MathAction SandBoxGrassmannIsometry

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Isometry from Grassmann Multivectors

Representation

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

fricas

K:=SparseMultivariatePolynomial(Fraction Integer,Symbol)

$\label{eq1}\hbox{\axiomType{SparseMultivariatePolynomial}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{Symbol}\ })$

(1)

Type: Type

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.

fricas

E:=Expression K

$\label{eq2}\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{SparseMultivariatePolynomial}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{Symbol}\ }))$

(2)

Type: Type

fricas

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

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

fricas

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

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

Grassmann Algebra Operators

Symmetric inner product

fricas

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

Type: BasicOperator?

fricas

Dot(A:E,B:E):E == idot(A,B)

   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

fricas

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

fricas

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

fricas

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

$\label{eq3} \mbox{\rm true}$

(3)

Type: Boolean

Exterior product

fricas

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

Type: BasicOperator?

fricas

Hat(A:E,B:E):E == ihat(A,B)

   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

fricas

hat(A:E,B:E):E ==
  A=B=>0
  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

fricas

test(hat(P, Q)=-hat(Q,P)) and test(hat(P,P)=0)

fricas

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

$\label{eq4} \mbox{\rm true}$

(4)

Type: Boolean

fricas

combine:=rule
  Dot(-A,B) == -dot(A,B)
  Dot(A,-B) == -dot(A,B)
  Dot(A,B)^2-Dot(A,A)*Dot(B,B) == hat(A,B)^2
  -Dot(A,B)^2+Dot(A,A)*Dot(B,B) == -hat(A,B)^2
  Dot(A,B)*Dot(A,C)-Dot(A,A)*Dot(B,C) == dot(hat(A,B),hat(A,C))
  Dot(A,B)*Dot(B,C)-Dot(B,B)*Dot(A,C) == dot(hat(B,A),hat(B,C))
  Dot(B,C)*Dot(A,C)-Dot(C,C)*Dot(A,B) == dot(hat(C,A),hat(C,B))

fricas

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

$\label{eq5}\begin{array}{@{}l} \displaystyle \left\{{= = \left({{\left(- A{\cdot}B \right)}, \: -{\left(A{\cdot}B \right)}}\right)}, \: \right. \ \ \displaystyle \left.{= = \left({{\left(A{\cdot}- B \right)}, \: -{\left(A{\cdot}B \right)}}\right)}, \: \right. \ \ \displaystyle \left.{= = \left({{-{{{A}^{2}}\ {{B}^{2}}}+{{\left(A{\cdot}B \right)}^{2}}+ \%B}, \:{{{\left(A{\wedge}B \right)}^{2}}+ \%B}}\right)}, \: \right. \ \ \displaystyle \left.{= = \left({{{{{A}^{2}}\ {{B}^{2}}}-{{\left(A{\cdot}B \right)}^{2}}+ \%C}, \:{-{{\left(A{\wedge}B \right)}^{2}}+ \%C}}\right)}, \: \right. \ \ \displaystyle \left.{= = \left({{-{{{A}^{2}}\ {\left(B{\cdot}C \right)}}+{{\left(A{\cdot}B \right)}\ {\left(A{\cdot}C \right)}}+ \%D}, \:{{\left({\left(A{\wedge}B \right)}{\cdot}{\left(A{\wedge}C \right)}\right)}+ \%D}}\right)}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{= = \left({{{{\left(A{\cdot}B \right)}\ {\left(B{\cdot}C \right)}}-{{\left(A{\cdot}C \right)}\ {{B}^{2}}}+ \%E}, \:{{\left(-{\left(A{\wedge}B \right)}{\cdot}{\left(B{\wedge}C \right)}\right)}+ \%E}}\right)}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{= = \left({{-{{\left(A{\cdot}B \right)}\ {{C}^{2}}}+{{\left(A{\cdot}C \right)}\ {\left(B{\cdot}C \right)}}+ \%F}, \:{{\left(-{\left(B{\wedge}C \right)}{\cdot}-{\left(A{\wedge}C \right)}\right)}+ \%F}}\right)}\right\}$

(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 $E^{\wedge 2} \subset \mathcal{Cl}(E,g)$ .

