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SandBoxObserverAsIdempotent

Edit detail for SandBoxObserverAsIdempotent revision 13 of 21

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Editor: Bill page Time: 2014/08/08 14:05:50 GMT+0
Note:

changed:
-R:=FRAC DMP(concat [[x1,x2,x3,x4,x5,x6,x7,x8,x9,x10],[m(i) for i in gens],concat [[γ(j,i) for i in gens] for j in gens]], INT)
R:=FRAC DMP(concat [[m(i) for i in gens],concat [[γ(j,i) for i in gens] for j in gens]], INT)

changed:
-Lie derivation
-\begin{axiom}
-Dp(q) == p*q - q*p
-test(Dp(q*r) = Dp(q)*r + q*Dp(r))
-Dp(p)
-Dp(q)
-Dp(Dp(q))
-p*Dp(q)
-Dq(r) == q*r - r*q
-Dp(Dq(r))
-Dq(Dp(r))
-Dp(Dq(Dp(r)))
-Dq(Dp(Dq(r)))
-Dp(Dp(Dp(r)))
-Dp(Dp(Dq(r)))
-Dp(Dq(Dq(r)))
-\end{axiom}
Lie derivations
\begin{axiom}
D(p:A):(A->A) == (q:A):A +-> (p*q - q*p)
test(D(p)(q*r) = D(p)(q)*r + q*D(p)(r))
D(p)(p)
D(p)(q)
D(p)(D(p)(q))
p*D(p)(q)
--
D(p)(D(q)(r))
D(q)(D(p)(r))
D(p)(D(q)(D(p)(r)))
D(q)(D(p)(D(q)(r)))
test( D(p)(D(p)(D(p)(r))) = m(gens(1))^2*D(p)(r) )
test( D(p)(D(p)(D(p)(D(q)(r)))) = m(gens(1))^2*D(p)(D(q)(r)) )
D(p)(D(p)(D(q)(r)))
D(p)(D(q)(D(q)(r)))
\end{axiom}

Obs(3) is a 9 dimensional Frobenius Algrebra

fricas

)set output abbreviate on

V := OrderedVariableList [p,q,r]

$\label{eq1}\hbox{\axiomType{OVAR}\ } ([ p , q , r ])$

(1)

Type: TYPE

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M := FreeMonoid V

$\label{eq2}\hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r ]))$

(2)

Type: TYPE

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gens:List M := enumerate()$V

$\label{eq3}\left[ p , \: q , \: r \right]$

(3)

Type: LIST(FMONOID(OVAR([p,q,r])))

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divisible := Record(lm: M,rm: M)

$\label{eq4}\mbox{\rm \hbox{\axiomType{Record}\ } (lm : \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r ])) , rm : \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r ])))}$

(4)

Type: TYPE

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leftDiv(k:Union(divisible,"failed")):M == (k::divisible).lm

   Function declaration leftDiv : Union(Record(lm: FMONOID(OVAR([p,q,r]
      )),rm: FMONOID(OVAR([p,q,r]))),"failed") -> FMONOID(OVAR([p,q,r])
      ) has been added to workspace.

Type: VOID

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rightDiv(k:Union(divisible,"failed")):M == (k::divisible).rm

   Function declaration rightDiv : Union(Record(lm: FMONOID(OVAR([p,q,r
      ])),rm: FMONOID(OVAR([p,q,r]))),"failed") -> FMONOID(OVAR([p,q,r]
      )) has been added to workspace.

Type: VOID

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K := FRAC POLY INT

$\label{eq5}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{POLY}\ } (\hbox{\axiomType{INT}\ }))$

(5)

Type: TYPE

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MK := FreeModule(K,M)

$\label{eq6}FM (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{POLY}\ } (\hbox{\axiomType{INT}\ })) , \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q , r ])))$

(6)

Type: TYPE

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coeff(x:MK):K == leadingCoefficient(x)

   Function declaration coeff : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]
      ))) -> FRAC(POLY(INT)) has been added to workspace.

Type: VOID

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monomial(x:MK):M == leadingMonomial(x)

   Function declaration monomial : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q
      ,r]))) -> FMONOID(OVAR([p,q,r])) has been added to workspace.

Type: VOID

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m(x:M):K == subscript('m,[retract(x)::Symbol])

   Function declaration m : FMONOID(OVAR([p,q,r])) -> FRAC(POLY(INT)) 
      has been added to workspace.

Type: VOID

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γ(x:M,y:M):K == subscript('γ,[concat(string retract x, string retract y)::Symbol])

   Function declaration γ : (FMONOID(OVAR([p,q,r])),FMONOID(OVAR([p,q,r
      ]))) -> FRAC(POLY(INT)) has been added to workspace.

Type: VOID

Basis

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basis := concat(gens,concat [[j*i for i in gens | i~=j] for j in gens])

$\label{eq7}\left[ p , \: q , \: r , \:{p \ q}, \:{p \ r}, \:{q \ p}, \:{q \ r}, \:{r \ p}, \:{r \ q}\right]$

(7)

Type: LIST(FMONOID(OVAR([p,q,r])))

Idempotent

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rule1(ij:MK):MK ==
  for k in gens repeat
    kk := divide(monomial(ij),k*k)
    if kk case divisible then
      ij:=(coeff(ij) * m(k)*γ(k,k)) * (leftDiv(kk) * k * rightDiv(kk))
  return(ij)

   Function declaration rule1 : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]
      ))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]))) has been added 
      to workspace.

