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	SandBox Quaternion Algebra is Frobenius in Many Ways
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Snake Relation

Edit detail for Snake Relation revision 1 of 11

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Editor: Bill Page Time: 2011/04/21 20:39:17 GMT-7
Note: dimension

changed:
-
Non-degeneracy of the pairing

Ref:

- http://mat.uab.es/~kock/TQFT.html

  Frobenius algebras and 2D topological quantum field theories

  Section 2.3.11, page 112.

  *Joachim Kock*

We need the Axiom LinearOperator library.
\begin{axiom}
)library MONAL PROP LIN
\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 4-dimensional linear operators
\begin{axiom}
dim:=2
macro ℒ == List
macro ℚ == Expression Integer
𝐋 := LinearOperator(dim, OVAR [], ℚ)
𝐞:ℒ 𝐋      := basisVectors()
𝐝:ℒ 𝐋      := basisForms()
I:𝐋:=[1]   -- identity for composition
X:𝐋:=[2,1] -- twist
\end{axiom}

A scalar product (pairing) is denoted by
\begin{axiom}
U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)
Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/U, i,1..dim), j,1..dim)
\end{axiom}

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)/(U*I);
d2:=(Ω*I)/(I*U);
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)
-- compare
inverse map(retract,Um)
\end{axiom}

Check "dimension": It depends on parameters!
\begin{axiom}

d:𝐋:=
    (   Ω    )  /
    (   U    )

test
    (  I Ω   )  /
    (   U I  )  =  I

test
    (   Ω I  )  /
    (  I U   )  =  I

\end{axiom}

Non-degeneracy of the pairing

Ref:

http://mat.uab.es/~kock/TQFT.html
Frobenius algebras and 2D topological quantum field theories

Section 2.3.11, page 112.

Joachim Kock

We need the Axiom LinearOperator? library.

fricas

(1) -> )library MONAL PROP LIN

   )library cannot find the file LIN.
   Monoidal is now explicitly exposed in frame initial 
   Monoidal will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/MONAL.NRLIB/MONAL
   Prop is now explicitly exposed in frame initial 
   Prop will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/PROP.NRLIB/PROP

Use the following macros for convenient notation

fricas

-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])

Type: Void

fricas

-- list
macro Ξ(f,i,n)==[f for i in n]

Type: Void

fricas

-- subscript
macro sb == subscript

Type: Void

𝐋 is the domain of 4-dimensional linear operators

fricas

dim:=2

$\label{eq1}2$

(1)

Type: PositiveInteger?

fricas

macro ℒ == List

Type: Void

fricas

macro ℚ == Expression Integer

Type: Void

fricas

𝐋 := LinearOperator(dim, OVAR [], ℚ)

   There are no library operations named LinearOperator 
      Use HyperDoc Browse or issue
                           )what op LinearOperator
      to learn if there is any operation containing " LinearOperator " 
      in its name.

   Cannot find a definition or applicable library operation named 
      LinearOperator with argument type(s) 
                               PositiveInteger
                                    Type
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

A scalar product (pairing) is denoted by

fricas

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

   There are no library operations named 𝐝 
      Use HyperDoc Browse or issue
                                 )what op 𝐝
      to learn if there is any operation containing " 𝐝 " in its name.
   Cannot find a definition or applicable library operation named 𝐝 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are no library operations named 𝐝 
      Use HyperDoc Browse or issue
                                 )what op 𝐝
      to learn if there is any operation containing " 𝐝 " in its name.

   Cannot find a definition or applicable library operation named 𝐝 
      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Co-pairing

Solve the "snake relation" as a system of linear equations.

fricas

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

   𝐋 is not a valid type.

Check "dimension": It depends on parameters!

fricas

d:𝐋:=
    (   Ω    )  /
    (   U    )

   𝐋 is not a valid type.