login  home  contents  what's new  discussion  bug reports     help  links  subscribe  changes  refresh  edit

Obs(2) is a 4 dimensional Frobenius Algebra

Generators of Obs(2)

fricas
)set output abbreviate on
 
fricas
)set message type off
V := OrderedVariableList [p,q]

\label{eq1}\hbox{\axiomType{OVAR}\ } ([ p , q ])(1)
fricas
vars:List V := enumerate()$V

\label{eq2}\left[ p , \: q \right](2)

--Representation

fricas
M := FreeMonoid V

\label{eq3}\hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q ]))(3)
fricas
divisible := Record(lm: M,rm: M)

\label{eq4}\mbox{\rm \hbox{\axiomType{Record}\ } (lm : \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q ])) , rm : \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q ])))}(4)
fricas
gamma(i:Symbol,j:Symbol):Symbol == 
  concat([string 'γ,string i,string j])::Symbol
Function declaration gamma : (SYMBOL,SYMBOL) -> SYMBOL has been added to workspace. --subscript('γ,[concat(string i, string j)::Symbol]) mass(i:Symbol):Symbol == concat("m",string i)::Symbol
Function declaration mass : SYMBOL -> SYMBOL has been added to workspace. --subscript('m,[i]) B := OrderedVariableList(concat [ ['x1,'x2,'x3,'x4], _ [mass i for i in vars], _ concat [[gamma(vars i ,vars j) for i in (j+1)..#vars] for j in 1..#vars] ])
fricas
Compiling function mass with type SYMBOL -> SYMBOL
fricas
Compiling function gamma with type (SYMBOL,SYMBOL) -> SYMBOL

\label{eq5}\hbox{\axiomType{OVAR}\ } ([ x 1, x 2, x 3, x 4, mp , mq , �� qp ])(5)
fricas
K := FRAC SMP(Integer,B)

\label{eq6}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{SMP}\ } (\hbox{\axiomType{INT}\ } , \hbox{\axiomType{OVAR}\ } ([ x 1, x 2, x 3, x 4, mp , mq , �� qp ])))(6)
fricas
MK := FreeModule(K,M)

\label{eq7}FM (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{SMP}\ } (\hbox{\axiomType{INT}\ } , \hbox{\axiomType{OVAR}\ } ([ x 1, x 2, x 3, x 4, mp , mq , �� qp ]))) , \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q ])))(7)
fricas
m(x:V):K == mass(x::Symbol)
Function declaration m : OVAR([p,q]) -> FRAC(SMP(INT,OVAR([x1,x2,x3, x4,mp,mq,γqp]))) has been added to workspace. m(vars 1)
fricas
Compiling function m with type OVAR([p,q]) -> FRAC(SMP(INT,OVAR([x1,
      x2,x3,x4,mp,mq,γqp])))

\label{eq8}mp(8)
fricas
γ(x:V,y:V):K ==
  if x<y then
    return variable(gamma(x::Symbol,y::Symbol))$B
  if x>y then
    return variable(gamma(y::Symbol,x::Symbol))$B
  return 1
Function declaration γ : (OVAR([p,q]),OVAR([p,q])) -> FRAC(SMP(INT, OVAR([x1,x2,x3,x4,mp,mq,γqp]))) has been added to workspace. Compiled code for gamma has been cleared. γ(vars 2,vars 1)
fricas
Compiling function gamma with type (SYMBOL,SYMBOL) -> SYMBOL
fricas
Compiling function γ with type (OVAR([p,q]),OVAR([p,q])) -> FRAC(SMP
      (INT,OVAR([x1,x2,x3,x4,mp,mq,γqp])))

\label{eq9}�� qp(9)
fricas
--Basis
basis:List M := concat(vars,concat [[i::M*j::M for j in vars | i~=j] for i in vars])

\label{eq10}\left[ p , \: q , \:{p \  q}, \:{q \  p}\right](10)

Idempotent: ii --> mᵢ γᵢᵢ i

fricas
idem(p:MK):MK ==
  -- p = c*q
  q := leadingSupport p
  c := leadingCoefficient p
  for i in vars::List M repeat
    f := divide(q, i*i)
    if f case divisible then -- q = f.lm * ii * f.rm
      return monomial(c * m i * γ(i,i), elt(f,lm) * i * elt(f,rm))
      --return monomial(c * γ(i,i), elt(f,lm) * i * elt(f,rm))
  return p
Function declaration idem : FM(FRAC(SMP(INT,OVAR([x1,x2,x3,x4,mp,mq, γqp]))),FMONOID(OVAR([p,q]))) -> FM(FRAC(SMP(INT,OVAR([x1,x2,x3, x4,mp,mq,γqp]))),FMONOID(OVAR([p,q]))) has been added to workspace. idem(basis(1)*basis(1))
fricas
Compiling function idem with type FM(FRAC(SMP(INT,OVAR([x1,x2,x3,x4,
      mp,mq,γqp]))),FMONOID(OVAR([p,q]))) -> FM(FRAC(SMP(INT,OVAR([x1,
      x2,x3,x4,mp,mq,γqp]))),FMONOID(OVAR([p,q])))

\label{eq11}mp \  p(11)