Type: VOID

Reduction

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rule2(ij:MK):MK ==
  for i in gens repeat
    for j in gens | j ~= i repeat
      for k in gens | k ~= j repeat
        ijk:=divide(leadingMonomial(ij),i*j*k)
        if ijk case divisible then
          if i=k then
            ij := (coeff(ij)*m(i)*m(j)*γ(i,j)*γ(j,i) ) * _
                  (leftDiv(ijk)*i*rightDiv(ijk))
          else
            ij := (coeff(ij)*m(j)*γ(i,j)*γ(j,k) / γ(i,k) ) * _
                  (leftDiv(ijk)*i*k*rightDiv(ijk))
  return(ij)

   Function declaration rule2 : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]
      ))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]))) has been added 
      to workspace.

Type: VOID

Modulo fixed point of applied rules

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mod(ij:MK):MK ==
  ijFix:MK := 1
  while ijFix~=ij repeat
    ijFix := ij
    ij := rule1(ij)
    ij := rule2(ij)
  return(ij)

   Function declaration mod : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]))
      ) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r]))) has been added to
      workspace.

Type: VOID

Matrix

Multiplication is monoidal concatenation modulo the fixed point

fricas

--MT := [[mod(i*j) for j in basis] for i in basis]
-- idempotent
MT := [[monomial(eval(coeff(mod(i*j)),[γ(gens(1),gens(1))=1,γ(gens(2),gens(2))=1,γ(gens(3),gens(3))=1,γ(gens(2),gens(1))=γ(gens(1),gens(2)),γ(gens(3),gens(2))=γ(gens(2),gens(3)),γ(gens(3),gens(1))=γ(gens(1),gens(3))]),monomial(mod(i*j)))$MK for j in basis] for i in basis]

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Compiling function monomial with type FM(FRAC(POLY(INT)),FMONOID(
      OVAR([p,q,r]))) -> FMONOID(OVAR([p,q,r]))

fricas

Compiling function coeff with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q,r]))) -> FRAC(POLY(INT))

fricas

Compiling function m with type FMONOID(OVAR([p,q,r])) -> FRAC(POLY(
      INT))

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Compiling function γ with type (FMONOID(OVAR([p,q,r])),FMONOID(OVAR(
      [p,q,r]))) -> FRAC(POLY(INT))

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Compiling function leftDiv with type Union(Record(lm: FMONOID(OVAR([
      p,q,r])),rm: FMONOID(OVAR([p,q,r]))),"failed") -> FMONOID(OVAR([p
      ,q,r]))

fricas

Compiling function rightDiv with type Union(Record(lm: FMONOID(OVAR(
      [p,q,r])),rm: FMONOID(OVAR([p,q,r]))),"failed") -> FMONOID(OVAR([
      p,q,r]))

fricas

Compiling function rule1 with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q,r]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r])))

fricas

Compiling function rule2 with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q,r]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r])))

fricas

Compiling function mod with type FM(FRAC(POLY(INT)),FMONOID(OVAR([p,
      q,r]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r])))

$\label{eq8}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{{m_{p}}\ p}, \:{p \ q}, \:{p \ r}, \:{{m_{p}}\ p \ q}, \:{{m_{p}}\ p \ r}, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ p}, \:{{{{m_{q}}\ {��_{pq}}\ {��_{qr}}}\over{��_{pr}}}\ p \ r}, \: \right. \ \ \displaystyle \left.{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}\ p}, \:{{{{m_{r}}\ {��_{pr}}\ {��_{qr}}}\over{��_{pq}}}\ p \ q}\right]$

(8)

Type: LIST(LIST(FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q,r])))))

Structure Constants

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R:=FRAC DMP(concat [[m(i) for i in gens],concat [[γ(j,i) for i in gens] for j in gens]], INT)

$\label{eq9}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ 01 mp , 01 mq , 01 mr , 01 �� pp , * 01 �� pq , * 01 �� pr , * 01 �� qp , * 01 �� qq , * 01 �� qr , * 01 �� rp , * 0 1 �� rq , * 01 �� rr ] , \hbox{\axiomType{INT}\ }))$

(9)

Type: TYPE

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mat3(y:M):List List R == map(z+->map(x+->coefficient(x,y)::FRAC POLY INT,z),MT)

   Function declaration mat3 : FMONOID(OVAR([p,q,r])) -> LIST(LIST(FRAC
      (DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr
      ,*01γrp,*01γrq,*01γrr],INT)))) has been added to workspace.