fricas

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

$\label{eq6}\left[ \begin{array}{cc} -{{P}^{2}}&{{\left(P{\cdot}Q \right)}+ c} \ {-{\left(P{\cdot}Q \right)}+ c}&{{Q}^{2}}$

(6)

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

fricas

eq35 := inverse(eq33)

$\label{eq7}\left[ \begin{array}{cc} -{{{Q}^{2}}\over{{{{P}^{2}}\ {{Q}^{2}}}-{{\left(P{\cdot}Q \right)}^{2}}+{{c}^{2}}}}&{{{\left(P{\cdot}Q \right)}+ c}\over{{{{P}^{2}}\ {{Q}^{2}}}-{{\left(P{\cdot}Q \right)}^{2}}+{{c}^{2}}}} \ {{-{\left(P{\cdot}Q \right)}+ c}\over{{{{P}^{2}}\ {{Q}^{2}}}-{{\left(P{\cdot}Q \right)}^{2}}+{{c}^{2}}}}&{{{P}^{2}}\over{{{{P}^{2}}\ {{Q}^{2}}}-{{\left(P{\cdot}Q \right)}^{2}}+{{c}^{2}}}}$

(7)

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

fricas

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

$\label{eq8}\left[ \begin{array}{cc} {{{Q}^{2}}\over{{{\left(A{\wedge}B \right)}^{2}}-{{c}^{2}}}}&{{-{\left(P{\cdot}Q \right)}- c}\over{{{\left(A{\wedge}B \right)}^{2}}-{{c}^{2}}}} \ {{{\left(P{\cdot}Q \right)}- c}\over{{{\left(A{\wedge}B \right)}^{2}}-{{c}^{2}}}}& -{{{P}^{2}}\over{{{\left(A{\wedge}B \right)}^{2}}-{{c}^{2}}}}$

(8)

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

Isometry from Trivector
Consider the following endomorphism,

$\label{eq9} L \equiv id_E - P \otimes \alpha - Q \otimes \beta - R \otimes \gamma \ \ \in End_K E$

(9)

$\label{eq10} t \equiv P \wedge Q \wedge R \ne 0, \mbox{and} \alpha \wedge \beta \wedge \gamma \ne 0$

(10)

fricas

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)  ]]

$\label{eq11}\left[ \begin{array}{ccc} {{P}^{2}}&{{\left(P{\cdot}Q \right)}- a}&{{\left(P{\cdot}R \right)}- b} \ {{\left(P{\cdot}Q \right)}+ a}&{{Q}^{2}}&{{\left(Q{\cdot}R \right)}- c} \ {{\left(P{\cdot}R \right)}+ b}&{{\left(Q{\cdot}R \right)}+ c}&{{R}^{2}}$

(11)

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

fricas

eq47a := adjoint(eq44);

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

fricas

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

$\label{eq12}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{{{{Q}^{2}}\ {{R}^{2}}}-{{\left(Q{\cdot}R \right)}^{2}}+{{c}^{2}}}, \: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle {{\left(-{\left(P{\cdot}Q \right)}+ a \right)}\ {{R}^{2}}}+{{\left({\left(P{\cdot}R \right)}- b \right)}\ {\left(Q{\cdot}R \right)}}+ \ \ \displaystyle {c \ {\left(P{\cdot}R \right)}}-{b \ c}$

(12)

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

fricas

--)set output tex on
--)set output algebra off
eq47a.detMat

$\label{eq13}\begin{array}{@{}l} \displaystyle {{\left({{{P}^{2}}\ {{Q}^{2}}}-{{\left(P{\cdot}Q \right)}^{2}}+{{a}^{2}}\right)}\ {{R}^{2}}}-{{{P}^{2}}\ {{\left(Q{\cdot}R \right)}^{2}}}+ \ \ \displaystyle {{\left({2 \ {\left(P{\cdot}Q \right)}\ {\left(P{\cdot}R \right)}}-{2 \ a \ b}\right)}\ {\left(Q{\cdot}R \right)}}+ \ \ \displaystyle {{\left(-{{\left(P{\cdot}R \right)}^{2}}+{{b}^{2}}\right)}\ {{Q}^{2}}}+{2 \ a \ c \ {\left(P{\cdot}R \right)}}-{2 \ b \ c \ {\left(P{\cdot}Q \right)}}+ \ \ \displaystyle {{{c}^{2}}\ {{P}^{2}}}$

