Edit detail for SandBox Grassmann Algebra Is Frobenius In Many Ways revision 7 of 8

changed:
-$2^n$-dimensional vector space representing Grassmann algebra with $n$ generators
-
-An algebra is represented by a (2,1)-tensor
-$Y=\{ {y^k}_{ij} \ i,j,k =1,2, ... dim \}$
-viewed as a linear operator with two inputs $i,j$ and one
-output $k$. For example:
-\begin{axiom}
-n:=2
-dim:=2^n
-T := CartesianTensor(1,dim,EXPR INT)
---T:=CartesianTensor(1,dim,FRAC POLY INT)
-X:List T := [unravel [(i=j => 1;0) for j in 1..dim] for i in 1..dim]
-X(1),X(2)
-Y:T := unravel(concat concat
-  [[[script(y,[[i,j],[k]])
-    for i in 1..dim]
-      for j in 1..dim]
-        for k in 1..dim]
-          )
-\end{axiom}
Grassmann Algebra Is Frobenius In Many Ways

A $2^n$-dimensional vector space represents Grassmann algebra with $n$ generators

Linear operators over a 4-dimensional vector space representing Grassmann
algebra with two generators.

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

- http://en.wikipedia.org/wiki/Grassmann_algebra

We need the Axiom LinearOperator library.
\begin{axiom}
)library CARTEN ARITY CMONAL CPROP CLOP 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 and superscripts
macro sb == subscript
macro sp == superscript
\end{axiom}

𝐋 is the domain of 4-dimensional linear operators over the
rational functions ℚ (Expression Integer), i.e. ratio of
polynomials with integer coefficients.
\begin{axiom}
dim:=4
macro ℒ == List
macro ℂ == CaleyDickson
macro ℚ == Expression Integer
𝐋 := ClosedLinearOperator(OVAR ['1,'i,'j,'k], ℚ)
𝐞:ℒ 𝐋      := basisOut()
𝐝:ℒ 𝐋      := basisIn()
I:𝐋:=[1]   -- identity for composition
X:𝐋:=[2,1] -- twist
V:𝐋:=ev(1) -- evaluation
Λ:𝐋:=co(1) -- co-evaluation
equate(eq)==map((x,y)+->(x=y),ravel lhs eq, ravel rhs eq);
\end{axiom}

changed:
-\begin{axiom}
--- Construct a basis for the Grassmann algebra
-GA:=AntiSymm(INT,[subscript(g,[i]) for i in 1..n])
-B:=[exp(reverse concat([0 for i in 1..n-length(x)],wholeRagits(x::RADIX(2))))$GA for x in 0..dim-1]
--- Compute the multiplication table
-M:=matrix [[B.i * B.j for j in 1..dim] for i in 1..dim]
--- The structure constants of the algebra are given by the coefficients
--- of the polynomials in the multiplication table with respect to the basis
-S(y)==map(x+->coefficient(x,y),M)
-Yg:T:=unravel concat concat(map(S,B)::List List List FRAC POLY INT)
-\end{axiom}

The structure constants can be obtained by dividing each matrix entry by the list of basis vectors.

Grassmann algebra will be specified by setting the Caley-Dickson parameters (i2, j2) to zero.

\begin{axiom}
i2:=sp('i,[2])
j2:=sp('j,[2])
QQ:=CliffordAlgebra(2,ℚ,matrix [[i2,0],[0,j2]])
B:ℒ QQ := [monomial(1,[]),monomial(1,[1]),monomial(1,[2]),monomial(1,[1,2])]
M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)
S(y) == map(x +-> coefficient(recip(y)*x,[]),M)
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)
Y := eval(Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*𝐝.k, i,1..dim), j,1..dim), k,1..dim),[i2=0,j2=0])
matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim)
\end{axiom}

Units
\begin{axiom}
e:=𝐞.1; i:=𝐞.2; j:=𝐞.3; k:=𝐞.4;
\end{axiom}

Multiplication of arbitrary Grassmann numbers $a$ 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}

