Edit detail for Snake Relation revision 9 of 11

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.

• 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

)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

and 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(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.
𝐞:ℒ 𝐋      := basisVectors()

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

   Cannot find a no-argument definition or library operation named 
      basisVectors .
𝐝:ℒ 𝐋      := basisForms()

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

   Cannot find a no-argument definition or library operation named 
      basisForms .
I:𝐋:=[1]   -- identity for composition

(2)

X:𝐋:=[2,1] -- twist

(3)

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)

   >> System error:
   The index 3 is too large.

In general we do not require that it be symmetric.

Co-pairing

Solve the "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)

   >> System error:
   The index 3 is too large.

This is equivalent to a matrix inverse (transposed!)

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

   >> System error:
   The index 3 is too large.

Check that the snake relation holds

test
    (  I Ω   )  /
    (   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.

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

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

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

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

Dimension

This quantity depends on !

d:=
    Ω /
    U

(4)

This "twisted" quantity does not.

d':=
     Ω /
     X /
     U

   There are 13 exposed and 12 unexposed library operations named / 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op /
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

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

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

Symmetric Pairing

Repeat the calculation, assuming that U is symmetric.

sym:=groebner ravel(U-X/U)

   There are 13 exposed and 12 unexposed library operations named / 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op /
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named / 
      with argument type(s) 
                            List(PositiveInteger)
                                 Variable(U)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
vars:=map(x+->kernels(x).1,sym)::List Symbol

   >> Error detected within library code:
   index out of range

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

   >> System error:
   The index 3 is too large.

Check that the snake relation holds

test
    (  I Ω   )  /
    (   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.

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

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

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

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

These quantities no longer depends on !

d:=
    Ω /
    U

(5)

d':=
     Ω /
     X /
     U

   There are 13 exposed and 12 unexposed library operations named / 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op /
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

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

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