Type: VOID

fricas

ss:=map(mat3, basis)

fricas

Compiling function mat3 with type FMONOID(OVAR([p,q,r])) -> LIST(
      LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γ
      qq,*01γqr,*01γrp,*01γrq,*01γrr],INT))))

$\label{eq10}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{\left[{m_{p}}, \: 0, \: 0, \: 0, \: 0, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \: 0, \:{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle \left[{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \: 0, \: 0, \: 0, \: 0, \:{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}, \: 0, \:{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}, \: 0 \right]$

(10)

Type: LIST(LIST(LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))))

Algebra

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cats(m:M):Symbol==concat(map(x+->string(x.gen::Symbol),factors m))::Symbol

   Function declaration cats : FMONOID(OVAR([p,q,r])) -> SYMBOL has 
      been added to workspace.

Type: VOID

fricas

A:=AlgebraGivenByStructuralConstants(R,#(basis)::PI,map(cats,basis),ss::Vector(Matrix R))

fricas

Compiling function cats with type FMONOID(OVAR([p,q,r])) -> SYMBOL

$\label{eq11}\hbox{\axiomType{ALGSC}\ } (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ 01 mp , 01 mq , 01 mr , 01 �� pp , * 01 �� pq , * 01 �� pr , * 01 �� qp , * 01 �� qq , * 01 �� qr , * 01 �� rp , * 01 �� rq , * 01 �� rr ] , \hbox{\axiomType{INT}\ })) , 9, [ p , q , r , pq , pr , qp , qr , rp , rq ] , [ \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } ])$

(11)

Type: TYPE

fricas

alternative?()$A

   algebra satisfies 2*associator(a,b,b) = 0 = 2*associator(a,a,b) = 0

$\label{eq12} \mbox{\rm true}$

(12)

Type: BOOLEAN

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antiAssociative?()$A

   algebra is not anti-associative

$\label{eq13} \mbox{\rm false}$

(13)

Type: BOOLEAN

fricas

antiCommutative?()$A

   algebra is not anti-commutative

$\label{eq14} \mbox{\rm false}$

(14)

Type: BOOLEAN

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associative?()$A

   algebra is associative

$\label{eq15} \mbox{\rm true}$

(15)

Type: BOOLEAN

fricas

commutative?()$A

   algebra is not commutative

$\label{eq16} \mbox{\rm false}$

(16)

Type: BOOLEAN

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flexible?()$A

   algebra is flexible

$\label{eq17} \mbox{\rm true}$

(17)

Type: BOOLEAN

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jacobiIdentity?()$A

   Jacobi identity does not hold

$\label{eq18} \mbox{\rm false}$

(18)

Type: BOOLEAN

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jordanAdmissible?()$A

   algebra is not Jordan admissible

$\label{eq19} \mbox{\rm false}$

(19)

Type: BOOLEAN

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jordanAlgebra?()$A

   algebra is not commutative
   this is not a Jordan algebra

$\label{eq20} \mbox{\rm false}$

(20)

Type: BOOLEAN

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leftAlternative?()$A

   algebra is left alternative

$\label{eq21} \mbox{\rm true}$

(21)

Type: BOOLEAN

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rightAlternative?()$A

   algebra is right alternative

$\label{eq22} \mbox{\rm true}$

(22)

Type: BOOLEAN

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lieAdmissible?()$A

   algebra is Lie admissible

$\label{eq23} \mbox{\rm true}$

(23)

Type: BOOLEAN

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lieAlgebra?()$A

   algebra is not anti-commutative
   this is not a Lie algebra

$\label{eq24} \mbox{\rm false}$

(24)

Type: BOOLEAN

fricas

powerAssociative?()$A

   Internal Error
   The function powerAssociative? with signature hashcode is missing 
      from domain AlgebraGivenByStructuralConstants
      (Fraction (DistributedMultivariatePolynomial ((*01m p) (*01m q) (*01m r) (*01γ pp) (*01γ pq) (*01γ pr) (*01γ qp) (*01γ qq) (*01γ qr) (*01γ rp) (*01γ rq) (*01γ rr)) (Integer)))
      9(p q r pq pr qp qr rp rq)UNPRINTABLE

Check Multiplication

fricas

AB := entries basis()$A

$\label{eq25}\left[ p , \: q , \: r , \: pq , \: pr , \: qp , \: qr , \: rp , \: rq \right]$

(25)

Type: LIST(ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX]))

fricas

p:=AB(1); q:=AB(2); r:=AB(3);

Type: ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX])

fricas

A2MK(z:A):MK==reduce(+,map((x:R,y:M):MK+->(x::K)*y,coordinates(z),basis))

   Function declaration A2MK : ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp
      ,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)),9
      ,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX]) -> FM(FRAC(POLY(INT)),FMONOID(OVAR(
      [p,q,r]))) has been added to workspace.

Type: VOID

fricas

test(MT=map(x+->map(A2MK,x),[[i*j for j in AB] for i in AB]))

fricas

Compiling function A2MK with type ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,
      *01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],
      INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX,MATRIX]) -> FM(FRAC(POLY(INT)),
      FMONOID(OVAR([p,q,r])))

$\label{eq26} \mbox{\rm true}$

(26)

Type: BOOLEAN

fricas

p*p

$\label{eq27}{m_{p}}\ p$

(27)

fricas

p*q*p

$\label{eq28}{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ p$

(28)

fricas

p*q*r

$\label{eq29}{{{m_{q}}\ {��_{pq}}\ {��_{qr}}}\over{��_{pr}}}\ pr$

(29)

Lie derivations

fricas

D(p:A):(A->A) == (q:A):A +-> (p*q - q*p)

   Function declaration D : ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,
      *01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)),9,
      [p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX]) -> (ALGSC(FRAC(DMP([*01mp,*01mq,
      *01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,
      *01γrr],INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX]) -> ALGSC(FRAC(DMP([
      *01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γ
      rp,*01γrq,*01γrr],INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX])) has 
      been added to workspace.