Multiplication is Associative
\begin{axiom}
test(
  ( I Y ) / _
  (  Y  ) = _
  ( Y I ) / _
  (  Y  ) )
\end{axiom}

changed:
-U:T := unravel(concat
-  [[script(u,[[],[j,i]])
-    for i in 1..dim]
-      for j in 1..dim]
-        )
-\end{axiom}
U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)
\end{axiom}


changed:
-  In other words, if the (3,0)-tensor::
-
-    i  j  k   i  j  k   i  j  k
-     \ | /     \/  /     \  \/
-      \|/   =   \ /   -   \ /
-       0         0         0
  In other words, if the (3,0)-tensor:
$$
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\rput(1.6948438,0.205){=}
\end{pspicture} 
}
$$

changed:
-\begin{axiom}
-ω := reindex(reindex(U,[2,1])*reindex(Yg,[1,3,2]),[3,2,1])-U*Yg
-\end{axiom}
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 :=
     (    Y I    )  /
           U        -
     (    I Y    )  /
           U

\end{axiom}


changed:
-  is called *pre-Frobenius*.
  is called a [Frobenius Algebra].

The Cartan-Killing Trace
\begin{axiom}

Ú:=
   (  Y Λ  ) / _
   (   Y I ) / _
        V
Ù:=
   (  Λ Y  ) / _
   ( I Y   ) / _
      V

test(Ù=Ú)

\end{axiom}
forms is degenerate
\begin{axiom}
Ũ := Ù
test
     (    Y I    )  /
           Ũ        =
     (    I Y    )  /
           Ũ
determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim)
\end{axiom}

General Solution

added:


changed:
-J := jacobian(ravel ω,concat(map(variables,ravel U))::List Symbol);
-uu := transpose matrix [concat(map(variables,ravel(U)))::List Symbol];
-J::OutputForm * uu::OutputForm = 0
-nrows(J)
-ncols(J)
-\end{axiom}
J := jacobian(ravel ω,concat map(variables,ravel U)::ℒ Symbol);
nrows(J),ncols(J)
\end{axiom}


changed:
-\begin{axiom}
-NJ:=nullSpace(J)
-SS:=map((x,y)+->x=y,concat map(variables,ravel U),
-  entries reduce(+,[p[i]*NJ.i for i in 1..#NJ]))
-Ug:T := unravel(map(x+->subst(x,SS),ravel U))
-\end{axiom}

\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}

changed:
-test(unravel(map(x+->subst(x,SS),ravel ω))$T=0*ω)
-\end{axiom}
zero? eval(ω,ℰ)
Ų:𝐋 := eval(U,ℰ)
\end{axiom}

Frobenius Form (co-unit)
\begin{axiom}
d:=ε1*𝐝.1+εi*𝐝.2+εj*𝐝.3+εk*𝐝.4
𝔇:=equate(d=
    (    e I   ) / _
          Ų    )
\end{axiom}

Express scalar product in terms of Frobenius form
\begin{axiom}
𝔓:=solve(𝔇,Ξ(sb('p,[i]), i,1..#Ñ)).1
Ų:=eval(Ų,𝔓)
test
        Y     /
        d     =  Ų

\end{axiom}

In general the pairing is not symmetric!
\begin{axiom}
u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim)
--eigenvectors(u1::Matrix FRAC POLY INT)
\end{axiom}

changed:
-Ud := determinant [[Ug[i,j] for j in 1..dim] for i in 1..dim]
-\end{axiom}
Ů:=determinant u1
factor(numer Ů)/factor(denom Ů)
\end{axiom}

Frobenius scalar product of "vectors" $a$ and $b$
\begin{axiom}
a:=sb('a,[1])*i+sb('a,[2])*j
b:=sb('b,[1])*i+sb('b,[2])*j
(a,a)/Ų
(a,b)/Ų
\end{axiom}

changed:
-  Co-pairing
-\begin{axiom}
-Ωg:T:=unravel concat(transpose(1/Ud*adjoint([[Ug[i,j] for j in 1..dim] for i in 1..dim]).adjMat)::List List FRAC POLY INT)
-\end{axiom}
-<center><pre>
-dimension
-Ω
-U
-</pre></center>
-\begin{axiom}
-contract(contract(Ωg,1,Ug,1),1,2)
-\end{axiom}
  Co-scalar product

