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last edited 1 year ago by Bill Page |
Edit detail for The Algebra of Complex Numbers Is Frobenius In Many Ways revision 1 of 11
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Editor: Bill Page
Time: 2011/04/27 23:40:10 GMT-7 |
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| Note: three-point function | ||
changed: - The Algebra of Complex Numbers Is Frobenius In Many Ways Linear operators over a 2-dimensional vector space representing the algebra of complex numbers Ref: - http://arxiv.org/abs/1103.5113 $S_3$-permuted Frobenius Algebras *Zbigniew Oziewicz (UNAM), Gregory Peter Wene (UTSA)* - http://mat.uab.es/~kock/TQFT.html Frobenius algebras and 2D topological quantum field theories *Joachim Kock* - http://en.wikipedia.org/wiki/Frobenius_algebra We need the Axiom LinearOperator library. \begin{axiom} )library CARTEN MONAL PROP LIN CALEY \end{axiom} Use the following macros for convenient notation \begin{axiom} -- summation macro Σ(x,i,n)==reduce(+,[x for i in n]) -- list macro Ξ(f,i,n)==[f for i in n] -- subscript macro sb == subscript \end{axiom} 𝐋 is the domain of 2-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients. \begin{axiom} dim:=2 macro ℒ == List macro ℂ == CaleyDickson macro ℚ == Expression Integer 𝐋 := LinearOperator(dim, OVAR [], ℚ) 𝐞:ℒ 𝐋 := basisVectors() 𝐝:ℒ 𝐋 := basisForms() o:𝐋:=1 -- identity for product I:𝐋:=[1] -- identity for composition X:𝐋:=[2,1] -- twist \end{axiom} Now generate structure constants for Quaternion Algebra The basis consists of the real and imaginary units. We use quaternion multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors. Split-complex can be specified by Caley-Dickson parameter (q0 = -1) \begin{axiom} q0:=sb('q,[0]) --q0:=1 -- split-complex QQ := ℂ(ℚ,'i,q0); \end{axiom} Basis: Each B.i is a quaternion number \begin{axiom} B:ℒ QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::ℒ ℒ ℚ) -- Multiplication table: M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim) -- Function to divide the matrix entries by a basis element S(y) == map(x +-> real(x/y),M) -- The result is a nested list ѕ :=map(S,B)::ℒ ℒ ℒ ℚ -- structure constants form a tensor operator Y := Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*𝐝.k, i,1..dim), j,1..dim), k,1..dim) arity Y matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim) \end{axiom} Multiplication of arbitrary quaternions $a$ and $b$ \begin{axiom} a:=Σ(sb('a,[i])*𝐞.i, i,1..dim) b:=Σ(sb('b,[i])*𝐞.i, i,1..dim) (a*b)/Y \end{axiom} A scalar product is denoted by the (2,0)-tensor $U = \{ u_{ij} \}$ \begin{axiom} U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim) \end{axiom} Definition 1 We say that the scalar product is *associative* if the tensor equation holds:: Y = Y U U In other words, if the (3,0)-tensor: $$ \scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-0.92)(4.82,0.92) \psbezier[linewidth=0.04](2.2,0.9)(2.2,0.1)(2.6,0.1)(2.6,0.9) \psline[linewidth=0.04cm](2.4,0.3)(2.4,-0.1) \psbezier[linewidth=0.04](2.4,-0.1)(2.4,-0.9)(3.0,-0.9)(3.0,-0.1) \psline[linewidth=0.04cm](3.0,-0.1)(3.0,0.9) \psbezier[linewidth=0.04](4.8,0.9)(4.8,0.1)(4.4,0.1)(4.4,0.9) \psline[linewidth=0.04cm](4.6,0.3)(4.6,-0.1) \psbezier[linewidth=0.04](4.6,-0.1)(4.6,-0.9)(4.0,-0.9)(4.0,-0.1) \psline[linewidth=0.04cm](4.0,-0.1)(4.0,0.9) \usefont{T1}{ptm}{m}{n} \rput(3.4948437,0.205){-} \psline[linewidth=0.04cm](0.6,-0.7)(0.6,0.9) \psbezier[linewidth=0.04](0.0,-0.1)(0.0,-0.9)(1.2,-0.9)(1.2,-0.1) \psline[linewidth=0.04cm](0.0,-0.1)(0.0,0.9) \psline[linewidth=0.04cm](1.2,-0.1)(1.2,0.9) \usefont{T1}{ptm}{m}{n} \rput(1.6948438,0.205){=} \end{pspicture} } $$ \begin{equation} \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \} \end{equation} (three-point function) is zero. Using the LinearOperator domain in Axiom and