Reductions: ijk --> mᵢmⱼ γᵢⱼγⱼₖ/γᵢₖ ik

fricas
reduct(p:MK):MK ==
  q := leadingSupport p
  c := leadingCoefficient p
  for i in vars repeat
    for j in vars::List M | j ~= i repeat
      for k in vars::List M | k ~= j repeat
        f:=divide(q, i*j*k)
        if f case divisible then
          return monomial(c * m j * γ(i,j) * γ(j,k) / γ(i,k), _
          --return monomial(c * γ(i,j) * γ(j,k) / γ(i,k), _
                     elt(f,lm) * i * k * elt(f,rm))
  return p
Function declaration reduct : FM(FRAC(SMP(INT,OVAR([x1,x2,x3,x4,mp, mq,γqp]))),FMONOID(OVAR([p,q]))) -> FM(FRAC(SMP(INT,OVAR([x1,x2, x3,x4,mp,mq,γqp]))),FMONOID(OVAR([p,q]))) has been added to workspace. reduct(basis(1)*basis(2)*basis(1))
fricas
Compiling function reduct with type FM(FRAC(SMP(INT,OVAR([x1,x2,x3,
      x4,mp,mq,γqp]))),FMONOID(OVAR([p,q]))) -> FM(FRAC(SMP(INT,OVAR([
      x1,x2,x3,x4,mp,mq,γqp]))),FMONOID(OVAR([p,q])))

\label{eq12}{{�� qp}^{2}}\  mq \ {{p}^{2}}(12)

An endomorphism on the K-Module is defined by the fixed point of applied rules

fricas
Y(p:MK):MK ==
  repeat
    r := p; p := idem reduct r
    if r=p then return p
Function declaration Y : FM(FRAC(SMP(INT,OVAR([x1,x2,x3,x4,mp,mq,γqp ]))),FMONOID(OVAR([p,q]))) -> FM(FRAC(SMP(INT,OVAR([x1,x2,x3,x4, mp,mq,γqp]))),FMONOID(OVAR([p,q]))) has been added to workspace. Y(basis(1)*basis(2))
fricas
Compiling function Y with type FM(FRAC(SMP(INT,OVAR([x1,x2,x3,x4,mp,
      mq,γqp]))),FMONOID(OVAR([p,q]))) -> FM(FRAC(SMP(INT,OVAR([x1,x2,
      x3,x4,mp,mq,γqp]))),FMONOID(OVAR([p,q])))

\label{eq13}p \  q(13)

Matrix

Algebra is the free algebra product modulo the fixed point

fricas
MT := [[Y(i*j) for j in basis] for i in basis]; matrix MT

\label{eq14}\left[ 
\begin{array}{cccc}
{mp \  p}&{p \  q}&{mp \  p \  q}&{{{�� qp}^{2}}\  mq \  mp \  p}
\
{q \  p}&{mq \  q}&{{{�� qp}^{2}}\  mq \  mp \  q}&{mq \  q \  p}
\
{{{�� qp}^{2}}\  mq \  mp \  p}&{mq \  p \  q}&{{{�� qp}^{2}}\  mq \  mp \  p \  q}&{{{�� qp}^{2}}\ {{mq}^{2}}\  mp \  p}
\
{mp \  q \  p}&{{{�� qp}^{2}}\  mq \  mp \  q}&{{{�� qp}^{2}}\  mq \ {{mp}^{2}}\  q}&{{{�� qp}^{2}}\  mq \  mp \  q \  p}
(14)

Structure Constants

fricas
mat3(y:M):List List K == map(z+->map(x+->coefficient(x,y),z),MT)
Function declaration mat3 : FMONOID(OVAR([p,q])) -> LIST(LIST(FRAC( SMP(INT,OVAR([x1,x2,x3,x4,mp,mq,γqp]))))) has been added to workspace. ss:=map(mat3, basis); map(matrix,ss)
fricas
Compiling function mat3 with type FMONOID(OVAR([p,q])) -> LIST(LIST(
      FRAC(SMP(INT,OVAR([x1,x2,x3,x4,mp,mq,γqp])))))

\label{eq15}\begin{array}{@{}l}
\displaystyle
\left[{\left[ 
\begin{array}{cccc}
mp & 0 & 0 &{{{�� qp}^{2}}\  mq \  mp}
\
0 & 0 & 0 & 0 
\
{{{�� qp}^{2}}\  mq \  mp}& 0 & 0 &{{{�� qp}^{2}}\ {{mq}^{2}}\  mp}
\
0 & 0 & 0 & 0 
(15)

Algebra

fricas
cats(m:M):Symbol==concat(map(x+->string(x.gen::Symbol),factors m))::Symbol
Function declaration cats : FMONOID(OVAR([p,q])) -> SYMBOL has been added to workspace. A:=AlgebraGivenByStructuralConstants(K,#(basis)::PI,map(cats,basis),ss::Vector(Matrix K))
fricas
Compiling function cats with type FMONOID(OVAR([p,q])) -> SYMBOL