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)
ΩX:=Ω/X;
UXΩ:=(I*ΩX)/(Ų*I);
ΩXU:=(ΩX*I)/(I*Ų);
eq1:=equate(UXΩ=I);
eq2:=equate(ΩXU=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))
ΩX:=Ω/X;
\end{axiom}

\begin{axiom}
matrix Ξ(Ξ(retract(Ω/(𝐝.i*𝐝.j)), i,1..dim), j,1..dim)
\end{axiom}

Check "dimension" and the snake relations.
\begin{axiom}

O:𝐋:=
       Ω    /
       Ų

test
    (    I ΩX     )  /
    (     Ų I     )  =  I

test
    (     ΩX I    )  /
    (    I Ų      )  =  I

\end{axiom}

changed:
-  Co-multiplication
-\begin{axiom}
-λg:=reindex(contract(contract(Ug*Yg,1,Ωg,1),1,Ωg,1),[2,3,1]);
--- just for display
-reindex(λg,[3,1,2])
-\end{axiom}
-<center><pre>
-i  
-λ=Ω
-</pre></center>
-\begin{axiom}
-test(λg*X(1)=Ωg)
-\end{axiom}
-
-Definition 5
-
-  <center>Co-unit<pre>
-  i 
-  U
-  </pre></center>
-
-\begin{axiom}
-ιg:=X(1)*Ug
-\end{axiom}
-<center><pre>
-Y=U
-ι  
-</pre></center>
-\begin{axiom}
-test(ιg * Yg = Ug)
-\end{axiom}
-
-For example:
-\begin{axiom}
-Ug0:T:=unravel eval(ravel Ug,[p[1]=1,p[2]=0,p[3]=0,p[4]=1])
-Ωg0:T:=unravel eval(ravel Ωg,[p[1]=1,p[2]=0,p[3]=0,p[4]=1])
-λg0:T:=unravel eval(ravel λg,[p[1]=1,p[2]=0,p[3]=0,p[4]=1]);
-reindex(λg0,[3,1,2])
-\end{axiom}
  Co-algebra

Compute the "three-point" function and use it to define co-multiplication.
\begin{axiom}

W:=
  (Y I) /
    Ų

λ:=
  (  ΩX I ΩX  ) /
  (  I  W  I  )

\end{axiom}

\begin{axiom}

test
     (    I ΩX     )  /
     (     Y I     )  =  λ

test
     (     ΩX I    )  /
     (    I  Y     )  =  λ

\end{axiom}

Co-associativity
\begin{axiom}
test(
  (  λ  ) / _
  ( I λ ) = _
  (  λ  ) / _
  ( λ I ) )
\end{axiom}

\begin{axiom}

test
         e     /
         λ     =    ΩX

\end{axiom}

Frobenius Condition (fork)
\begin{axiom}

H :=
         Y    /
         λ

test
     (   λ I   )  /
     (  I Y    )  =  H

test
     (   I λ   )  /
     (    Y I  )  =  H

\end{axiom}

Handle
\begin{axiom}

Φ :=
         λ     /
         Y

\end{axiom}

Figure 12
\begin{axiom}

φφ:=          _
  ( Ω  Ω  ) / _
  ( X I I ) / _
  ( I X I ) / _
  ( I I X ) / _
  (  Y  Y )

\end{axiom}

Bi-algebra conditions
\begin{axiom}
ΦΦ:=          _
  (  λ λ  ) / _
  ( I I X ) / _
  ( I X I ) / _
  ( I I X ) / _
  (  Y  Y )
test((e,e)/ΦΦ=φφ)
\end{axiom}

Grassmann Algebra Is Frobenius In Many Ways

A -dimensional vector space represents Grassmann algebra with generators

Linear operators over a 4-dimensional vector space representing Grassmann algebra with two generators.

Ref:

• http://arxiv.org/abs/1103.5113

  -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
• http://en.wikipedia.org/wiki/Grassmann_algebra

We need the Axiom LinearOperator? library.

axiom

)library CARTEN ARITY CMONAL CPROP CLOP CALEY

   CartesianTensor is now explicitly exposed in frame initial 
   CartesianTensor will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/CARTEN.NRLIB/CARTEN
   Arity is now explicitly exposed in frame initial 
   Arity will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/ARITY.NRLIB/ARITY
   ClosedMonoidal is now explicitly exposed in frame initial 
   ClosedMonoidal will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/CMONAL.NRLIB/CMONAL
   ClosedProp is now explicitly exposed in frame initial 
   ClosedProp will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/CPROP.NRLIB/CPROP
   ClosedLinearOperator is now explicitly exposed in frame initial 
   ClosedLinearOperator will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/CLOP.NRLIB/CLOP
   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 and superscripts
macro sb == subscript