some carefully chosen symbols we can easily enter expressions that are both readable and interpreted by Axiom as "graphical calculus" diagrams describing complex products and compositions of linear operators. \begin{axiom} ω:𝐋 := o Y I / o U - o I Y / o U o \end{axiom} Note: The only purpose of the o symbols on the left above is to serve as a constant left-side margin as required by Axiom. The symbols on the right describe the relation between row. Definition 2 An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]. We may consider the problem where multiplication Y is given, and look for all associative scalar products $U = U(Y)$ This problem can be solved using linear algebra. \begin{axiom} )expose MCALCFN J := jacobian(ravel ω,concat map(variables,ravel U)::ℒ Symbol); u := transpose matrix [concat map(variables,ravel U)::ℒ Symbol]; J::OutputForm * u::OutputForm = 0 nrows(J),ncols(J) \end{axiom} The matrix 'J' transforms the coefficients of the tensor $U$ into coefficients of the tensor $\Phi$. We are looking for the general linear family of tensors $U=U(Y,p_i)$ such that 'J' transforms $U$ into $\Phi=0$ for any such $U$. If the null space of the 'J' matrix is not empty we can use the basis to find all non-trivial solutions for U: \begin{axiom} Ñ:=nullSpace(J) ℰ:=map((x,y)+->x=y, concat map(variables,ravel U), entries Σ(sb('p,[i])*Ñ.i, i,1..#Ñ) ) \end{axiom} This defines a family of pre-Frobenius algebras: \begin{axiom} zero? eval(ω,ℰ) \end{axiom} In two dimensions the pairing is necessarily symmetric! \begin{axiom} Ų:𝐋 := eval(U,ℰ) matrix Ξ(Ξ((𝐞.i 𝐞.j)/Ų, i,1..dim), j,1..dim) \end{axiom} This is the most general form of the "dot product" of two complex numbers \begin{axiom} (a*b)/Ų (a*a)/Ų \end{axiom} The scalar product must be non-degenerate: \begin{axiom} Ů:=determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ų), j,1..dim), i,1..dim) factor Ů \end{axiom} Definition 3 Co-pairing Solve the [Snake Relation] as a system of linear equations. \begin{axiom} Ω:𝐋:=Σ(Σ(script('u,[[i,j]])*𝐞.i*𝐞.j, i,1..dim), j,1..dim) d1:=(I*Ω)/(Ų*I); d2:=(Ω*I)/(I*Ų); equate(f,g)==map((x,y)+->(x=y),ravel f, ravel g); eq1:=equate(d1,I); eq2:=equate(d2,I); snake:=solve(concat(eq1,eq2),concat Ξ(Ξ(script('u,[[i,j]]), i,1..dim), j,1..dim)); if #snake ~= 1 then error "no solution" Ω:=eval(Ω,snake(1)) matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j), i,1..dim), j,1..dim) \end{axiom} Check "dimension" and the snake relations. \begin{axiom} d:𝐋:= o Ω / o X / o Ų o test ( I Ω ) / ( Ų I ) = I test ( Ω I ) / ( I Ų ) = I \end{axiom} Definition 4 Co-algebra Compute the "three-point" function and use it to define co-multiplication. \begin{axiom} W:=(Y,I)/Ų λ:=(Ω,I,Ω)/(I,W,I) \end{axiom} \begin{axiom} test ( I Ω ) / ( Y I ) = λ test ( Ω I ) / ( I Y ) = λ \end{axiom} Frobenius Condition \begin{axiom} Χ := Y / λ test ( λ I ) / ( I Y ) = Χ test ( I λ ) / ( Y I ) = Χ \end{axiom} i = Unit of the algebra \begin{axiom} i:=𝐞.1 test o i / o λ = Ω \end{axiom} Handle \begin{axiom} H:𝐋 := λ / X / Y \end{axiom} Definition 5 <center>Co-unit<pre> i U </pre></center> \begin{axiom} ι:𝐋:= o i I / o Ų o \end{axiom} <center><pre> Y=U ι </pre></center> \begin{axiom} test o Y / o ι o = Ų \end{axiom} For example: \begin{axiom} ex1:=[q[0]=1,p[1]=0,p[2]=1] Ų0:𝐋 :=eval(Ų,ex1) Ω0:𝐋 :=eval(Ω,ex1)$𝐋 λ0:𝐋 :=eval(λ,ex1)$𝐋 H0:𝐋 :=eval(H,ex1)$𝐋 \end{axiom}
The Algebra of Complex Numbers Is Frobenius In Many Ways
Linear operators over a 2-dimensional vector space representing the algebra of complex numbers
Ref:
- http://arxiv.org/abs/1103.5113
-permuted Frobenius AlgebrasZbigniew Oziewicz (UNAM), Gregory Peter Wene (UTSA)
- http://mat.uab.es/~kock/TQFT.html
Frobenius algebras and 2D topological quantum field theories
Joachim Kock
- http://en.wikipedia.org/wiki/Frobenius_algebra
We need the Axiom LinearOperator? library.