\label{eq16}\hbox{\axiomType{ALGSC}\ } (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{SMP}\ } (\hbox{\axiomType{INT}\ } , \hbox{\axiomType{OVAR}\ } ([ x 1, x 2, x 3, x 4, mp , mq , �� qp ]))) , 4, [ p , q , pq , qp ] , [ [ [ mp , 0, 0, �� qp^2 <em> mq </em> mp ] , [ 0, 0, 0, 0 ] , [ �� qp^2 <em> mq </em> mp , 0, 0, �� qp^2 <em> mq^2 </em> mp ] , [ 0, 0, 0, 0 ] ] , [ [ 0, 0, 0, 0 ] , [ 0, mq , �� qp^2 <em> mq </em> mp , 0 ] , [ 0, 0, 0, 0 ] , [ 0, �� qp^2 <em> mq </em> mp , �� qp^2 <em> mq </em> mp^2, 0 ] ] , [ [ 0, 1, mp , 0 ] , [ 0, 0, 0, 0 ] , [ 0, mq , �� qp^2 <em> mq </em> mp , 0 ] , [ 0, 0, 0, 0 ] ] , [ [ 0, 0, 0, 0 ] , [ 1, 0, 0, mq ] , [ 0, 0, 0, 0 ] , [ mp , 0, 0, �� qp^2 <em> mq </em> mp ] ] ])(16)
fricas
alternative?()$A
algebra satisfies 2*associator(a,b,b) = 0 = 2*associator(a,a,b) = 0

\label{eq17} \mbox{\rm true} (17)
fricas
antiAssociative?()$A
algebra is not anti-associative

\label{eq18} \mbox{\rm false} (18)
fricas
antiCommutative?()$A
algebra is not anti-commutative

\label{eq19} \mbox{\rm false} (19)
fricas
associative?()$A
algebra is associative

\label{eq20} \mbox{\rm true} (20)
fricas
commutative?()$A
algebra is not commutative

\label{eq21} \mbox{\rm false} (21)
fricas
flexible?()$A
algebra is flexible

\label{eq22} \mbox{\rm true} (22)
fricas
jacobiIdentity?()$A
Jacobi identity does not hold

\label{eq23} \mbox{\rm false} (23)
fricas
jordanAdmissible?()$A
algebra is not Jordan admissible

\label{eq24} \mbox{\rm false} (24)
fricas
jordanAlgebra?()$A
algebra is not commutative this is not a Jordan algebra

\label{eq25} \mbox{\rm false} (25)
fricas
leftAlternative?()$A
algebra is left alternative

\label{eq26} \mbox{\rm true} (26)
fricas
lieAdmissible?()$A
algebra is Lie admissible

\label{eq27} \mbox{\rm true} (27)
fricas
lieAlgebra?()$A
algebra is not anti-commutative this is not a Lie algebra

\label{eq28} \mbox{\rm false} (28)
fricas
--powerAssociative?()$A
rightAlternative?()$A
algebra is right alternative

\label{eq29} \mbox{\rm true} (29)

Check Multiplication

fricas
AB := entries basis()$A

\label{eq30}\left[ p , \: q , \: pq , \: qp \right](30)
fricas
A2MK(z:A):MK==reduce(+,map((x:K,y:M):MK+->(x::K)*y,coordinates(z),basis))
Function declaration A2MK : ALGSC(FRAC(SMP(INT,OVAR([x1,x2,x3,x4,mp, mq,γqp]))),4,[p,q,pq,qp],[[[mp,0,0,γqp^2*mq*mp],[0,0,0,0],[γqp^2* mq*mp,0,0,γqp^2*mq^2*mp],[0,0,0,0]],[[0,0,0,0],[0,mq,γqp^2*mq*mp, 0],[0,0,0,0],[0,γqp^2*mq*mp,γqp^2*mq*mp^2,0]],[[0,1,mp,0],[0,0,0, 0],[0,mq,γqp^2*mq*mp,0],[0,0,0,0]],[[0,0,0,0],[1,0,0,mq],[0,0,0,0 ],[mp,0,0,γqp^2*mq*mp]]]) -> FM(FRAC(SMP(INT,OVAR([x1,x2,x3,x4,mp ,mq,γqp]))),FMONOID(OVAR([p,q]))) has been added to workspace. 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(SMP(INT,OVAR([x1,x2,x3,
      x4,mp,mq,γqp]))),4,[p,q,pq,qp],[[[mp,0,0,γqp^2*mq*mp],[0,0,0,0],[
      γqp^2*mq*mp,0,0,γqp^2*mq^2*mp],[0,0,0,0]],[[0,0,0,0],[0,mq,γqp^2*
      mq*mp,0],[0,0,0,0],[0,γqp^2*mq*mp,γqp^2*mq*mp^2,0]],[[0,1,mp,0],[
      0,0,0,0],[0,mq,γqp^2*mq*mp,0],[0,0,0,0]],[[0,0,0,0],[1,0,0,mq],[0
      ,0,0,0],[mp,0,0,γqp^2*mq*mp]]]) -> FM(FRAC(SMP(INT,OVAR([x1,x2,x3
      ,x4,mp,mq,γqp]))),FMONOID(OVAR([p,q])))

\label{eq31} \mbox{\rm true} (31)

Trace

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

\label{eq32}\left[{2 \  mp}, \:{2 \  mq}, \:{2 \ {{�� qp}^{2}}\  mq \  mp}, \:{2 \ {{�� qp}^{2}}\  mq \  mp}\right](32)
fricas
[leftTrace(i)$A for i in AB]