(13)

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

Simplifications

fricas

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

$\label{eq14}{{a \ R}-{b \ Q}+{c \ P}}= v$

(14)

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

fricas

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

$\label{eq15}\begin{array}{@{}l} \displaystyle \left\{{= = \left({{{a \ {{R}^{2}}}-{b \ {\left(Q{\cdot}R \right)}}+{c \ {\left(P{\cdot}R \right)}}+ \%G}, \:{{\left(R{\cdot}v \right)}+ \%G}}\right)}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{= = \left({{{a \ {\left(Q{\cdot}R \right)}}-{b \ {{Q}^{2}}}+{c \ {\left(P{\cdot}Q \right)}}+ \%H}, \:{{\left(Q{\cdot}v \right)}+ \%H}}\right)}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{= = \left({{{a \ {\left(P{\cdot}R \right)}}-{b \ {\left(P{\cdot}Q \right)}}+{c \ {{P}^{2}}}+ \%I}, \:{{\left(P{\cdot}v \right)}+ \%I}}\right)}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{= = \left({{-{a \ {{R}^{2}}}+{b \ {\left(Q{\cdot}R \right)}}-{c \ {\left(P{\cdot}R \right)}}+ \%J}, \:{-{\left(R{\cdot}v \right)}+ \%J}}\right)}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{= = \left({{-{a \ {\left(Q{\cdot}R \right)}}+{b \ {{Q}^{2}}}-{c \ {\left(P{\cdot}Q \right)}}+ \%K}, \:{-{\left(Q{\cdot}v \right)}+ \%K}}\right)}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{= = \left({{-{a \ {\left(P{\cdot}R \right)}}+{b \ {\left(P{\cdot}Q \right)}}-{c \ {{P}^{2}}}+ \%L}, \:{-{\left(P{\cdot}v \right)}+ \%L}}\right)}\right\}$

(15)

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

fricas

for i in 3..3 repeat
  for j in 2..2 repeat
    x:=(eq47a.adjMat)(i,j); outputAsTex ["x",i,j,x]
    y:=combine(x); outputAsTex ["y",i,j,y]
    z:=eq45a(y)
    outputAsTex ["z",i,j,z]

$\label{eq16}\begin{array}{@{}l} \displaystyle \verb#"x"# \ 3 \ 2 \ { \begin{array}{@{}l} \displaystyle -{{{P}^{2}}\ {\left(Q{\cdot}R \right)}}+{{\left({\left(P{\cdot}Q \right)}- a \right)}\ {\left(P{\cdot}R \right)}}+ \ \ \displaystyle {b \ {\left(P{\cdot}Q \right)}}-{c \ {{P}^{2}}}-{a \ b}$

(16)

$\label{eq17}\begin{array}{@{}l} \displaystyle \verb#"y"# \ 3 \ 2 \ { \begin{array}{@{}l} \displaystyle {\left({\left(A{\wedge}B \right)}{\cdot}{\left(A{\wedge}C \right)}\right)}-{a \ {\left(P{\cdot}R \right)}}+ \ \ \displaystyle {b \ {\left(P{\cdot}Q \right)}}-{c \ {{P}^{2}}}-{a \ b}$

(17)

$\label{eq18}\verb#"z"# \ 3 \ 2 \ {{\left({\left(A{\wedge}B \right)}{\cdot}{\left(A{\wedge}C \right)}\right)}+{\left(P{\cdot}v \right)}-{a \ b}}$

(18)

Type: Void

fricas

eq45x := rule
  -a*Dot(A,C)+b*Dot(A,B)-c*Dot(A,A) == -dot(A,v)

$\label{eq19}= = \left({{-{a \ {\left(A{\cdot}C \right)}}+{b \ {\left(A{\cdot}B \right)}}-{c \ {{A}^{2}}}+ \%M}, \:{-{\left(A{\cdot}v \right)}+ \%M}}\right)$

