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	SandBox Quaternion Algebra is Frobenius in Many Ways
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Snake Relation

Edit detail for Snake Relation revision 11 of 11

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Editor: Bill Page Time: 2013/04/23 21:47:01 GMT+0
Note: missing CARTEN library

changed:
-)library MONAL PROP LOP
)library CARTEN MONAL PROP LOP

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

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We use the Axiom LinearOperator library

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(1) -> )library CARTEN MONAL PROP LOP

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

and convenient notation

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

Type: Void

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macro Ξ(f,i,n)==[f for i in n]

Type: Void

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macro sb == subscript

Type: Void

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macro sp == superscript

Type: Void

Let 𝐋 be the domain of 2-dimensional linear operators

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dim:=2

$\label{eq1}2$

(1)

Type: PositiveInteger?

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macro ℒ == List

Type: Void

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macro ℚ == Expression Integer

Type: Void

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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

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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 "snake relation" as a system of linear equations.

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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!)

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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 snake relation holds

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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.

Dimension

This quantity depends on !

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d:=
    Ω /
    U

$\label{eq2}\frac{��}{U}$

(2)

Type: Fraction(Polynomial(Integer))

This "twisted" quantity does not.

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d':=
     Ω /
     X /
     U

$\label{eq3}\frac{��}{U \ X}$

(3)

Type: Fraction(Polynomial(Integer))

Symmetric Pairing

Repeat the calculation, assuming that U is symmetric.

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

   There are 1 exposed and 0 unexposed library operations named ravel 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op ravel
      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 ravel
      with argument type(s) 
                        Fraction(Polynomial(Integer))

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

fricas

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 snake relation holds

fricas

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.

These quantities no longer depends on !

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d:=
    Ω /
    U

$\label{eq4}\frac{��}{U}$

(4)

Type: Fraction(Polynomial(Integer))

fricas

d':=
     Ω /
     X /
     U

$\label{eq5}\frac{��}{U \ X}$

(5)

Type: Fraction(Polynomial(Integer))

Twist dimension or twist snake? --Bill Page, Sun, 08 May 2011 14:16:39 -0700 reply

TwistedSnakeRelation