Type: VOID

fricas

test(D(p)(q*r) = D(p)(q)*r + q*D(p)(r))

fricas

Compiling function D with type ALGSC(FRAC(DMP([*01mp,*01mq,*01mr,
      *01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],
      INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX,MATRIX]) -> (ALGSC(FRAC(DMP([*01mp,
      *01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γ
      rq,*01γrr],INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX]) -> ALGSC(FRAC(
      DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,
      *01γrp,*01γrq,*01γrr],INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX,MATRIX]))

$\label{eq30} \mbox{\rm true}$

(30)

Type: BOOLEAN

fricas

D(p)(p)

$\label{eq31}0$

(31)

fricas

D(p)(q)

$\label{eq32}- qp + pq$

(32)

fricas

D(p)(D(p)(q))

$\label{eq33}{{m_{p}}\ qp}+{{m_{p}}\ pq}-{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ p}$

(33)

fricas

p*D(p)(q)

$\label{eq34}{{m_{p}}\ pq}-{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ p}$

(34)

fricas

--
D(p)(D(q)(r))

$\label{eq35}{{{{m_{q}}\ {��_{pq}}\ {��_{qr}}}\over{��_{pr}}}\ rp}-{{{{m_{r}}\ {��_{pr}}\ {��_{qr}}}\over{��_{pq}}}\ qp}+{{{{m_{q}}\ {��_{pq}}\ {��_{qr}}}\over{��_{pr}}}\ pr}-{{{{m_{r}}\ {��_{pr}}\ {��_{qr}}}\over{��_{pq}}}\ pq}$

(35)

fricas

D(q)(D(p)(r))

$\label{eq36}{{{{m_{p}}\ {��_{pq}}\ {��_{pr}}}\over{��_{qr}}}\ rq}+{{{{m_{p}}\ {��_{pq}}\ {��_{pr}}}\over{��_{qr}}}\ qr}-{{{{m_{r}}\ {��_{pr}}\ {��_{qr}}}\over{��_{pq}}}\ qp}-{{{{m_{r}}\ {��_{pr}}\ {��_{qr}}}\over{��_{pq}}}\ pq}$

(36)

fricas

D(p)(D(q)(D(p)(r)))

$\label{eq37}\begin{array}{@{}l} \displaystyle -{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ rp}+{{{-{{m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}}+{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}}\over{��_{pq}}}\ qp}+ \ \ \displaystyle {{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ pr}+{{{{{m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}}-{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}}\over{��_{pq}}}\ pq}$

(37)

fricas

D(q)(D(p)(D(q)(r)))

$\label{eq38}\begin{array}{@{}l} \displaystyle -{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ rq}+{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ qr}+ \ \ \displaystyle {{{{{m_{q}}\ {m_{r}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}}\over{��_{pq}}}\ qp}+ \ \ \displaystyle {{{-{{m_{q}}\ {m_{r}}\ {��_{pq}}\ {{��_{qr}}^{2}}}+{{m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}}\over{��_{pq}}}\ pq}$

(38)

fricas

test( D(p)(D(p)(D(p)(r))) = m(gens(1))^2*D(p)(r) )

$\label{eq39} \mbox{\rm true}$

(39)

Type: BOOLEAN

fricas

test( D(p)(D(p)(D(p)(D(q)(r)))) = m(gens(1))^2*D(p)(D(q)(r)) )

$\label{eq40} \mbox{\rm true}$

(40)

Type: BOOLEAN

fricas

D(p)(D(p)(D(q)(r)))

$\label{eq41}\begin{array}{@{}l} \displaystyle -{{{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qr}}}\over{��_{pr}}}\ rp}+{{{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}\over{��_{pq}}}\ qp}+{{{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qr}}}\over{��_{pr}}}\ pr}- \ \ \displaystyle {{{{m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}\over{��_{pq}}}\ pq}$

(41)

fricas

D(p)(D(q)(D(q)(r)))

$\label{eq42}\begin{array}{@{}l} \displaystyle -{{{{{m_{q}}^{2}}\ {��_{pq}}\ {��_{qr}}}\over{��_{pr}}}\ rp}+{{{{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}}\over{��_{pq}}}\ qp}+ \ \ \displaystyle {{{{{m_{q}}^{2}}\ {��_{pq}}\ {��_{qr}}}\over{��_{pr}}}\ pr}+{{{-{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {{��_{qr}}^{2}}}+{{m_{q}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}}\over{��_{pq}}}\ pq}$

(42)

Trace

fricas

[rightTrace(i)$A for i in AB]

