Non-degeneracy of the pairing (snake relation)

or equivalently

Ref:

• http://mat.uab.es/~kock/TQFT.html

  Frobenius algebras and 2D topological quantum field theories

  Section 2.3.11, page 112.

• Also related notes: http://mat.uab.es/~kock/TQFT/FS.pdf

  Section 2.2.9, page 23.

  Joachim Kock

• http://arxiv.org/abs/math/0512103

  Categorical Aspects of Topological Quantum Field Theories

  Section 2.3.3, page 27.

  Bruce H. Bartlett

We use the Axiom LinearOperator library

(1) -> )library CARTEN MONAL PROP LOP

   >> System error:
   The value
  15684
is not of type
  LIST

and some convenient notation

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

macro Ξ(f,i,n)==[f for i in n]

macro sb == subscript

macro sp == superscript

Let 𝐋 be the domain of 2-dimensional linear operators

dim:=2

(1)

macro ℒ == List

macro ℚ == Expression Integer

𝐋 := LinearOperator(OVAR ['1,'2], 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) 
                                    Type
                                    Type

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

Pairing

A scalar product (pairing) is represented by

U:=Σ(Σ(sp('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.

In general we do not require that it be symmetric.

Co-pairing

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

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

   𝐋 is not a valid type.

This is equivalent to a matrix inverse (transposed!)

Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/U, i,1..dim), j,1..dim)

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

   Cannot find a definition or applicable library operation named 𝐞 
      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.

Check that the twisted snake relation holds

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

   There are no exposed library operations named I but there is one 
      unexposed operation with that name. Use HyperDoc Browse or issue
                                )display op I
      to learn more about the available operation.

   Cannot find a definition or applicable library operation named I 
      with argument type(s) 
                                 Variable(Ω)

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

Dimension

Since the "snake" is twisted, dimension is as expected.

d:=
    Ω /
    U

(2)

This "twisted dimension " depends on !

d':=
     Ω /
     X /
     U

(3)