
 rightRecip : % -> Union(%,"failed")
 rightRegularRepresentation : (%,Vector %) -> Matrix Fraction Polynomial Fraction Integer
 rightRegularRepresentation : % -> Matrix Fraction Polynomial Fraction Integer
 rightTrace : % -> Fraction Polynomial Fraction Integer
 rightTraceMatrix : Vector % -> Matrix Fraction Polynomial Fraction Integer
 rightTraceMatrix : () -> Matrix Fraction Polynomial Fraction Integer
 rightUnit : () -> Union(%,"failed")
 rightUnits : () -> Union(Record(particular: %,basis: List %),"failed")
 structuralConstants : Vector % -> Vector Matrix Fraction Polynomial Fraction Integer
 structuralConstants : () -> Vector Matrix Fraction Polynomial Fraction Integer
 subtractIfCan : (%,%) -> Union(%,"failed")

axiom
associatorDependence()$G

(29)

Type: List Vector Fraction Polynomial Fraction Integer

axiom
conditionsForIdempotents()$G

(30)

Type: List Polynomial Fraction Integer

axiom
leftRankPolynomial()$G

(31)

Type: SparseUnivariatePolynomial? Fraction Polynomial Fraction Integer

axiomrightUnit()$G
   this algebra has no right unit

(32)

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

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

(33)

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

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

(34)

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

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

(35)

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

axiom
leftRecip(p1)$G

(36)

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

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

(37)

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

axiom
leftRegularRepresentation(p1)

(38)

Type: Matrix Fraction Polynomial Fraction Integer

axiom
rightRegularRepresentation(p1)

(39)

Type: Matrix Fraction Polynomial Fraction Integer

axiom
associator(p1,p2,p3)$G

(40)

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

axiomfor x in basis()$V repeat
  for y in basis()$V repeat
    output(["commuator",x,y,"=",commutator(x,y)])
   FriCAS cannot iterate with x over your form now. Perhaps you should 
      try using a conversion to make sure your form is a list or 
      stream, for example.
associator(a.2,a.3,a.4)

(41)

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