axiom
)library CARTEN MONAL PROP LIN CALEY
CartesianTensor is now explicitly exposed in frame initial CartesianTensor will be automatically loaded when needed from /var/zope2/var/LatexWiki/CARTEN.NRLIB/CARTEN Monoidal is now explicitly exposed in frame initial Monoidal will be automatically loaded when needed from /var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL Prop is now explicitly exposed in frame initial Prop will be automatically loaded when needed from /var/zope2/var/LatexWiki/PROP.NRLIB/PROP LinearOperator is now explicitly exposed in frame initial LinearOperator will be automatically loaded when needed from /var/zope2/var/LatexWiki/LIN.NRLIB/LIN CaleyDickson is now explicitly exposed in frame initial CaleyDickson will be automatically loaded when needed from /var/zope2/var/LatexWiki/CALEY.NRLIB/CALEY
Use the following macros for convenient notation
axiom
-- summation macro Σ(x,i, n)==reduce(+, [x for i in n])
Type: Void
axiom
-- list macro Ξ(f,i, n)==[f for i in n]
Type: Void
axiom
-- subscript macro sb == subscript
Type: Void
𝐋 is the domain of 2-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.
axiom
dim:=2
 | (1) |
Type: PositiveInteger?
axiom
macro ℒ == List
Type: Void
axiom
macro ℂ == CaleyDickson
Type: Void
axiom
macro ℚ == Expression Integer
Type: Void
axiom
𝐋 := LinearOperator(dim,OVAR [], ℚ)
![\label{eq2}\hbox{\axiomType{LinearOperator}\ } (2, \hbox{\axiomType{OrderedVariableList}\ } ([ ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))](images/6205923391440641029-16.0px.png "\label{eq2}\hbox{\axiomType{LinearOperator}\ } (2, \hbox{\axiomType{OrderedVariableList}\ } ([ ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))") | (2) |
Type: Type
axiom
𝐞:ℒ 𝐋 := basisVectors()
![\label{eq3}\left[{|_{1}}, \:{|_{2}}\right]](images/3103936605298351352-16.0px.png "\label{eq3}\left[{|_{1}}, \:{|_{2}}\right]") | (3) |
axiom
𝐝:ℒ 𝐋 := basisForms()
![\label{eq4}\left[{|_{\ }^{1}}, \:{|_{\ }^{2}}\right]](images/6443194698690308847-16.0px.png "\label{eq4}\left[{|_{\ }^{1}}, \:{|_{\ }^{2}}\right]") | (4) |
axiom
o:𝐋:=1 -- identity for product
 | (5) |
axiom
I:𝐋:=[1] -- identity for composition
 | (6) |
axiom
X:𝐋:=[2,1] -- twist
 | (7) |
Now generate structure constants for Quaternion Algebra
The basis consists of the real and imaginary units. We use quaternion multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors.