\label{eq33}\left[{2 \  mp}, \:{2 \  mq}, \:{2 \ {{�� qp}^{2}}\  mq \  mp}, \:{2 \ {{�� qp}^{2}}\  mq \  mp}\right](33)
fricas
trace(i)==rightTrace(i) / #vars
[trace(i) for i in AB]
fricas
Compiling function trace with type ALGSC(FRAC(SMP(INT,OVAR([x1,x2,x3
      ,x4,mp,mq,γqp]))),4,[p,q,pq,qp],[[[mp,0,0,γqp^2*mq*mp],[0,0,0,0],
      [γqp^2*mq*mp,0,0,γqp^2*mq^2*mp],[0,0,0,0]],[[0,0,0,0],[0,mq,γqp^2
      *mq*mp,0],[0,0,0,0],[0,γqp^2*mq*mp,γqp^2*mq*mp^2,0]],[[0,1,mp,0],
      [0,0,0,0],[0,mq,γqp^2*mq*mp,0],[0,0,0,0]],[[0,0,0,0],[1,0,0,mq],[
      0,0,0,0],[mp,0,0,γqp^2*mq*mp]]]) -> FRAC(SMP(INT,OVAR([x1,x2,x3,
      x4,mp,mq,γqp])))

\label{eq34}\left[ mp , \: mq , \:{{{�� qp}^{2}}\  mq \  mp}, \:{{{�� qp}^{2}}\  mq \  mp}\right](34)

fricas
p:=AB(1); q:=AB(2);
test(p*p=trace(p)*p)

\label{eq35} \mbox{\rm true} (35)
fricas
test(q*q=trace(q)*q)

\label{eq36} \mbox{\rm true} (36)

Center

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

\label{eq37}1(37)
fricas
c:=C(1)

\label{eq38}qp + pq -{mp \  q}-{mq \  p}(38)
fricas
[c*i-i*c for i in AB]

\label{eq39}\left[ 0, \: 0, \: 0, \: 0 \right](39)
fricas
c*c

\label{eq40}\begin{array}{@{}l}
\displaystyle
{{\left({{�� qp}^{2}}- 1 \right)}\  mq \  mp \  qp}+{{\left({{�� qp}^{2}}- 1 \right)}\  mq \  mp \  pq}+ 
\
\
\displaystyle
{{\left(-{{�� qp}^{2}}+ 1 \right)}\  mq \ {{mp}^{2}}\  q}+{{\left(-{{�� qp}^{2}}+ 1 \right)}\ {{mq}^{2}}\  mp \  p}
(40)
fricas
test(c*c=c)

\label{eq41} \mbox{\rm false} (41)

Unit

fricas
n := #vars/trace(c) * c

\label{eq42}\begin{array}{@{}l}
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq \  mp}}\  qp}+{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq \  mp}}\  pq}- 
\
\
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq}}\  q}-{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mp}}\  p}
(42)
fricas
test(n = unit()$A)

\label{eq43} \mbox{\rm true} (43)
fricas
trace(n)

\label{eq44}2(44)
fricas
test(n*n=n)

\label{eq45} \mbox{\rm true} (45)
fricas
f:=gcd map(x+->denom x,coordinates(n))

\label{eq46}{{�� qp}^{2}}- 1(46)
fricas
--Silberstein symmetric matrix
ff:= matrix [[(i=j => 1$K; γ(i,j)) for j in vars] for i in vars]

\label{eq47}\left[ 
\begin{array}{cc}
1 & �� qp 
\
�� qp & 1 
(47)
fricas
test(f = - determinant(ff))

\label{eq48} \mbox{\rm true} (48)
fricas
(f*n)::OutputForm / f::OutputForm

\label{eq49}{{{1 \over{mq \  mp}}\  qp}+{{1 \over{mq \  mp}}\  pq}-{{1 \over mq}\  q}-{{1 \over mp}\  p}}\over{{{�� qp}^{2}}- 1}(49)

Orthogonal Observers

fricas
dual(p) == trace(p)*n - p
--dual(p) == n - (1/trace(p))*p p' := dual p
fricas
Compiling function dual with type ALGSC(FRAC(SMP(INT,OVAR([x1,x2,x3,
      x4,mp,mq,γqp]))),4,[p,q,pq,qp],[[[mp,0,0,γqp^2*mq*mp],[0,0,0,0],[
      γqp^2*mq*mp,0,0,γqp^2*mq^2*mp],[0,0,0,0]],[[0,0,0,0],[0,mq,γqp^2*
      mq*mp,0],[0,0,0,0],[0,γqp^2*mq*mp,γqp^2*mq*mp^2,0]],[[0,1,mp,0],[
      0,0,0,0],[0,mq,γqp^2*mq*mp,0],[0,0,0,0]],[[0,0,0,0],[1,0,0,mq],[0
      ,0,0,0],[mp,0,0,γqp^2*mq*mp]]]) -> ALGSC(FRAC(SMP(INT,OVAR([x1,x2
      ,x3,x4,mp,mq,γqp]))),4,[p,q,pq,qp],[[[mp,0,0,γqp^2*mq*mp],[0,0,0,
      0],[γqp^2*mq*mp,0,0,γqp^2*mq^2*mp],[0,0,0,0]],[[0,0,0,0],[0,mq,
      γqp^2*mq*mp,0],[0,0,0,0],[0,γqp^2*mq*mp,γqp^2*mq*mp^2,0]],[[0,1,
      mp,0],[0,0,0,0],[0,mq,γqp^2*mq*mp,0],[0,0,0,0]],[[0,0,0,0],[1,0,0
      ,mq],[0,0,0,0],[mp,0,0,γqp^2*mq*mp]]])

\label{eq50}\begin{array}{@{}l}
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq}}\  qp}+{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq}}\  pq}-{{mp \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq}}\  q}- 
\
\
\displaystyle
{{{{�� qp}^{2}}\over{{{�� qp}^{2}}- 1}}\  p}
(50)
fricas
trace p'