(19)

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

fricas

$\label{eq20}{\left({\left(A{\wedge}B \right)}{\cdot}{\left(A{\wedge}C \right)}\right)}+{\left(P{\cdot}v \right)}-{a \ b}$

(20)

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

fricas

eq45x z

$\label{eq21}{\left({\left(A{\wedge}B \right)}{\cdot}{\left(A{\wedge}C \right)}\right)}+{\left(P{\cdot}v \right)}-{a \ b}$

(21)

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

fricas

variables(z)

$\label{eq22}\left[ \right]$

(22)

Type: List(Symbol)

fricas

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

$\label{eq23}\left[ \begin{array}{ccc} {-{{\left(A{\wedge}B \right)}^{2}}+{{c}^{2}}}&{{a \ {{R}^{2}}}-{b \ {\left(Q{\cdot}R \right)}}+{c \ {\left(P{\cdot}R \right)}}-{\left(A{\cdot}B \right)}-{b \ c}}&{-{\left(Q{\cdot}v \right)}-{\left(A{\cdot}B \right)}+{a \ c}} \ {-{a \ {{R}^{2}}}+{b \ {\left(Q{\cdot}R \right)}}-{c \ {\left(P{\cdot}R \right)}}-{\left(A{\cdot}B \right)}-{b \ c}}&{-{{\left(A{\wedge}B \right)}^{2}}+{{b}^{2}}}&{{\left({\left(A{\wedge}B \right)}{\cdot}{\left(A{\wedge}C \right)}\right)}+{\left(P{\cdot}v \right)}-{a \ b}} \ {{\left(Q{\cdot}v \right)}-{\left(A{\cdot}B \right)}+{a \ c}}&{{\left({\left(A{\wedge}B \right)}{\cdot}{\left(A{\wedge}C \right)}\right)}+{\left(P{\cdot}v \right)}-{a \ b}}&{-{{\left(A{\wedge}B \right)}^{2}}+{{a}^{2}}}$

(23)

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

fricas

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

$\label{eq24}\begin{array}{@{}l} \displaystyle {{{{a}^{2}}\ {{R}^{2}}}+{{\left(-{2 \ a \ b \ Q}+{2 \ a \ c \ P}\right)}\ R}+{{{b}^{2}}\ {{Q}^{2}}}-{2 \ b \ c \ P \ Q}+{{{c}^{2}}\ {{P}^{2}}}}= \ \ \displaystyle {{v}^{2}}$

(24)

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

fricas

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(P,R) - 2*a*b*Dot(Q,R) == v^2

$\label{eq25}= = \left({{{{{a}^{2}}\ {{R}^{2}}}-{2 \ a \ b \ {\left(Q{\cdot}R \right)}}+{{{b}^{2}}\ {{Q}^{2}}}+{2 \ a \ c \ {\left(P{\cdot}R \right)}}-{2 \ b \ c \ {\left(P{\cdot}Q \right)}}+{{{c}^{2}}\ {{P}^{2}}}+ \%N}, \:{{{v}^{2}}+ \%N}}\right)$

(25)

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

fricas

eq47d(eq47a.detMat)

$\label{eq26}\begin{array}{@{}l} \displaystyle {{\left({{{P}^{2}}\ {{Q}^{2}}}-{{\left(P{\cdot}Q \right)}^{2}}+{{a}^{2}}\right)}\ {{R}^{2}}}-{{{P}^{2}}\ {{\left(Q{\cdot}R \right)}^{2}}}+ \ \ \displaystyle {{\left({2 \ {\left(P{\cdot}Q \right)}\ {\left(P{\cdot}R \right)}}-{2 \ a \ b}\right)}\ {\left(Q{\cdot}R \right)}}+ \ \ \displaystyle {{\left(-{{\left(P{\cdot}R \right)}^{2}}+{{b}^{2}}\right)}\ {{Q}^{2}}}+{2 \ a \ c \ {\left(P{\cdot}R \right)}}-{2 \ b \ c \ {\left(P{\cdot}Q \right)}}+ \ \ \displaystyle {{{c}^{2}}\ {{P}^{2}}}$

(26)

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

Subject: Be Bold !!

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