$\label{eq43}\begin{array}{@{}l} \displaystyle \left[{3 \ {m_{p}}}, \:{3 \ {m_{q}}}, \:{3 \ {m_{r}}}, \:{3 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{3 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \: \right. \ \ \displaystyle \left.{3 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{3 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}, \:{3 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \:{3 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}\right]$

(43)

Type: LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

[leftTrace(i)$A for i in AB]

$\label{eq44}\begin{array}{@{}l} \displaystyle \left[{3 \ {m_{p}}}, \:{3 \ {m_{q}}}, \:{3 \ {m_{r}}}, \:{3 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{3 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \: \right. \ \ \displaystyle \left.{3 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{3 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}, \:{3 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \:{3 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}\right]$

(44)

Type: LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

trace(i)==rightTrace(i) / #gens

Type: VOID

fricas

[trace(i) for i in AB]

fricas

Compiling function trace with type ALGSC(FRAC(DMP([*01mp,*01mq,*01mr
      ,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],
      INT)),9,[p,q,r,pq,pr,qp,qr,rp,rq],[MATRIX,MATRIX,MATRIX,MATRIX,
      MATRIX,MATRIX,MATRIX,MATRIX,MATRIX]) -> FRAC(DMP([*01mp,*01mq,
      *01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,
      *01γrr],INT))

$\label{eq45}\begin{array}{@{}l} \displaystyle \left[{m_{p}}, \:{m_{q}}, \:{m_{r}}, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \:{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \:{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}, \: \right. \ \ \displaystyle \left.{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}, \:{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}, \:{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}\right]$

(45)

Type: LIST(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

Scalar Product

fricas

matrix [[trace(i*j) for j in [AB.1,AB.2,AB.3]] for i in [AB.1,AB.2,AB.3]]

$\label{eq46}\left[ \begin{array}{ccc} {{m_{p}}^{2}}&{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}} \ {{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{q}}^{2}}&{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}} \ {{m_{p}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{r}}^{2}}$

(46)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.1,AB.2,AB.3]] for i in [AB.4,AB.5,AB.6]]

$\label{eq47}\left[ \begin{array}{ccc} {{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}} \ {{{m_{p}}^{2}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}} \ {{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}$

(47)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.1,AB.2,AB.3]] for i in [AB.7,AB.8,AB.9]]

$\label{eq48}\left[ \begin{array}{ccc} {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}} \ {{{m_{p}}^{2}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}} \ {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}}$

(48)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.4,AB.5,AB.6]] for i in [AB.1,AB.2,AB.3]]

$\label{eq49}\left[ \begin{array}{ccc} {{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}} \ {{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}} \ {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}$

(49)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.4,AB.5,AB.6]] for i in [AB.4,AB.5,AB.6]]

$\label{eq50}\left[ \begin{array}{ccc} {{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{4}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}} \ {{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{4}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}} \ {{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{4}}}$

(50)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.4,AB.5,AB.6]] for i in [AB.7,AB.8,AB.9]]

$\label{eq51}\left[ \begin{array}{ccc} {{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{qr}}^{2}}} \ {{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}} \ {{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}$

(51)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.7,AB.8,AB.9]] for i in [AB.1,AB.2,AB.3]]

$\label{eq52}\left[ \begin{array}{ccc} {{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {m_{r}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}} \ {{{m_{q}}^{2}}\ {m_{r}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {m_{r}}\ {{��_{qr}}^{2}}} \ {{m_{q}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{q}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}}$

(52)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.7,AB.8,AB.9]] for i in [AB.4,AB.5,AB.6]]

$\label{eq53}\left[ \begin{array}{ccc} {{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{qr}}^{2}}} \ {{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qr}}^{2}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}} \ {{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{qr}}^{2}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {{��_{pr}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}$

(53)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

matrix [[trace(i*j) for j in [AB.7,AB.8,AB.9]] for i in [AB.7,AB.8,AB.9]]

$\label{eq54}\left[ \begin{array}{ccc} {{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{4}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}} \ {{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}&{{{m_{p}}^{2}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{4}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qr}}^{2}}} \ {{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{2}}}&{{m_{p}}\ {m_{q}}\ {{m_{r}}^{2}}\ {{��_{pr}}^{2}}\ {{��_{qr}}^{2}}}&{{{m_{q}}^{2}}\ {{m_{r}}^{2}}\ {{��_{qr}}^{4}}}$

(54)

Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

Center

fricas

C:=basisOfCenter()$AlgebraPackage(R,A); # C

$\label{eq55}1$

(55)

Type: PI

fricas

c:=C(1)