Split-complex can be specified by Caley-Dickson parameter (q0 = -1)
axiom
q0:=sb('q,
 | (8) |
Type: Symbol
axiom
--q0:=1 -- split-complex QQ := ℂ(ℚ,'i, q0);
Type: Type
Basis: Each B.i is a quaternion number
axiom
B:ℒ QQ := map(x +-> hyper x,1$SQMATRIX(dim, ℚ)::ℒ ℒ ℚ)
![\label{eq9}\left[ 1, \: i \right]](images/5501638393175894660-16.0px.png "\label{eq9}\left[ 1, \: i \right]") | (9) |
axiom
-- Multiplication table: M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j,i, 1..dim), j, 1..dim)
![\label{eq10}\left[
\begin{array}{cc}
1 & i
\
i & -{q_{0}}](images/6521528484919710153-16.0px.png "\label{eq10}\left[
\begin{array}{cc}
1 & i
\
i & -{q_{0}}") | (10) |
axiom
-- Function to divide the matrix entries by a basis element S(y) == map(x +-> real(x/y),M)
Type: Void
axiom
-- The result is a nested list ѕ :=map(S,B)::ℒ ℒ ℒ ℚ
axiom
Compiling function S with type CaleyDickson(Expression(Integer),i, *01q(0)) -> Matrix(Expression(Integer))
![\label{eq11}\left[{\left[{\left[ 1, \: 0 \right]}, \:{\left[ 0, \: -{q_{0}}\right]}\right]}, \:{\left[{\left[ 0, \: 1 \right]}, \:{\left[ 1, \: 0 \right]}\right]}\right]](images/6165186774850846677-16.0px.png "\label{eq11}\left[{\left[{\left[ 1, \: 0 \right]}, \:{\left[ 0, \: -{q_{0}}\right]}\right]}, \:{\left[{\left[ 0, \: 1 \right]}, \:{\left[ 1, \: 0 \right]}\right]}\right]") | (11) |
Type: List(List(List(Expression(Integer))))
axiom
-- structure constants form a tensor operator Y := Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*𝐝.k,i, 1..dim), j, 1..dim), k, 1..dim)
 | (12) |
axiom
arity Y
 | (13) |
axiom
matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y,i, 1..dim), j, 1..dim)
![\label{eq14}\left[
\begin{array}{cc}
{|_{1}}&{|_{2}}
\
{|_{2}}& -{{q_{0}}\ {|_{1}}}](images/1125495401171150979-16.0px.png "\label{eq14}\left[
\begin{array}{cc}
{|_{1}}&{|_{2}}
\
{|_{2}}& -{{q_{0}}\ {|_{1}}}") | (14) |
Multiplication of arbitrary quaternions 
and 
axiom
a:=Σ(sb('a,
 | (15) |
axiom
b:=Σ(sb('b,
 | (16) |
axiom
(a*b)/Y
 | (17) |
A scalar product is denoted by the (2,0)-tensor
axiom
U:=Σ(Σ(script('u,
 | (18) |
Definition 1
We say that the scalar product is associative if the tensor equation holds:
Y = Y
U U
In other words, if the (3,0)-tensor:
(2.2,0.1)(2.6,0.1)(2.6,0.9)
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\psbezier[linewidth=0.04](4.8,0.9)(4.8,0.1)(4.4,0.1)(4.4,0.9)
\psline[linewidth=0.04cm](4.6,0.3)(4.6,-0.1)
\psbezier[linewidth=0.04](4.6,-0.1)(4.6,-0.9)(4.0,-0.9)(4.0,-0.1)
\psline[linewidth=0.04cm](4.0,-0.1)(4.0,0.9)
\usefont{T1}{ptm}{m}{n}
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\psline[linewidth=0.04cm](0.0,-0.1)(0.0,0.9)
\psline[linewidth=0.04cm](1.2,-0.1)(1.2,0.9)
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\end{pspicture}
}](images/2375189246716000159-16.0px.png "\scalebox{1} % Change this value to rescale the drawing.
{
\begin{pspicture}(0,-0.92)(4.82,0.92)
\psbezier[linewidth=0.04](2.2,0.9)(2.2,0.1)(2.6,0.1)(2.6,0.9)
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\psbezier[linewidth=0.04](2.4,-0.1)(2.4,-0.9)(3.0,-0.9)(3.0,-0.1)
\psline[linewidth=0.04cm](3.0,-0.1)(3.0,0.9)
\psbezier[linewidth=0.04](4.8,0.9)(4.8,0.1)(4.4,0.1)(4.4,0.9)
\psline[linewidth=0.04cm](4.6,0.3)(4.6,-0.1)
\psbezier[linewidth=0.04](4.6,-0.1)(4.6,-0.9)(4.0,-0.9)(4.0,-0.1)
\psline[linewidth=0.04cm](4.0,-0.1)(4.0,0.9)
\usefont{T1}{ptm}{m}{n}
\rput(3.4948437,0.205){-}
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\psline[linewidth=0.04cm](0.0,-0.1)(0.0,0.9)
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\usefont{T1}{ptm}{m}{n}
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\end{pspicture}
}") |
 | (19) |
Using the LinearOperator? domain in Axiom and some carefully chosen symbols we can easily enter expressions that are both readable and interpreted by Axiom as "graphical calculus" diagrams describing complex products and compositions of linear operators.