\label{eq51}mp(51)
fricas
p'' := dual p'

\label{eq52}p(52)
fricas
trace p''

\label{eq53}mp(53)
fricas
test(p' * p' = trace(p')*p')

\label{eq54} \mbox{\rm true} (54)
fricas
p * p'

\label{eq55}0(55)
fricas
p' * p

\label{eq56}0(56)
fricas
q' := dual q

\label{eq57}\begin{array}{@{}l}
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mp}}\  qp}+{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mp}}\  pq}-{{{{�� qp}^{2}}\over{{{�� qp}^{2}}- 1}}\  q}- 
\
\
\displaystyle
{{mq \over{{\left({{�� qp}^{2}}- 1 \right)}\  mp}}\  p}
(57)
fricas
trace(q')

\label{eq58}mq(58)
fricas
test(q' * q' = trace(q')*q')

\label{eq59} \mbox{\rm true} (59)
fricas
q * q'

\label{eq60}0(60)
fricas
q' * q

\label{eq61}0(61)
fricas
p' * q'

\label{eq62}{{1 \over{{{�� qp}^{2}}- 1}}\  qp}+{{{{�� qp}^{2}}\over{{{�� qp}^{2}}- 1}}\  pq}-{{{{{�� qp}^{2}}\  mp}\over{{{�� qp}^{2}}- 1}}\  q}-{{{{{�� qp}^{2}}\  mq}\over{{{�� qp}^{2}}- 1}}\  p}(62)
fricas
q' * p'

\label{eq63}{{{{�� qp}^{2}}\over{{{�� qp}^{2}}- 1}}\  qp}+{{1 \over{{{�� qp}^{2}}- 1}}\  pq}-{{{{{�� qp}^{2}}\  mp}\over{{{�� qp}^{2}}- 1}}\  q}-{{{{{�� qp}^{2}}\  mq}\over{{{�� qp}^{2}}- 1}}\  p}(63)
fricas
p' * q

\label{eq64}- pq +{mp \  q}(64)
fricas
q * p'

\label{eq65}- qp +{mp \  q}(65)
fricas
p * q'

\label{eq66}- pq +{mq \  p}(66)
fricas
q' * p

\label{eq67}- qp +{mq \  p}(67)

Orthogonal Observers are Derivations if there are only two observers

fricas
test(p'*(p*q) = (p'*p)*q + p*(p'*q))

\label{eq68} \mbox{\rm true} (68)
fricas
test(q'*(p*q) = (q'*p)*q + p*(q'*q))

\label{eq69} \mbox{\rm true} (69)
fricas
test((p*q)*p' = (p*p')*q + p*(q*p'))

\label{eq70} \mbox{\rm true} (70)
fricas
test((p*q)*q' = (p*q')*q + p*(q*q'))

\label{eq71} \mbox{\rm true} (71)

Momentum

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

\label{eq72}{{1 \over �� qp}\  qp}+{{1 \over �� qp}\  pq}+{mq \  q}+{mp \  p}(72)
fricas
trace(P)

\label{eq73}{{mp}^{2}}+{2 \  �� qp \  mq \  mp}+{{mq}^{2}}(73)
fricas
u:=1/trace(P)*P

\label{eq74}\begin{array}{@{}l}
\displaystyle
{{1 \over{{�� qp \ {{mp}^{2}}}+{2 \ {{�� qp}^{2}}\  mq \  mp}+{�� qp \ {{mq}^{2}}}}}\  qp}+ 
\
\
\displaystyle
{{1 \over{{�� qp \ {{mp}^{2}}}+{2 \ {{�� qp}^{2}}\  mq \  mp}+{�� qp \ {{mq}^{2}}}}}\  pq}+ 
\
\
\displaystyle
{{mq \over{{{mp}^{2}}+{2 \  �� qp \  mq \  mp}+{{mq}^{2}}}}\  q}+{{mp \over{{{mp}^{2}}+{2 \  �� qp \  mq \  mp}+{{mq}^{2}}}}\  p}
(74)
fricas
u*u-u

\label{eq75}0(75)
fricas
trace(u)

\label{eq76}1(76)

All idempotents

fricas
x:=x1*p+x2*q+x3*p*q+x4*q*p
Compiled code for A2MK has been cleared. Compiled code for cats has been cleared. Compiled code for mat3 has been cleared.

\label{eq77}{x 4 \  qp}+{x 3 \  pq}+{x 2 \  q}+{x 1 \  p}(77)
fricas
ideq:=conditionsForIdempotents()$GCNAALG(K,#(basis)::PI,map(cats,basis),ss::Vector(Matrix K))
fricas
Compiling function cats with type FMONOID(OVAR([p,q])) -> SYMBOL

\label{eq78}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
{{\left({{{�� qp}^{2}}\ {{mq}^{2}}\  mp \  \%x 3}+{{{�� qp}^{2}}\  mq \  mp \  \%x 1}\right)}\  \%x 4}+ 
\
\
\displaystyle
{{{�� qp}^{2}}\  mq \  mp \  \%x 1 \  \%x 3}+{mp \ {{\%x 1}^{2}}}- \%x 1 
(78)
fricas
gbs:=groebnerFactorize ideq;
#gbs