$\label{eq56}\begin{array}{@{}l} \displaystyle rq +{{{{{m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{q}}\ {��_{pr}}\ {��_{qr}}}}\over{{{m_{p}}\ {��_{pq}}\ {{��_{pr}}^{2}}}-{{m_{p}}\ {��_{pr}}\ {��_{qr}}}}}\ rp}+ qr + \ \ \displaystyle {{{-{{m_{r}}\ {��_{pq}}\ {��_{qr}}}+{{m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}}\over{{{m_{p}}\ {{��_{pq}}^{2}}\ {��_{pr}}}-{{m_{p}}\ {��_{pq}}\ {��_{qr}}}}}\ qp}+{{{{{m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{{m_{q}}\ {��_{pr}}\ {��_{qr}}}}\over{{{m_{p}}\ {��_{pq}}\ {{��_{pr}}^{2}}}-{{m_{p}}\ {��_{pr}}\ {��_{qr}}}}}\ pr}+ \ \ \displaystyle {{{-{{m_{r}}\ {��_{pq}}\ {��_{qr}}}+{{m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}}\over{{{m_{p}}\ {{��_{pq}}^{2}}\ {��_{pr}}}-{{m_{p}}\ {��_{pq}}\ {��_{qr}}}}}\ pq}+{{{-{{m_{q}}\ {{��_{pq}}^{2}}\ {��_{qr}}}+{{m_{q}}\ {��_{qr}}}}\over{{{��_{pq}}\ {��_{pr}}}-{��_{qr}}}}\ r}+ \ \ \displaystyle {{{-{{m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{r}}\ {��_{qr}}}}\over{{{��_{pq}}\ {��_{pr}}}-{��_{qr}}}}\ q}+{{{-{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}+{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\over{{{m_{p}}\ {��_{pq}}\ {��_{pr}}}-{{m_{p}}\ {��_{qr}}}}}\ p}$

(56)

fricas

[c*i-i*c for i in AB]

$\label{eq57}\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]$

(57)

fricas

test(c*c=c)

$\label{eq58} \mbox{\rm false}$

(58)

Type: BOOLEAN

Unit

fricas

rightTrace(c)

$\label{eq59}{\left( \begin{array}{@{}l} \displaystyle -{9 \ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}+{{18}\ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}- \ \ \displaystyle {9 \ {m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}-{9 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}+{9 \ {m_{q}}\ {m_{r}}\ {��_{qr}}}$

(59)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

n := #basis / rightTrace(c) * c

$\label{eq60}\begin{array}{@{}l} \displaystyle {{{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}\over{\left( \begin{array}{@{}l} \displaystyle {{m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}+ \ \ \displaystyle {{m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}-{{m_{q}}\ {m_{r}}\ {��_{qr}}}$

(60)

fricas

trace(n)

$\label{eq61}3$

(61)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

test(n*n=n)

$\label{eq62} \mbox{\rm true}$

(62)

Type: BOOLEAN

fricas

[n*i-i for i in AB]

$\label{eq63}\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]$

(63)

fricas

[i*n-i for i in AB]

$\label{eq64}\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]$

(64)

fricas

test(n=unit()$A)

$\label{eq65} \mbox{\rm true}$

(65)

Type: BOOLEAN

fricas

f:=gcd map(x+->denom x,coordinates(n))

$\label{eq66}{{��_{pq}}^{2}}-{2 \ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{��_{pr}}^{2}}+{{��_{qr}}^{2}}- 1$

(66)

Type: DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)

fricas

--ff:= matrix [[γ(i,j)::R for j in gens] for i in gens]
ff:= matrix [[eval(γ(i,j)::R,[γ(gens(1),gens(1))=1,γ(gens(2),gens(2))=1,γ(gens(3),gens(3))=1,γ(gens(2),gens(1))=γ(gens(1),gens(2)),γ(gens(3),gens(2))=γ(gens(2),gens(3)),γ(gens(3),gens(1))=γ(gens(1),gens(3))]) for j in gens] for i in gens]

$\label{eq67}\left[ \begin{array}{ccc} 1 &{��_{pq}}&{��_{pr}} \ {��_{pq}}& 1 &{��_{qr}} \ {��_{pr}}&{��_{qr}}& 1$

(67)

Type: MATRIX(FRAC(POLY(INT)))

fricas

--ff:= matrix [[eval(γ(i,j)::R,[γ(gens(1),gens(1))=0,γ(gens(2),gens(2))=0,γ(gens(3),gens(3))=0,γ(gens(2),gens(1))=γ(gens(1),gens(2)),γ(gens(3),gens(2))=γ(gens(2),gens(3)),γ(gens(3),gens(1))=γ(gens(1),gens(3))]) for j in gens] for i in gens]
-determinant(ff)

$\label{eq68}{{��_{qr}}^{2}}-{2 \ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{��_{pr}}^{2}}+{{��_{pq}}^{2}}- 1$

(68)

Type: FRAC(POLY(INT))

fricas

test(f = %)

$\label{eq69} \mbox{\rm true}$

(69)

Type: BOOLEAN

fricas

(f*n)::OutputForm / f::OutputForm

$\label{eq70}{\left( \begin{array}{@{}l} \displaystyle {{{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}\over{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ rq}+{{{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}\over{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ rp}+ \ \ \displaystyle {{{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}\over{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ qr}+{{{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}\over{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ qp}+{{{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}\over{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ pr}+ \ \ \displaystyle {{{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}\over{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ pq}+{{{{{��_{pq}}^{2}}- 1}\over{m_{r}}}\ r}+{{{{{��_{pr}}^{2}}- 1}\over{m_{q}}}\ q}+{{{{{��_{qr}}^{2}}- 1}\over{m_{p}}}\ p}$

(70)

