|
|
|
last edited 16 years ago by Bill Page |
Edit detail for SandBoxNonAssociativeAlgebra revision 13 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 19:21:01 GMT-7 |
||
| Note: structural constants | ||
changed: -matrix([[(a.i * a.j)::OutputForm for j in 1..4] for i in 1..4])$OutputForm matrix([[(a.i * a.j) for j in 1..4] for i in 1..4])$OutputForm changed: -(e*e=e)$V -(i*i=e)$V -(j*j=e)$V -(k*k=e)$V -(e*i=i)$V -(e*j=j)$V -(e*k=k)$V -(i*e=-i)$V -(j*e=-j)$V -(k*e=-k)$V -(i*j=k)$V -(j*i=-k)$V -(k*i=j)$V -(i*k=-j)$V -(j*k=i)$V -(k*j=-i)$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
axiome:Vector INT:=[1,0,0,0]
 | (1) |
Type: Vector Integer
axiomi:Vector INT:=[0,1,0,0]
 | (2) |
Type: Vector Integer
axiomj:Vector INT:=[0,0,1,0]
 | (3) |
Type: Vector Integer
axiomk: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
axiomV:=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")
axioma:=basis()$V
 | (7) |
Type: Vector AlgebraGivenByStructuralConstants?(Fraction Integer,4,[e,i,j,k]?,[MATRIX,MATRIX,MATRIX,MATRIX]?)
axiommatrix([[(a.i * a.j) for j in 1..4] for i in 1..4])$OutputForm
 | (8) |
Type: OutputForm?
axiomleftUnit()$V
 | (9) |
Type: Union(AlgebraGivenByStructuralConstants?(Fraction Integer,4,[e,i,j,k]?,[MATRIX,MATRIX,MATRIX,MATRIX]?),...)
axiomrightUnit()$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