\label{eq79}9(79)

fricas
gbs.9

\label{eq80}\begin{array}{@{}l}
\displaystyle
\left[{\%x 4 -{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq \  mp}}}, \:{\%x 3 -{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq \  mp}}}, \: \right.
\
\
\displaystyle
\left.{\%x 2 +{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq}}}, \:{\%x 1 +{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mp}}}\right] (80)
fricas
s9:=solve(gbs.9);
i9:=represents(reverse map(rhs,s9.1))$A

\label{eq81}\begin{array}{@{}l}
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq \  mp}}\  qp}+{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq \  mp}}\  pq}- 
\
\
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq}}\  q}-{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mp}}\  p}
(81)
fricas
test(i9=n)

\label{eq82} \mbox{\rm true} (82)

fricas
gbs.8

\label{eq83}\begin{array}{@{}l}
\displaystyle
\left[{\%x 4 -{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq \  mp}}}, \:{\%x 3 -{1 \over{{\left({{�� qp}^{4}}-{{�� qp}^{2}}\right)}\  mq \  mp}}}, \: \right.
\
\
\displaystyle
\left.{\%x 2 +{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq}}}, \:{\%x 1 +{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mp}}}\right] (83)
fricas
s8:=solve(gbs.8);
i8:=represents(reverse map(rhs,s8.1))$A

\label{eq84}\begin{array}{@{}l}
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq \  mp}}\  qp}+{{1 \over{{\left({{�� qp}^{4}}-{{�� qp}^{2}}\right)}\  mq \  mp}}\  pq}- 
\
\
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq}}\  q}-{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mp}}\  p}
(84)
fricas
test(i8=n-1/trace(p*q)*p*q)

\label{eq85} \mbox{\rm true} (85)

fricas
gbs.7

\label{eq86}\begin{array}{@{}l}
\displaystyle
\left[{\%x 4 -{1 \over{{\left({{�� qp}^{4}}-{{�� qp}^{2}}\right)}\  mq \  mp}}}, \:{\%x 3 -{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq \  mp}}}, \: \right.
\
\
\displaystyle
\left.{\%x 2 +{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq}}}, \:{\%x 1 +{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mp}}}\right] (86)
fricas
s7:=solve(gbs.7);
i7:=represents(reverse map(rhs,s7.1))$A

\label{eq87}\begin{array}{@{}l}
\displaystyle
{{1 \over{{\left({{�� qp}^{4}}-{{�� qp}^{2}}\right)}\  mq \  mp}}\  qp}+{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq \  mp}}\  pq}- 
\
\
\displaystyle
{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mq}}\  q}-{{1 \over{{\left({{�� qp}^{2}}- 1 \right)}\  mp}}\  p}
(87)
fricas
test(i7=n-1/trace(q*p)*q*p)

\label{eq88} \mbox{\rm true} (88)

fricas
gbs.6

\label{eq89}\left[{\%x 4 -{1 \over{{{�� qp}^{2}}\  mq \  mp}}}, \: \%x 3, \: \%x 2, \: \%x 1 \right](89)
fricas
s6:=solve(gbs.6);
i6:=represents(reverse map(rhs,s6.1))$A

\label{eq90}{1 \over{{{�� qp}^{2}}\  mq \  mp}}\  qp(90)
fricas
test(i6=1/trace(q*p)*q*p)

\label{eq91} \mbox{\rm true} (91)

fricas
gbs.5

\label{eq92}\left[ 1 \right](92)
fricas
s5:=solve(gbs.5)

\label{eq93}\left[ \right](93)

fricas
gbs.4

\label{eq94}\left[ \%x 4, \: \%x 3, \: \%x 2, \: \%x 1 \right](94)
fricas
s4:=solve(gbs.4);
i4:=represents(reverse map(rhs,s4.1))$A

\label{eq95}0(95)

fricas
gbs.3

\label{eq96}\left[ \%x 4, \:{\%x 3 -{1 \over{{{�� qp}^{2}}\  mq \  mp}}}, \: \%x 2, \: \%x 1 \right](96)
fricas
s3:=solve(gbs.3);
i3:=represents(reverse map(rhs,s3.1))$A

\label{eq97}{1 \over{{{�� qp}^{2}}\  mq \  mp}}\  pq(97)
fricas
test(i3=1/trace(p*q)*p*q)

\label{eq98} \mbox{\rm true} (98)

fricas
gbs.2

\label{eq99}\begin{array}{@{}l}
\displaystyle
\left[{\%x 4 + \%x 3 +{{2 \over{{{�� qp}^{2}}\  mq}}\  \%x 1}-{1 \over{{{�� qp}^{2}}\  mq \  mp}}}, \: \right.
\
\
\displaystyle
\left.{{{\%x 3}^{2}}+{{\left({{2 \over{{{�� qp}^{2}}\  mq}}\  \%x 1}-{1 \over{{{�� qp}^{2}}\  mq \  mp}}\right)}\  \%x 3}+{{1 \over{{{�� qp}^{2}}\ {{mq}^{2}}}}\ {{\%x 1}^{2}}}}, \right.
\
\
\displaystyle
\left.\:{\%x 2 -{{mp \over mq}\  \%x 1}}\right] 
(99)
fricas
-- apparently we need to look for solutions in a larger ring
ex2:=map(x+->interpret(x::InputForm)$InputFormFunctions1(FRAC POLY INT),concat(gbs.2,[%x3-%x4]));
s2:=solve(ex2,[%x1,%x2,%x3,%x4]);
#s2