Type: OUTFORM

Orthogonal Observers

fricas

p' := n - (1/trace(p))*p

$\label{eq71}\begin{array}{@{}l} \displaystyle {{{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}\over{\left( \begin{array}{@{}l} \displaystyle {{m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}+ \ \ \displaystyle {{m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}-{{m_{q}}\ {m_{r}}\ {��_{qr}}}$

(71)

fricas

trace(p')

$\label{eq72}2$

(72)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

p*p'

$\label{eq73}0$

(73)

fricas

test(p'*p=p*p')

$\label{eq74} \mbox{\rm true}$

(74)

Type: BOOLEAN

fricas

p'*p' - p'

$\label{eq75}0$

(75)

fricas

f*p'

$\label{eq76}\begin{array}{@{}l} \displaystyle {{{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}\over{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ rq}+{{{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}\over{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ rp}+{{{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}\over{{m_{q}}\ {m_{r}}\ {��_{qr}}}}\ qr}+ \ \ \displaystyle {{{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}\over{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ qp}+{{{-{{��_{pq}}\ {��_{qr}}}+{��_{pr}}}\over{{m_{p}}\ {m_{r}}\ {��_{pr}}}}\ pr}+{{{{��_{pq}}-{{��_{pr}}\ {��_{qr}}}}\over{{m_{p}}\ {m_{q}}\ {��_{pq}}}}\ pq}+ \ \ \displaystyle {{{{{��_{pq}}^{2}}- 1}\over{m_{r}}}\ r}+{{{{{��_{pr}}^{2}}- 1}\over{m_{q}}}\ q}+{{{-{{��_{pq}}^{2}}+{2 \ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}-{{��_{pr}}^{2}}}\over{m_{p}}}\ p}$

(76)

Lie Bracket

fricas

pq:=p*q-q*p

$\label{eq77}- qp + pq$

(77)

fricas

trace(pq)

$\label{eq78}0$

(78)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

pqr:=pq*r

$\label{eq79}-{{{{m_{p}}\ {��_{pq}}\ {��_{pr}}}\over{��_{qr}}}\ qr}+{{{{m_{q}}\ {��_{pq}}\ {��_{qr}}}\over{��_{pr}}}\ pr}$

(79)

fricas

trace(pqr)

$\label{eq80}0$

(80)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

pqr*pqr

$\label{eq81}0$

(81)

fricas

q' := n - (1/trace(q))*q

$\label{eq82}\begin{array}{@{}l} \displaystyle {{{-{{��_{pq}}\ {��_{pr}}}+{��_{qr}}}\over{\left( \begin{array}{@{}l} \displaystyle {{m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}+ \ \ \displaystyle {{m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}-{{m_{q}}\ {m_{r}}\ {��_{qr}}}$

(82)

fricas

p'q' := p'*q' - q'*p'

$\label{eq83}-{{1 \over{{m_{p}}\ {m_{q}}}}\ qp}+{{1 \over{{m_{p}}\ {m_{q}}}}\ pq}$

(83)

fricas

trace(p'q')

$\label{eq84}0$

(84)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

p'q'r:=p'q'*r

$\label{eq85}-{{{{��_{pq}}\ {��_{pr}}}\over{{m_{q}}\ {��_{qr}}}}\ qr}+{{{{��_{pq}}\ {��_{qr}}}\over{{m_{p}}\ {��_{pr}}}}\ pr}$

(85)

fricas

trace(p'q'r)

$\label{eq86}0$

(86)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

p'q'r * p'q'r

$\label{eq87}0$

(87)

fricas

p'q'r * pqr

$\label{eq88}0$

(88)

fricas

pq' := p*q' - q'*p

$\label{eq89}{{1 \over{m_{q}}}\ qp}-{{1 \over{m_{q}}}\ pq}$

(89)

fricas

trace(pq')

$\label{eq90}0$

(90)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

pq'r:=pq'*r

$\label{eq91}{{{{m_{p}}\ {��_{pq}}\ {��_{pr}}}\over{{m_{q}}\ {��_{qr}}}}\ qr}-{{{{��_{pq}}\ {��_{qr}}}\over{��_{pr}}}\ pr}$

(91)

fricas

pq'r * pq'r

$\label{eq92}0$

(92)

fricas

p'q := p'*q - q*p'

$\label{eq93}{{1 \over{m_{p}}}\ qp}-{{1 \over{m_{p}}}\ pq}$

(93)

fricas

trace(p'q)

$\label{eq94}0$

(94)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

p'qr:=p'q*r

$\label{eq95}{{{{��_{pq}}\ {��_{pr}}}\over{��_{qr}}}\ qr}-{{{{m_{q}}\ {��_{pq}}\ {��_{qr}}}\over{{m_{p}}\ {��_{pr}}}}\ pr}$

(95)

fricas

p'qr*p'qr

$\label{eq96}0$

(96)

Momentum

fricas

P:=reduce(+,concat [[(1/ ( i<j=>γ(basis(i),basis(j)); i>j=>γ(basis(j),basis(i));1) )::R*AB(i)*AB(j) for j in 1..size()$V] for i in 1..size()$V])

