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SandBoxNonAssociativeAlgebra

last edited 15 years ago by Bill Page

Edit detail for SandBoxNonAssociativeAlgebra revision 9 of 25

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Editor: Bill Page Time: 2008/04/24 15:08:22 GMT-7
Note: structural constants

changed:
-Author: Diego Saa
-
-(Submitted on 20 Nov 2007)
Author: Diego Saa (Submitted on 20 Nov 2007)

removed:
-\begin{axiom}
-all:=[a1,a2,a3,a4,b1,b2,b3,b4,c1,c2,c3,c4]
-a:Vector DMP(all,INT):=[a1,a2,a3,a4]
-b:Vector DMP(all,INT):=[b1,b2,b3,b4]
-c:Vector DMP(all,INT):=[c1,c2,c3,c4]
--- Jordan?
-a ** (b ** c) + c ** (a ** b) + b ** (c ** a)
-a0:Vector DMP(all,INT):=[0,a2,a3,a4]
-b0:Vector DMP(all,INT):=[0,b2,b3,b4]
-c0:Vector DMP(all,INT):=[0,c2,c3,c4]
--- Jordan?
-a0 ** (b0 ** c0) + c0 ** (a0 ** b0) + b0 ** (c0 ** a0)
-(a0 ** b0) ** c0 + (c0 ** a0) ** b0 + (b0 ** c0) ** a0
-\end{axiom}
-

added:
)clear all

added:

The algebra above can be given by structural constants.
\begin{axiom}
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]]]
V:=AlgebraGivenByStructuralConstants(Fraction Integer, 4, [e,i,j,k],sc)
\end{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

axiom
test(e**e=e)

axiom

Compiling function ** with type (Vector Integer,Vector Integer) -> 
      Vector Integer

(5)

Type: Boolean

axiom
test(i**i=e)

(6)

Type: Boolean

axiom
test(j**j=e)

(7)

Type: Boolean

axiom
test(k**k=e)

(8)

Type: Boolean

axiom
test(e**i=i)

(9)

Type: Boolean

axiom
test(e**j=j)

(10)

Type: Boolean

axiom
test(e**k=k)

(11)

Type: Boolean

axiom
test(i**e=-i)

(12)

Type: Boolean

axiom
test(j**e=-j)

(13)

Type: Boolean

axiom
test(k**e=-k)

(14)

Type: Boolean

axiom
test(i**j=k)

(15)

Type: Boolean

axiom
test(j**i=-k)

(16)

Type: Boolean

axiom
test(k**i=j)

(17)

Type: Boolean

axiom
test(i**k=-j)

(18)

Type: Boolean

axiom
test(j**k=i)

(19)

Type: Boolean

axiom
test(k**j=-i)

(20)

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.

axiom)clear all
   All user variables and function definitions have been cleared.
)show NonAssociativeAlgebra
 NonAssociativeAlgebra R: CommutativeRing  is a category constructor
 Abbreviation for NonAssociativeAlgebra is NAALG 
 This constructor is exposed in this frame.
 Issue )edit /usr/local/lib/axiom/target/x86_64-unknown-linux/../../src/algebra/NAALG.spad to see algebra source code for NAALG 
------------------------------- Operations --------------------------------
 ?*? : (R,%) -> %                      ?*? : (%,R) -> %
 ?*? : (%,%) -> %                      ?*? : (Integer,%) -> %
 ?*? : (PositiveInteger,%) -> %        ?**? : (%,PositiveInteger) -> %
 ?+? : (%,%) -> %                      ?-? : (%,%) -> %
 -? : % -> %                           ?=? : (%,%) -> Boolean
 0 : () -> %                           antiCommutator : (%,%) -> %
 associator : (%,%,%) -> %             coerce : % -> OutputForm
 commutator : (%,%) -> %               hash : % -> SingleInteger
 latex : % -> String                   sample : () -> %
 zero? : % -> Boolean                  ?~=? : (%,%) -> Boolean
 ?*? : (NonNegativeInteger,%) -> %
 leftPower : (%,PositiveInteger) -> %
 plenaryPower : (%,PositiveInteger) -> %
 rightPower : (%,PositiveInteger) -> %
 subtractIfCan : (%,%) -> Union(%,"failed")

The algebra above can be given by structural constants.

axiomsc: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]]]

(21)

Type: Vector Matrix Fraction Integer

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

(22)

Type: Domain