\label{eq100}2(100)
fricas
-- need this to convert solution back to K
(mp,mq,γqp):K
i2:=represents(map(x+->interpret(rhs(x)::InputForm)$InputFormFunctions1(K),s2.1))$A

\label{eq101}\begin{array}{@{}l}
\displaystyle
{{1 \over{{\left({2 \ {{�� qp}^{2}}}-{2 \  �� qp}\right)}\  mq \  mp}}\  qp}+{{1 \over{{\left({2 \ {{�� qp}^{2}}}-{2 \  �� qp}\right)}\  mq \  mp}}\  pq}- 
\
\
\displaystyle
{{1 \over{{\left({2 \  �� qp}- 2 \right)}\  mq}}\  q}-{{1 \over{{\left({2 \  �� qp}- 2 \right)}\  mp}}\  p}
(101)
fricas
i2':=represents(map(x+->interpret(rhs(x)::InputForm)$InputFormFunctions1(K),s2.2))$A

\label{eq102}\begin{array}{@{}l}
\displaystyle
{{1 \over{{\left({2 \ {{�� qp}^{2}}}+{2 \  �� qp}\right)}\  mq \  mp}}\  qp}+{{1 \over{{\left({2 \ {{�� qp}^{2}}}+{2 \  �� qp}\right)}\  mq \  mp}}\  pq}+ 
\
\
\displaystyle
{{1 \over{{\left({2 \  �� qp}+ 2 \right)}\  mq}}\  q}+{{1 \over{{\left({2 \  �� qp}+ 2 \right)}\  mp}}\  p}
(102)
fricas
test(n=i2+i2')

\label{eq103} \mbox{\rm true} (103)
fricas
i2*i2'

\label{eq104}0(104)
fricas
i2'*i2

\label{eq105}0(105)
fricas
-- decomposition
i2*x

\label{eq106}\begin{array}{@{}l}
\displaystyle
{{{- x 1 +{�� qp \  mq \  x 4}}\over{2 \  �� qp \  mq}}\  qp}+{{{- x 2 +{�� qp \  mp \  x 3}}\over{2 \  �� qp \  mp}}\  pq}+ 
\
\
\displaystyle
{{{x 2 -{�� qp \  mp \  x 3}}\over 2}\  q}+{{{x 1 -{�� qp \  mq \  x 4}}\over 2}\  p}
(106)
fricas
i2'*x

\label{eq107}\begin{array}{@{}l}
\displaystyle
{{{x 1 +{�� qp \  mq \  x 4}}\over{2 \  �� qp \  mq}}\  qp}+{{{x 2 +{�� qp \  mp \  x 3}}\over{2 \  �� qp \  mp}}\  pq}+{{{x 2 +{�� qp \  mp \  x 3}}\over 2}\  q}+ 
\
\
\displaystyle
{{{x 1 +{�� qp \  mq \  x 4}}\over 2}\  p}
(107)
fricas
test(i2*p+i2'*p=p)

\label{eq108} \mbox{\rm true} (108)
fricas
test(i2*q+i2'*q=q)

\label{eq109} \mbox{\rm true} (109)
fricas
test(i2*(p*q)+i2'*(p*q)=p*q)

\label{eq110} \mbox{\rm true} (110)
fricas
test(i2*(q*p)+i2'*(q*p)=q*p)

\label{eq111} \mbox{\rm true} (111)

fricas
expr2:=map(x+->interpret(x::InputForm)$InputFormFunctions1(EXPR INT)=0,concat(gbs.2,[]));
s2b:=solve(expr2,[%x1,%x2,%x3]);
#s2b

\label{eq112}2(112)
fricas
s2b.1

\label{eq113}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
\%x 1 ={{\left(
\begin{array}{@{}l}
\displaystyle
{\sqrt{-{{16}\ {{\%x 4}^{2}}\ {{mp}^{4}}\ {{mq}^{4}}\ {{�� qp}^{6}}}+{{\left({{16}\ {{\%x 4}^{2}}\ {{mp}^{4}}\ {{mq}^{4}}}+{{1
6}\  \%x 4 \ {{mp}^{3}}\ {{mq}^{3}}}\right)}\ {{�� qp}^{4}}}}}- 
\
\
\displaystyle
{4 \  \%x 4 \ {{mp}^{2}}\ {{mq}^{2}}\ {{�� qp}^{2}}}
(113)
fricas
s2b.2

\label{eq114}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
\%x 1 ={{\left(
\begin{array}{@{}l}
\displaystyle
- 
\
\
\displaystyle
{\sqrt{-{{16}\ {{\%x 4}^{2}}\ {{mp}^{4}}\ {{mq}^{4}}\ {{�� qp}^{6}}}+{{\left({{16}\ {{\%x 4}^{2}}\ {{mp}^{4}}\ {{mq}^{4}}}+{{1
6}\  \%x 4 \ {{mp}^{3}}\ {{mq}^{3}}}\right)}\ {{�� qp}^{4}}}}}- 
\
\
\displaystyle
{4 \  \%x 4 \ {{mp}^{2}}\ {{mq}^{2}}\ {{�� qp}^{2}}}
(114)