$\label{eq97}\begin{array}{@{}l} \displaystyle {{1 \over{��_{qr}}}\ rq}+{{1 \over{��_{pr}}}\ rp}+{{1 \over{��_{qr}}}\ qr}+{{1 \over{��_{pq}}}\ qp}+{{1 \over{��_{pr}}}\ pr}+{{1 \over{��_{pq}}}\ pq}+{{m_{r}}\ r}+ \ \ \displaystyle {{m_{q}}\ q}+{{m_{p}}\ p}$

(97)

fricas

M2:=trace(P)

$\label{eq98}{{m_{p}}^{2}}+{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}}+{2 \ {m_{p}}\ {m_{r}}\ {��_{pr}}}+{{m_{q}}^{2}}+{2 \ {m_{q}}\ {m_{r}}\ {��_{qr}}}+{{m_{r}}^{2}}$

(98)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

P*P-trace(P)*P

$\label{eq99}0$

(99)

fricas

P' := n - (1/trace(P))*P;

fricas

trace(P')

$\label{eq100}2$

(100)

Type: FRAC(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

P'*P' - P'

$\label{eq101}0$

(101)

fricas

P*P'

$\label{eq102}0$

(102)

fricas

fP':=gcd map(x+->denom x,coordinates(P'))

$\label{eq103}\begin{array}{@{}l} \displaystyle {{{m_{p}}^{2}}\ {{��_{pq}}^{2}}}-{2 \ {{m_{p}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{{m_{p}}^{2}}\ {{��_{pr}}^{2}}}+{{{m_{p}}^{2}}\ {{��_{qr}}^{2}}}-{{m_{p}}^{2}}+ \ \ \displaystyle {2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{3}}}-{4 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ {��_{pr}}\ {��_{qr}}}+{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{pr}}^{2}}}+ \ \ \displaystyle {2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {{��_{qr}}^{2}}}-{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}}+{2 \ {m_{p}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{pr}}}- \ \ \displaystyle {4 \ {m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{2 \ {m_{p}}\ {m_{r}}\ {{��_{pr}}^{3}}}+{2 \ {m_{p}}\ {m_{r}}\ {��_{pr}}\ {{��_{qr}}^{2}}}- \ \ \displaystyle {2 \ {m_{p}}\ {m_{r}}\ {��_{pr}}}+{{{m_{q}}^{2}}\ {{��_{pq}}^{2}}}-{2 \ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{{m_{q}}^{2}}\ {{��_{pr}}^{2}}}+ \ \ \displaystyle {{{m_{q}}^{2}}\ {{��_{qr}}^{2}}}-{{m_{q}}^{2}}+{2 \ {m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}\ {��_{qr}}}-{4 \ {m_{q}}\ {m_{r}}\ {��_{pq}}\ {��_{pr}}\ {{��_{qr}}^{2}}}+ \ \ \displaystyle {2 \ {m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}\ {��_{qr}}}+{2 \ {m_{q}}\ {m_{r}}\ {{��_{qr}}^{3}}}-{2 \ {m_{q}}\ {m_{r}}\ {��_{qr}}}+{{{m_{r}}^{2}}\ {{��_{pq}}^{2}}}- \ \ \displaystyle {2 \ {{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+{{{m_{r}}^{2}}\ {{��_{pr}}^{2}}}+{{{m_{r}}^{2}}\ {{��_{qr}}^{2}}}-{{m_{r}}^{2}}$

(103)

Type: DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)

fricas

factor(fP')

$\label{eq104}{\left({ \begin{array}{@{}l} \displaystyle {{��_{pq}}^{2}}-{2 \ {��_{pq}}\ {��_{pr}}\ {��_{qr}}}+ \ \ \displaystyle {{��_{pr}}^{2}}+{{��_{qr}}^{2}}- 1$

(104)

Type: FR(DMP([*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT))

fricas

test(fP'=f*M2)

$\label{eq105} \mbox{\rm true}$

(105)

Type: BOOLEAN

fricas

fP' * P'

$\label{eq106}\begin{array}{@{}l} \displaystyle {{{\left( \begin{array}{@{}l} \displaystyle -{{{m_{p}}^{2}}\ {��_{pq}}\ {��_{pr}}}+{{{m_{p}}^{2}}\ {��_{qr}}}-{2 \ {m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ {��_{pr}}}+ \ \ \displaystyle {2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qr}}}-{2 \ {m_{p}}\ {m_{r}}\ {��_{pq}}\ {{��_{pr}}^{2}}}+ \ \ \displaystyle {2 \ {m_{p}}\ {m_{r}}\ {��_{pr}}\ {��_{qr}}}-{{{m_{q}}^{2}}\ {��_{pq}}\ {��_{pr}}}+{{{m_{q}}^{2}}\ {��_{qr}}}- \ \ \displaystyle {{m_{q}}\ {m_{r}}\ {{��_{pq}}^{2}}}-{{m_{q}}\ {m_{r}}\ {{��_{pr}}^{2}}}+{{m_{q}}\ {m_{r}}\ {{��_{qr}}^{2}}}+{{m_{q}}\ {m_{r}}}- \ \ \displaystyle {{{m_{r}}^{2}}\ {��_{pq}}\ {��_{pr}}}+{{{m_{r}}^{2}}\ {��_{qr}}}$

(106)