Type: Void

axiom

macro sp == superscript

Type: Void

𝐋 is the domain of 4-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.

axiom

dim:=4

(1)

Type: PositiveInteger?

axiom

macro ℒ == List

Type: Void

axiom

macro ℂ == CaleyDickson

Type: Void

axiom

macro ℚ == Expression Integer

Type: Void

axiom

𝐋 := ClosedLinearOperator(OVAR ['1,'i,'j,'k], ℚ)

(2)

Type: Type

axiom

𝐞:ℒ 𝐋      := basisOut()

(3)

Type: List(ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer)))

axiom

𝐝:ℒ 𝐋      := basisIn()

(4)

Type: List(ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer)))

axiom

I:𝐋:=[1]   -- identity for composition

(5)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

X:𝐋:=[2,1] -- twist

(6)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

V:𝐋:=ev(1) -- evaluation

(7)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

Λ:𝐋:=co(1) -- co-evaluation

(8)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

equate(eq)==map((x,y)+->(x=y),ravel lhs eq, ravel rhs eq);

Type: Void

Generate structure constants for Grassmann Algebra

The structure constants can be obtained by dividing each matrix entry by the list of basis vectors.

Grassmann algebra will be specified by setting the Caley-Dickson parameters (i2, j2) to zero.

axiom

i2:=sp('i,[2])

(9)

Type: Symbol

axiom

j2:=sp('j,[2])

(10)

Type: Symbol

axiom

QQ:=CliffordAlgebra(2,ℚ,matrix [[i2,0],[0,j2]])

(11)

Type: Type

axiom

B:ℒ QQ := [monomial(1,[]),monomial(1,[1]),monomial(1,[2]),monomial(1,[1,2])]

(12)

Type: List(CliffordAlgebra?(2,Expression(Integer),[[*001i(2),0],[0,*001j(2)]]))

axiom

M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)

(13)

Type: Matrix(CliffordAlgebra?(2,Expression(Integer),[[*001i(2),0],[0,*001j(2)]]))

axiom

S(y) == map(x +-> coefficient(recip(y)*x,[]),M)

Type: Void

axiom

ѕ :=map(S,B)::ℒ ℒ ℒ ℚ

axiom

Compiling function S with type CliffordAlgebra(2,Expression(Integer)
      ,[[*001i(2),0],[0,*001j(2)]]) -> Matrix(Expression(Integer))

(14)

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)
Y := eval(Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*𝐝.k, i,1..dim), j,1..dim), k,1..dim),[i2=0,j2=0])

(15)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim)

(16)

Type: Matrix(ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer)))

Units

axiom

e:=𝐞.1; i:=𝐞.2; j:=𝐞.3; k:=𝐞.4;

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

Multiplication of arbitrary Grassmann numbers and

axiom

a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)

(17)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

b:=Σ(sb('b,[i])*𝐞.i, i,1..dim)

(18)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

(a*b)/Y

(19)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

Multiplication is Associative

axiom

test(
  ( I Y ) / _
  (  Y  ) = _
  ( Y I ) / _
  (  Y  ) )

(20)

Type: Boolean

A scalar product is denoted by the (2,0)-tensor

axiom

U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)

(21)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

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:

(22)

(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.

axiom

ω:𝐋 :=
     (    Y I    )  /
           U        -
     (    I Y    )  /
           U

(23)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

Definition 2

An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?.

The Cartan-Killing Trace

axiom

Ú:=
   (  Y Λ  ) / _
   (   Y I ) / _
        V

(24)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

Ù:=
   (  Λ Y  ) / _
   ( I Y   ) / _
      V

(25)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

test(Ù=Ú)

(26)

Type: Boolean

forms is degenerate

axiom

Ũ := Ù

(27)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

test
     (    Y I    )  /
           Ũ        =
     (    I Y    )  /
           Ũ

(28)

Type: Boolean

axiom

determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim)

(29)

Type: Expression(Integer)

General Solution

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

nrows(J),ncols(J)

(30)

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)

(31)

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) )

(32)

Type: List(Equation(Expression(Integer)))

This defines a family of pre-Frobenius algebras:

axiom

zero? eval(ω,ℰ)