fricas
gbs.1

\label{eq115}\begin{array}{@{}l}
\displaystyle
\left[{\%x 4 + \%x 3 +{{1 \over{{{�� qp}^{2}}\  mp}}\  \%x 2}+{{1 \over{{{�� qp}^{2}}\  mq}}\  \%x 1}-{1 \over{{{�� qp}^{2}}\  mq \  mp}}}, \: \right.
\
\
\displaystyle
\left.{
\begin{array}{@{}l}
\displaystyle
{{\%x 3}^{2}}+{{\left({{1 \over{{{�� qp}^{2}}\  mp}}\  \%x 2}+{{1 \over{{{�� qp}^{2}}\  mq}}\  \%x 1}-{1 \over{{{�� qp}^{2}}\  mq \  mp}}\right)}\  \%x 3}+ 
\
\
\displaystyle
{{1 \over{{{�� qp}^{2}}\  mq \  mp}}\  \%x 1 \  \%x 2}
(115)
fricas
s1:=solve(concat(gbs.1,[%x1-m('p)/trace(P),%x2-m('q)/trace(P)]));
i1:=represents(reverse map(rhs,s1.1))$A

\label{eq116}\begin{array}{@{}l}
\displaystyle
{{1 \over{{�� qp \ {{mp}^{2}}}+{2 \ {{�� qp}^{2}}\  mq \  mp}+{�� qp \ {{mq}^{2}}}}}\  qp}+ 
\
\
\displaystyle
{{1 \over{{�� qp \ {{mp}^{2}}}+{2 \ {{�� qp}^{2}}\  mq \  mp}+{�� qp \ {{mq}^{2}}}}}\  pq}+ 
\
\
\displaystyle
{{mq \over{{{mp}^{2}}+{2 \  �� qp \  mq \  mp}+{{mq}^{2}}}}\  q}+{{mp \over{{{mp}^{2}}+{2 \  �� qp \  mq \  mp}+{{mq}^{2}}}}\  p}
(116)
fricas
test(i1=u)

\label{eq117} \mbox{\rm true} (117)
fricas
eval(gbs.1,[x=e for x in [%x1,%x2,%x3,%x4] for e in entries coordinates(u)])

\label{eq118}\left[ 0, \: 0 \right](118)
fricas
eval(gbs.1,[x=e for x in [%x1,%x2,%x3,%x4] for e in entries coordinates(n-u)])

\label{eq119}\left[ 0, \: 0 \right](119)

fricas
)set output tex off
 
fricas
)set output algebra on

fricas
expr1a:=map(x+->interpret(x::InputForm)$InputFormFunctions1(EXPR INT)=0,concat(gbs.1,[]));
solve(expr1a,[%x1,%x2])
(147) [ [ %x1 = ROOT 2 2 2 2 4 (%x4 + 2%x3 %x4 + %x3 )mp mq γqp + 2 2 2 (- 4%x3 %x4 mp mq + (- 2%x4 - 2%x3)mp mq)γqp + 1 + 2 (- %x4 - %x3)mp mq γqp + 1 / 2mp ,
%x2 = - ROOT 2 2 2 2 4 (%x4 + 2%x3 %x4 + %x3 )mp mq γqp + 2 2 2 (- 4%x3 %x4 mp mq + (- 2%x4 - 2%x3)mp mq)γqp + 1 + 2 (- %x4 - %x3)mp mq γqp + 1 / 2mq ] ,
[ %x1 = - ROOT 2 2 2 2 4 (%x4 + 2%x3 %x4 + %x3 )mp mq γqp + 2 2 2 (- 4%x3 %x4 mp mq + (- 2%x4 - 2%x3)mp mq)γqp + 1 + 2 (- %x4 - %x3)mp mq γqp + 1 / 2mp ,
%x2 = ROOT 2 2 2 2 4 (%x4 + 2%x3 %x4 + %x3 )mp mq γqp + 2 2 2 (- 4%x3 %x4 mp mq + (- 2%x4 - 2%x3)mp mq)γqp + 1 + 2 (- %x4 - %x3)mp mq γqp + 1 / 2mq ] ]
expr1b:=map(x+->interpret(x::InputForm)$InputFormFunctions1(EXPR INT)=0,concat(gbs.1,[%x3-%x4]));
solve(expr1b,[%x1,%x2,%x3])
(149) [ [ %x1 = +------------------------------------------------------+ | 2 2 2 4 2 2 2 2 - \|4%x4 mp mq γqp + (- 4%x4 mp mq - 4%x4 mp mq)γqp + 1 + 2 - 2%x4 mp mq γqp + 1 / 2mp ,
%x2 = +------------------------------------------------------+ | 2 2 2 4 2 2 2 2 \|4%x4 mp mq γqp + (- 4%x4 mp mq - 4%x4 mp mq)γqp + 1 + 2 - 2%x4 mp mq γqp + 1 / 2mq , %x3 = %x4] ,
[ %x1 = +------------------------------------------------------+ | 2 2 2 4 2 2 2 2 \|4%x4 mp mq γqp + (- 4%x4 mp mq - 4%x4 mp mq)γqp + 1 + 2 - 2%x4 mp mq γqp + 1 / 2mp ,
%x2 = +------------------------------------------------------+ | 2 2 2 4 2 2 2 2 - \|4%x4 mp mq γqp + (- 4%x4 mp mq - 4%x4 mp mq)γqp + 1 + 2 - 2%x4 mp mq γqp + 1 / 2mq , %x3 = %x4] ]




  Subject:   Be Bold !!
  ( 13 subscribers )  
Please rate this page: