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SandBoxObserverAsIdempotent

Edit detail for SandBoxObserverAsIdempotent revision 9 of 21

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Editor: Bill Page Time: 2014/05/21 19:05:42 GMT+0
Note: nilpotent from idempotent

added:
pq*pq

changed:
-\end{axiom}
trace(pq)
\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

fricas

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

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--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]))

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

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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 [[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)

$\label{eq9}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9, x 10, 01 mp , 01 mq , 01 mr , 01 �� pp , * 01 �� pq , * 01 �� pr , * 01 �� qp , * 01 �� qq , * 01 �� qr , * 01 �� rp , * 01 �� rq , * 01 �� rr ] , \hbox{\axiomType{INT}\ }))$

(9)

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([x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,*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)

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Compiling function mat3 with type FMONOID(OVAR([p,q,r])) -> LIST(
      LIST(FRAC(DMP([x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,*01mp,*01mq,*01mr,
      *01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*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([x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,*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

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

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

(11)

Type: TYPE

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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

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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

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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 (x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 (*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([x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,*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([x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,*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([x1,x2,x3,x4,x5,x6,x7,x8,
      x9,x10,*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γ
      qr,*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([x1,x2,x3,x4,x5,x6,
      x7,x8,x9,x10,*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*r

$\label{eq28}{{{m_{q}}\ {��_{pq}}\ {��_{qr}}}\over{��_{pr}}}\ pr$

(28)

fricas

pq:=p*q-q*p

$\label{eq29}- qp + pq$

(29)

fricas

pq*pq

$\label{eq30}{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ qp}+{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ pq}-{{{m_{p}}^{2}}\ {m_{q}}\ {{��_{pq}}^{2}}\ q}-{{m_{p}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}\ p}$

(30)

fricas

pqp:=pq*p

$\label{eq31}-{{m_{p}}\ qp}+{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ p}$

(31)

fricas

ppq:=p*pq

$\label{eq32}{{m_{p}}\ pq}-{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}\ p}$

(32)

fricas

pqq:=pq*q

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

(33)

fricas

qpq:=q*pq

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

(34)

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pqr:=pq*r

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

(35)

fricas

rpq:=r*pq

$\label{eq36}{{{{m_{p}}\ {��_{pq}}\ {��_{pr}}}\over{��_{qr}}}\ rq}-{{{{m_{q}}\ {��_{pq}}\ {��_{qr}}}\over{��_{pr}}}\ rp}$

(36)

fricas

pqp*pqp

$\label{eq37}0$

(37)

fricas

ppq*ppq

$\label{eq38}0$

(38)

fricas

pqq*pqq

$\label{eq39}0$

(39)

fricas

qpq*qpq

$\label{eq40}0$

(40)

fricas

pqr*pqr

$\label{eq41}0$

(41)

fricas

rpq*rpq

$\label{eq42}0$

(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([x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,*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([x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,*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([x1,x2,x3,x4,x5,x6
      ,x7,x8,x9,x10,*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γ
      qq,*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([x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,*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([x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,*01mp,*01mq,*01mr,*01γpp,*01γpq,*01γpr,*01γqp,*01γqq,*01γqr,*01γrp,*01γrq,*01γrr],INT)))

fricas

trace(pq)

$\label{eq46}0$

(46)

Type: FRAC(DMP([x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,*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]]

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