(33)

Type: Boolean

axiom

Ų:𝐋 := eval(U,ℰ)

(34)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

Frobenius Form (co-unit)

axiom

d:=ε1*𝐝.1+εi*𝐝.2+εj*𝐝.3+εk*𝐝.4

(35)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

𝔇:=equate(d=
    (    e I   ) / _
          Ų    )

axiom

Compiling function equate with type Equation(ClosedLinearOperator(
      OrderedVariableList([1,i,j,k]),Expression(Integer))) -> List(
      Equation(Expression(Integer)))

(36)

Type: List(Equation(Expression(Integer)))

Express scalar product in terms of Frobenius form

axiom

𝔓:=solve(𝔇,Ξ(sb('p,[i]), i,1..#Ñ)).1

(37)

Type: List(Equation(Expression(Integer)))

axiom

Ų:=eval(Ų,𝔓)

(38)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

test
        Y     /
        d     =  Ų

(39)

Type: Boolean

In general the pairing is not symmetric!

axiom

u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim)

(40)

Type: Matrix(Expression(Integer))

The scalar product must be non-degenerate:

axiom

Ů:=determinant u1

(41)

Type: Expression(Integer)

axiom

factor(numer Ů)/factor(denom Ů)

(42)

Type: Fraction(Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer)))))

Frobenius scalar product of "vectors" and

axiom

a:=sb('a,[1])*i+sb('a,[2])*j

(43)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

b:=sb('b,[1])*i+sb('b,[2])*j

(44)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

(a,a)/Ų

(45)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

(a,b)/Ų

(46)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

Definition 3

Co-scalar product

Solve the [Snake Relation]? as a system of linear equations.

axiom

Ω:𝐋:=Σ(Σ(script('u,[[i,j]])*𝐞.i*𝐞.j, i,1..dim), j,1..dim)

(47)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

ΩX:=Ω/X;

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

UXΩ:=(I*ΩX)/(Ų*I);

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

ΩXU:=(ΩX*I)/(I*Ų);

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

eq1:=equate(UXΩ=I);

Type: List(Equation(Expression(Integer)))

axiom

eq2:=equate(ΩXU=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))

(48)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

ΩX:=Ω/X;

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

matrix Ξ(Ξ(retract(Ω/(𝐝.i*𝐝.j)), i,1..dim), j,1..dim)

(49)

Type: Matrix(Expression(Integer))

Check "dimension" and the snake relations.

axiom

O:𝐋:=
       Ω    /
       Ų

(50)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

test
    (    I ΩX     )  /
    (     Ų I     )  =  I

(51)

Type: Boolean

axiom

test
    (     ΩX I    )  /
    (    I Ų      )  =  I

(52)

Type: Boolean

Definition 4

Co-algebra

Compute the "three-point" function and use it to define co-multiplication.

axiom

W:=
  (Y I) /
    Ų

(53)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

λ:=
  (  ΩX I ΩX  ) /
  (  I  W  I  )

(54)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

test
     (    I ΩX     )  /
     (     Y I     )  =  λ

(55)

Type: Boolean

axiom

test
     (     ΩX I    )  /
     (    I  Y     )  =  λ

(56)

Type: Boolean

Co-associativity

axiom

test(
  (  λ  ) / _
  ( I λ ) = _
  (  λ  ) / _
  ( λ I ) )

(57)

Type: Boolean

axiom

test
         e     /
         λ     =    ΩX

(58)

Type: Boolean

Frobenius Condition (fork)

axiom

H :=
         Y    /
         λ

(59)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

test
     (   λ I   )  /
     (  I Y    )  =  H

(60)

Type: Boolean

axiom

test
     (   I λ   )  /
     (    Y I  )  =  H

(61)

Type: Boolean

Handle

axiom

Φ :=
         λ     /
         Y

(62)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

Figure 12

axiom

φφ:=          _
  ( Ω  Ω  ) / _
  ( X I I ) / _
  ( I X I ) / _
  ( I I X ) / _
  (  Y  Y )

(63)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

Bi-algebra conditions

axiom

ΦΦ:=          _
  (  λ λ  ) / _
  ( I I X ) / _
  ( I X I ) / _
  ( I I X ) / _
  (  Y  Y )

(64)

Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom

test((e,e)/ΦΦ=φφ)

(65)

Type: Boolean