axiom
ω:𝐋 :=
o Y I /
o U -
o I Y /
o U o
![\label{eq20}\begin{array}{@{}l}
\displaystyle
{{\left({u^{2, \: 1}}-{u^{1, \: 2}}\right)}\ {|_{\ }^{1 \ 2 \ 1}}}+{{\left({{u^{1, \: 1}}\ {q_{0}}}+{u^{2, \: 2}}\right)}\ {|_{\ }^{1 \ 2 \ 2}}}+
\
\
\displaystyle
{{\left(-{{u^{1, \: 1}}\ {q_{0}}}-{u^{2, \: 2}}\right)}\ {|_{\ }^{2 \ 2 \ 1}}}+{{\left({u^{2, \: 1}}-{u^{1, \: 2}}\right)}\ {q_{0}}\ {|_{\ }^{2 \ 2 \ 2}}}](images/2107500098681298555-16.0px.png "\label{eq20}\begin{array}{@{}l}
\displaystyle
{{\left({u^{2, \: 1}}-{u^{1, \: 2}}\right)}\ {|_{\ }^{1 \ 2 \ 1}}}+{{\left({{u^{1, \: 1}}\ {q_{0}}}+{u^{2, \: 2}}\right)}\ {|_{\ }^{1 \ 2 \ 2}}}+
\
\
\displaystyle
{{\left(-{{u^{1, \: 1}}\ {q_{0}}}-{u^{2, \: 2}}\right)}\ {|_{\ }^{2 \ 2 \ 1}}}+{{\left({u^{2, \: 1}}-{u^{1, \: 2}}\right)}\ {q_{0}}\ {|_{\ }^{2 \ 2 \ 2}}}") | (20) |
Note: The only purpose of the o symbols on the left above is to serve as a constant left-side margin as required by Axiom. The symbols on the right describe the relation between row.
Definition 2
An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?.
We may consider the problem where multiplication Y is given,
and look for all associative scalar products 
This problem can be solved using linear algebra.
axiom
)expose MCALCFN
MultiVariableCalculusFunctions is now explicitly exposed in frame initial J := jacobian(ravel ω,concat map(variables, ravel U)::ℒ Symbol);
Type: Matrix(Expression(Integer))
axiom
u := transpose matrix [concat map(variables,ravel U)::ℒ Symbol];
Type: Matrix(Polynomial(Integer))
axiom
J::OutputForm * u::OutputForm = 0
![\label{eq21}\begin{array}{@{}l}
\displaystyle
{{\left[
\begin{array}{cccc}
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0
\
0 & - 1 & 1 & 0
\
{q_{0}}& 0 & 0 & 1
\
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0
\
-{q_{0}}& 0 & 0 & - 1
\
0 & -{q_{0}}&{q_{0}}& 0](images/7468782644637304744-16.0px.png "\label{eq21}\begin{array}{@{}l}
\displaystyle
{{\left[
\begin{array}{cccc}
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0
\
0 & - 1 & 1 & 0
\
{q_{0}}& 0 & 0 & 1
\
0 & 0 & 0 & 0
\
0 & 0 & 0 & 0
\
-{q_{0}}& 0 & 0 & - 1
\
0 & -{q_{0}}&{q_{0}}& 0") | (21) |
Type: Equation(OutputForm?)
axiom
nrows(J),ncols(J)
![\label{eq22}\left[ 8, \: 4 \right]](images/8012068307576462172-16.0px.png "\label{eq22}\left[ 8, \: 4 \right]") | (22) |
Type: Tuple(PositiveInteger?)
The matrix J transforms the coefficients of the tensor 
into coefficients of the tensor 
. We are looking for
the general linear family of tensors 
such that
J transforms into 
for any such .
