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SandBoxGrassmannIsometry

Edit detail for SandBoxGrassmannIsometry revision 5 of 7

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Editor: page Time: 2009/09/11 17:19:15 GMT-7
Note: new representation

added:
Isometry from Grassmann Multivectors

  Representation

'K' is a unital associative and commutative ring represented by polynomials
with rational coefficients of a set of symbols.
\begin{axiom}
K:=SparseMultivariatePolynomial(Fraction Integer,Symbol)
\end{axiom}
The Grassmann Hopf K-algebra is represented by the Axiom domain **Expression**
consisting of rational functions with coefficients from K over an additional
set of symbols and common mathematical operators.
\begin{axiom}
E:=Expression K
a:=a::Symbol::K; b:=b::Symbol::K; c:=c::Symbol::K;
P:=P::Symbol::E; Q:=Q::Symbol::E; R:=R::Symbol::E;
\end{axiom}


changed:
-dot(A:EXPR INT,B:EXPR INT):EXPR INT == (smaller?(A,B)=>idot(A,B);idot(B,A))
-dot(P, Q)=dot(Q,P)
dot(A:E,B:E):E == (smaller?(A,B)=>idot(A,B);idot(B,A))
test(dot(P, Q)=dot(Q,P))

changed:
-hat(A:EXPR INT,B:EXPR INT):EXPR INT == (smaller?(A,B)=>ihat(A,B);-ihat(B,A))
-hat(P, Q)=-hat(Q,P)
hat(A:E,B:E):E == (smaller?(A,B)=>ihat(A,B);-ihat(B,A))
test(hat(P, Q)=-hat(Q,P))

changed:
-6 Proof of the main theorem 15 (isometry from bivector)
5 Isometry from Bivector

  In 1937 Elie Cartan observed that the Lie algebra of the isometry group
$ O_g = Aut(E,g) $, is given by bivectors $E^{\wedge 2} \subset \mathcal{Cl}(E,g)$.


changed:
-eq33 := matrix [[-dot(P,P), -dot(P,Q)+c], _
-                [dot(Q,P)+c, dot(Q,Q)]]
eq33 := matrix [[-dot(P,P),   dot(Q,P)+c], _
                [-dot(P,Q)+c, dot(Q,Q)  ]]

added:

  Consider the following endomorphism,

\begin{equation}
L \equiv id_E - P \otimes \alpha - Q \otimes \beta - R \otimes \gamma  \ \ \in End_K E
\end{equation}
\begin{equation}
t \equiv P \wedge Q \wedge R \ne 0, \mbox{and} \alpha \wedge \beta \wedge \gamma \ne 0
\end{equation}
 

changed:
-eq44 := matrix [[dot(P, P), dot(P, Q)+a, dot(P, R)+b], _
-                [dot(P, Q)-a, dot(Q, Q), dot(Q, R)+c], _
-                [dot(P, R)-b, dot(Q, R)-c, dot(R, R)]]
eq44 := matrix [[dot(P,P),   dot(Q,P)-a, dot(R,P)-b], _
                [dot(P,Q)+a, dot(Q,Q),   dot(R,Q)-c], _
                [dot(P,R)+b, dot(Q,R)+c, dot(R,R)  ]]

changed:
-eq47a.adjMat
eq47a.adjMat::List List E
)set output tex on

removed:
-)set output tex on

added:


changed:
-  a*dot(R, R)-b*dot(R, Q)+c*dot(R, P) == dot(R, v)
-  a*dot(Q, R)-b*dot(Q, Q)+c*dot(Q, P) == dot(Q, v)
-  a*dot(P, R)-b*dot(P, Q)+c*dot(P, P) == dot(P, v)
  a*dot(R,R)-b*dot(R,Q)+c*dot(R,P) == dot(R,v)
  a*dot(Q,R)-b*dot(Q,Q)+c*dot(Q,P) == dot(Q,v)
  a*dot(P,R)-b*dot(P,Q)+c*dot(P,P) == dot(P,v)

Isometry from Grassmann Multivectors

Representation

K is a unital associative and commutative ring represented by polynomials with rational coefficients of a set of symbols.

axiom

K:=SparseMultivariatePolynomial(Fraction Integer,Symbol)

(1)

Type: Domain

The Grassmann Hopf K-algebra is represented by the Axiom domain Expression consisting of rational functions with coefficients from K over an additional set of symbols and common mathematical operators.

axiom

E:=Expression K

(2)

Type: Domain

axiom

a:=a::Symbol::K; b:=b::Symbol::K;
c:=c::Symbol::K;

Type: SparseMultivariatePolynomial?(Fraction(Integer),Symbol)

axiom

P:=P::Symbol::E; Q:=Q::Symbol::E;
R:=R::Symbol::E;

Type: Expression(SparseMultivariatePolynomial?(Fraction(Integer),Symbol))

Grassmann Algebra Operators

Symmetric inner product

axiom

idot:=display(operator('dot,2), (x:List OutputForm):OutputForm +->
hconcat([x.1,_{_\cdot_} ,x.2]));

Type: BasicOperator?

axiom

dot(A:E,B:E):E == (smaller?(A,B)=>idot(A,B);idot(B,A))

   Function declaration dot : (Expression(SparseMultivariatePolynomial(
      Fraction(Integer),Symbol)),Expression(
      SparseMultivariatePolynomial(Fraction(Integer),Symbol))) ->
      Expression(SparseMultivariatePolynomial(Fraction(Integer),Symbol)
      ) has been added to workspace.

Type: Void

axiom

test(dot(P, Q)=dot(Q,P))

axiom

Compiling function dot with type (Expression(
      SparseMultivariatePolynomial(Fraction(Integer),Symbol)),
      Expression(SparseMultivariatePolynomial(Fraction(Integer),Symbol)
      )) -> Expression(SparseMultivariatePolynomial(Fraction(Integer),
      Symbol))

(3)

Type: Boolean

Exterior product

axiom

ihat:=display(operator('hat,2), (x:List OutputForm):OutputForm +->
hconcat([x.1,_{_\wedge_} ,x.2]));

Type: BasicOperator?

axiom

hat(A:E,B:E):E == (smaller?(A,B)=>ihat(A,B);-ihat(B,A))

   Function declaration hat : (Expression(SparseMultivariatePolynomial(
      Fraction(Integer),Symbol)),Expression(
      SparseMultivariatePolynomial(Fraction(Integer),Symbol))) ->
      Expression(SparseMultivariatePolynomial(Fraction(Integer),Symbol)
      ) has been added to workspace.

Type: Void

axiom

test(hat(P, Q)=-hat(Q,P))

axiom

Compiling function hat with type (Expression(
      SparseMultivariatePolynomial(Fraction(Integer),Symbol)),
      Expression(SparseMultivariatePolynomial(Fraction(Integer),Symbol)
      )) -> Expression(SparseMultivariatePolynomial(Fraction(Integer),
      Symbol))

(4)

Type: Boolean

axiom

simplifyHat:=rule
  dot(P, Q)^2-dot(P,P)*dot(Q,Q) == hat(P,Q)^2
  -dot(P,Q)^2+dot(P,P)*dot(Q,Q) == -hat(P,Q)^2
  dot(Q,R)*dot(P,R)-dot(R,R)*dot(P,Q) == dot(hat(R,Q),hat(R,P))

(5)

Type: Ruleset(Integer,SparseMultivariatePolynomial?(Fraction(Integer),Symbol),Expression(SparseMultivariatePolynomial?(Fraction(Integer),Symbol)))

Isometry from Bivector
In 1937 Elie Cartan observed that the Lie algebra of the isometry group , is given by bivectors .

axiom

eq33 := matrix [[-dot(P,P),   dot(Q,P)+c], _
                [-dot(P,Q)+c, dot(Q,Q)  ]]

(6)

Type: Matrix(Expression(SparseMultivariatePolynomial?(Fraction(Integer),Symbol)))

axiom

eq35 := inverse(eq33)

(7)

Type: Union(Matrix(Expression(SparseMultivariatePolynomial?(Fraction(Integer),Symbol))),...)

axiom

map(x+->simplifyHat(x),eq35)

(8)

Type: Matrix(Expression(SparseMultivariatePolynomial?(Fraction(Integer),Symbol)))

Isometry from Trivector
Consider the following endomorphism,

(9)

(10)

axiom

eq44 := matrix [[dot(P,P),   dot(Q,P)-a, dot(R,P)-b], _
                [dot(P,Q)+a, dot(Q,Q),   dot(R,Q)-c], _
                [dot(P,R)+b, dot(Q,R)+c, dot(R,R)  ]]

(11)

Type: Matrix(Expression(SparseMultivariatePolynomial?(Fraction(Integer),Symbol)))

axiom

eq47a := adjoint(eq44);

Type: Record(adjMat: Matrix(Expression(SparseMultivariatePolynomial?(Fraction(Integer),Symbol))),detMat: Expression(SparseMultivariatePolynomial?(Fraction(Integer),Symbol)))

axiom

)set output tex off

axiom

)set output algebra on

eq47a.adjMat::List List E

   (17)
   [
                                    2    2
     [Q{\cdot}QR{\cdot}R - Q{\cdot}R  + c ,
      (- P{\cdot}Q + a)R{\cdot}R + (P{\cdot}R - b)Q{\cdot}R + cP{\cdot}R -
b c,
      (P{\cdot}Q - a)Q{\cdot}R + (- P{\cdot}R + b)Q{\cdot}Q - cP{\cdot}Q +
a c]
     ,

     [(- P{\cdot}Q - a)R{\cdot}R + (P{\cdot}R + b)Q{\cdot}R - cP{\cdot}R -
b c,
                                    2    2
      P{\cdot}PR{\cdot}R - P{\cdot}R  + b ,

         - P{\cdot}PQ{\cdot}R + (P{\cdot}Q + a)P{\cdot}R - bP{\cdot}Q
       +
         cP{\cdot}P - a b
       ]
     ,

     [(P{\cdot}Q + a)Q{\cdot}R + (- P{\cdot}R - b)Q{\cdot}Q + cP{\cdot}Q +
a c,

         - P{\cdot}PQ{\cdot}R + (P{\cdot}Q - a)P{\cdot}R + bP{\cdot}Q
       +
         - cP{\cdot}P - a b
       ,
                                    2    2
      P{\cdot}PQ{\cdot}Q - P{\cdot}Q  + a ]
     ]

Type: List(List(Expression(SparseMultivariatePolynomial?(Fraction(Integer),Symbol))))

axiom

)set output tex on

axiom

)set output algebra off

eq47a.detMat

(12)

Type: Expression(SparseMultivariatePolynomial?(Fraction(Integer),Symbol))

Simplifications

axiom

eq45 := a*R-b*Q+c*P = v

(13)

Type: Equation(Expression(SparseMultivariatePolynomial?(Fraction(Integer),Symbol)))

axiom

eq45a := rule
  a*dot(R,R)-b*dot(R,Q)+c*dot(R,P) == dot(R,v)
  a*dot(Q,R)-b*dot(Q,Q)+c*dot(Q,P) == dot(Q,v)
  a*dot(P,R)-b*dot(P,Q)+c*dot(P,P) == dot(P,v)

(14)

Type: Ruleset(Integer,SparseMultivariatePolynomial?(Fraction(Integer),Symbol),Expression(SparseMultivariatePolynomial?(Fraction(Integer),Symbol)))

axiom

eq47b := map(x+->eq45a simplifyHat x,eq47a.adjMat)

(15)

Type: Matrix(Expression(SparseMultivariatePolynomial?(Fraction(Integer),Symbol)))

axiom

map(x+->x^2,eq45)

(16)

Type: Equation(Expression(SparseMultivariatePolynomial?(Fraction(Integer),Symbol)))

axiom

eq47d := rule
  dot(R,R)*a^2 + dot(Q,Q)*b^2 + dot(P,P)*c^2 - _
  2*c*b*dot(P,Q) + 2*a*c*dot(R,P) - 2*a*b*dot(R,Q) == v^2

(17)

Type: RewriteRule?(Integer,SparseMultivariatePolynomial?(Fraction(Integer),Symbol),Expression(SparseMultivariatePolynomial?(Fraction(Integer),Symbol)))

axiom

eq47d(eq47a.detMat)

(18)

Type: Expression(SparseMultivariatePolynomial?(Fraction(Integer),Symbol))