If the null space of the J matrix is not empty we can use
the basis to find all non-trivial solutions for U:
axiom
Ñ:=nullSpace(J)
![\label{eq23}\left[{\left[ 0, \: 1, \: 1, \: 0 \right]}, \:{\left[ -{1 \over{q_{0}}}, \: 0, \: 0, \: 1 \right]}\right]](images/9187658044477029535-16.0px.png "\label{eq23}\left[{\left[ 0, \: 1, \: 1, \: 0 \right]}, \:{\left[ -{1 \over{q_{0}}}, \: 0, \: 0, \: 1 \right]}\right]") | (23) |
Type: List(Vector(Expression(Integer)))
axiom
ℰ:=map((x,y)+->x=y, concat map(variables, ravel U), entries Σ(sb('p, [i])*Ñ.i, i, 1..#Ñ) )
![\label{eq24}\left[{{u^{1, \: 1}}= -{{p_{2}}\over{q_{0}}}}, \:{{u^{1, \: 2}}={p_{1}}}, \:{{u^{2, \: 1}}={p_{1}}}, \:{{u^{2, \: 2}}={p_{2}}}\right]](images/4299451107465611344-16.0px.png "\label{eq24}\left[{{u^{1, \: 1}}= -{{p_{2}}\over{q_{0}}}}, \:{{u^{1, \: 2}}={p_{1}}}, \:{{u^{2, \: 1}}={p_{1}}}, \:{{u^{2, \: 2}}={p_{2}}}\right]") | (24) |
Type: List(Equation(Expression(Integer)))
This defines a family of pre-Frobenius algebras:
axiom
zero? eval(ω,ℰ)
 | (25) |
Type: Boolean
In two dimensions the pairing is necessarily symmetric!
axiom
Ų:𝐋 := eval(U,ℰ)
 | (26) |
axiom
matrix Ξ(Ξ((𝐞.i 𝐞.j)/Ų,i, 1..dim), j, 1..dim)
![\label{eq27}\left[
\begin{array}{cc}
-{{p_{2}}\over{q_{0}}}&{p_{1}}
\
{p_{1}}&{p_{2}}](images/5825809348662629403-16.0px.png "\label{eq27}\left[
\begin{array}{cc}
-{{p_{2}}\over{q_{0}}}&{p_{1}}
\
{p_{1}}&{p_{2}}") | (27) |
This is the most general form of the "dot product" of two complex numbers
axiom
(a*b)/Ų
 | (28) |
axiom
(a*a)/Ų
 | (29) |
The scalar product must be non-degenerate:
axiom
Ů:=determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ų),j, 1..dim), i, 1..dim)
 | (30) |
Type: Expression(Integer)
axiom
factor Ů
 | (31) |
Type: Factored(Expression(Integer))
Definition 3
Co-pairing
Solve the [Snake Relation]? as a system of linear equations.
axiom
Ω:𝐋:=Σ(Σ(script('u,
 | (32) |
axiom
d1:=(I*Ω)/(Ų*I);
axiom
d2:=(Ω*I)/(I*Ų);
axiom
equate(f,g)==map((x, y)+->(x=y), ravel f, ravel g);
Type: Void
axiom
eq1:=equate(d1,I);
axiom
Compiling function equate with type (LinearOperator(2,OrderedVariableList([]), Expression(Integer)), LinearOperator(2, OrderedVariableList([]), Expression(Integer))) -> List(Equation( Expression(Integer)))
Type: List(Equation(Expression(Integer)))
axiom
eq2:=equate(d2,I);
Type: List(Equation(Expression(Integer)))
axiom
snake:=solve(concat(eq1,eq2), concat Ξ(Ξ(script('u, [[i, j]]), i, 1..dim), j, 1..dim));
Type: List(List(Equation(Expression(Integer))))
axiom
if #snake ~= 1 then error "no solution"
Type: Void
axiom
Ω:=eval(Ω,snake(1))
![\label{eq33}\begin{array}{@{}l}
\displaystyle
-{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 1}}}+{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 2}}}+
\
\
\displaystyle
{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 1}}}+{{{p_{2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 2}}}](images/6644729134506119865-16.0px.png "\label{eq33}\begin{array}{@{}l}
\displaystyle
-{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 1}}}+{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 2}}}+
\
\
\displaystyle
{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 1}}}+{{{p_{2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 2}}}") | (33) |
axiom
matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j),i, 1..dim), j, 1..dim)
![\label{eq34}\left[
\begin{array}{cc}
-{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}&{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}
\
{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}&{{p_{2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}](images/5680456234301441218-16.0px.png "\label{eq34}\left[
\begin{array}{cc}
-{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}&{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}
\
{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}&{{p_{2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}") | (34) |
Check "dimension" and the snake relations.
axiom
d:𝐋:=
o Ω /
o X /
o Ų o
 | (35) |
axiom
test
( I Ω ) /
( Ų I ) = I
 | (36) |
Type: Boolean
axiom
test
( Ω I ) /
( I Ų ) = I
 | (37) |
Type: Boolean
Definition 4
Co-algebra
Compute the "three-point" function and use it to define co-multiplication.
axiom
W:=(Y,I)/Ų
![\label{eq38}\begin{array}{@{}l}
\displaystyle
-{{{p_{2}}\over{q_{0}}}\ {|_{\ }^{1 \ 1 \ 1}}}+{{p_{1}}\ {|_{\ }^{1 \ 1 \ 2}}}+{{p_{1}}\ {|_{\ }^{1 \ 2 \ 1}}}+{{p_{2}}\ {|_{\ }^{1 \ 2 \ 2}}}+
\
\
\displaystyle
{{p_{1}}\ {|_{\ }^{2 \ 1 \ 1}}}+{{p_{2}}\ {|_{\ }^{2 \ 1 \ 2}}}+{{p_{2}}\ {|_{\ }^{2 \ 2 \ 1}}}-{{p_{1}}\ {q_{0}}\ {|_{\ }^{2 \ 2 \ 2}}}](images/4317172256605382430-16.0px.png "\label{eq38}\begin{array}{@{}l}
\displaystyle
-{{{p_{2}}\over{q_{0}}}\ {|_{\ }^{1 \ 1 \ 1}}}+{{p_{1}}\ {|_{\ }^{1 \ 1 \ 2}}}+{{p_{1}}\ {|_{\ }^{1 \ 2 \ 1}}}+{{p_{2}}\ {|_{\ }^{1 \ 2 \ 2}}}+
\
\
\displaystyle
{{p_{1}}\ {|_{\ }^{2 \ 1 \ 1}}}+{{p_{2}}\ {|_{\ }^{2 \ 1 \ 2}}}+{{p_{2}}\ {|_{\ }^{2 \ 2 \ 1}}}-{{p_{1}}\ {q_{0}}\ {|_{\ }^{2 \ 2 \ 2}}}") | (38) |
axiom
λ:=(Ω,I, Ω)/(I, W, I)
![\label{eq39}\begin{array}{@{}l}
\displaystyle
-{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 1}^{1}}}+{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 2}^{1}}}+
\
\
\displaystyle
{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 1}^{1}}}+{{{p_{2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 2}^{1}}}-
\
\
\displaystyle
{{{{p_{1}}\ {{q_{0}}^2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 1}^{2}}}-{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 2}^{2}}}-
\
\
\displaystyle
{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 1}^{2}}}+{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 2}^{2}}}](images/205055306048363317-16.0px.png "\label{eq39}\begin{array}{@{}l}
\displaystyle
-{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 1}^{1}}}+{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 2}^{1}}}+
\
\
\displaystyle
{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 1}^{1}}}+{{{p_{2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 2}^{1}}}-
\
\
\displaystyle
{{{{p_{1}}\ {{q_{0}}^2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 1}^{2}}}-{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 2}^{2}}}-
\
\
\displaystyle
{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 1}^{2}}}+{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 2}^{2}}}") | (39) |
axiom
test
( I Ω ) /
( Y I ) = λ
 | (40) |
Type: Boolean
axiom
test
( Ω I ) /
( I Y ) = λ
 | (41) |
Type: Boolean
Frobenius Condition
axiom
Χ :=
Y /
λ
![\label{eq42}\begin{array}{@{}l}
\displaystyle
-{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 1}^{1 \ 1}}}+{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 2}^{1 \ 1}}}+
\
\
\displaystyle
{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 1}^{1 \ 1}}}+{{{p_{2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 2}^{1 \ 1}}}-
\
\
\displaystyle
{{{{p_{1}}\ {{q_{0}}^2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 1}^{1 \ 2}}}-{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 2}^{1 \ 2}}}-
\
\
\displaystyle
{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 1}^{1 \ 2}}}+{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 2}^{1 \ 2}}}-
\
\
\displaystyle
{{{{p_{1}}\ {{q_{0}}^2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 1}^{2 \ 1}}}-{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 2}^{2 \ 1}}}-
\
\
\displaystyle
{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 1}^{2 \ 1}}}+{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 2}^{2 \ 1}}}+
\
\
\displaystyle
{{{{p_{2}}\ {{q_{0}}^2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 1}^{2 \ 2}}}-{{{{p_{1}}\ {{q_{0}}^2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 2}^{2 \ 2}}}-
\
\
\displaystyle
{{{{p_{1}}\ {{q_{0}}^2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 1}^{2 \ 2}}}-{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 2}^{2 \ 2}}}](images/8085198218851745543-16.0px.png "\label{eq42}\begin{array}{@{}l}
\displaystyle
-{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 1}^{1 \ 1}}}+{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 2}^{1 \ 1}}}+
\
\
\displaystyle
{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 1}^{1 \ 1}}}+{{{p_{2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 2}^{1 \ 1}}}-
\
\
\displaystyle
{{{{p_{1}}\ {{q_{0}}^2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 1}^{1 \ 2}}}-{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 2}^{1 \ 2}}}-
\
\
\displaystyle
{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 1}^{1 \ 2}}}+{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 2}^{1 \ 2}}}-
\
\
\displaystyle
{{{{p_{1}}\ {{q_{0}}^2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 1}^{2 \ 1}}}-{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 2}^{2 \ 1}}}-
\
\
\displaystyle
{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 1}^{2 \ 1}}}+{{{{p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 2}^{2 \ 1}}}+
\
\
\displaystyle
{{{{p_{2}}\ {{q_{0}}^2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 1}^{2 \ 2}}}-{{{{p_{1}}\ {{q_{0}}^2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1 \ 2}^{2 \ 2}}}-
\
\
\displaystyle
{{{{p_{1}}\ {{q_{0}}^2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 1}^{2 \ 2}}}-{{{{p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2 \ 2}^{2 \ 2}}}") | (42) |
axiom
test
( λ I ) /
( I Y ) = Χ
 | (43) |
Type: Boolean
axiom
test
( I λ ) /
( Y I ) = Χ
 | (44) |
Type: Boolean
i = Unit of the algebra
axiom
i:=𝐞.1
 | (45) |
axiom
test
o i /
o λ = Ω
 | (46) |
Type: Boolean
Handle
axiom
H:𝐋 :=
λ /
X /
Y
![\label{eq47}\begin{array}{@{}l}
\displaystyle
-{{{2 \ {p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1}^{1}}}+{{{2 \ {p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2}^{1}}}-{{{2 \ {p_{1}}\ {{q_{0}}^2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1}^{2}}}-
\
\
\displaystyle
{{{2 \ {p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2}^{2}}}](images/4063052312835129167-16.0px.png "\label{eq47}\begin{array}{@{}l}
\displaystyle
-{{{2 \ {p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1}^{1}}}+{{{2 \ {p_{1}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2}^{1}}}-{{{2 \ {p_{1}}\ {{q_{0}}^2}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{1}^{2}}}-
\
\
\displaystyle
{{{2 \ {p_{2}}\ {q_{0}}}\over{{{{p_{1}}^2}\ {q_{0}}}+{{p_{2}}^2}}}\ {|_{2}^{2}}}") | (47) |
Definition 5
i U
axiom
ι:𝐋:=
o i I /
o Ų o
 | (48) |
Y=U ι
axiom
test o Y / o ι o = Ų
 | (49) |
Type: Boolean
For example:
axiom
ex1:=[q[0]=1,p[1]=0, p[2]=1]
![\label{eq50}\left[{{q_{0}}= 1}, \:{{p_{1}}= 0}, \:{{p_{2}}= 1}\right]](images/4589732687609299584-16.0px.png "\label{eq50}\left[{{q_{0}}= 1}, \:{{p_{1}}= 0}, \:{{p_{2}}= 1}\right]") | (50) |
Type: List(Equation(Polynomial(Integer)))
axiom
Ų0:𝐋 :=eval(Ų,ex1)
 | (51) |
axiom
Ω0:𝐋 :=eval(Ω,ex1)$𝐋
 | (52) |
axiom
λ0:𝐋 :=eval(λ,ex1)$𝐋
 | (53) |
axiom
H0:𝐋 :=eval(H,ex1)$𝐋
 | (54) |