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	SandBox Quaternion Algebra is Frobenius in Many Ways
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Snake Relation

Edit detail for Snake Relation revision 5 of 11

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Editor: Bill Page Time: 2011/04/22 15:50:05 GMT-7
Note: assume U symmetric

added:

Repeat the calculation, assuming that U is symmetric.

\begin{axiom}
U:=eval(U,sp('u,[2,1])=sp('u,[1,2]))
\end{axiom}

\begin{axiom}
Um:=matrix Ξ(Ξ((𝐞.i*𝐞.j)/U, i,1..dim), j,1..dim)
mU:=transpose inverse map(retract,Um)
Ω:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i,1..dim), j,1..dim)
\end{axiom}

Check that the snake relation holds
\begin{axiom}

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

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

\end{axiom}

Dimension

  This quantity no longer depends on $U$!

\begin{axiom}

d:=
    Ω /
    U

\end{axiom}

This one apparently does not.
\begin{axiom}
d':=
     Ω /
     X /
     U
\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

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We use 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

and convenient notation

fricas

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

Type: Void

fricas

macro Ξ(f,i,n)==[f for i in n]

Type: Void

fricas

macro sb == subscript

Type: Void

fricas

macro sp == superscript

Type: Void

Let 𝐋 be the domain of 2-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(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.

Co-pairing

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

fricas

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!)

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.

Dimension

This quantity depends on !

fricas

d:=
    Ω /
    U

$\label{eq2}\frac{��}{U}$

(2)

Type: Fraction(Polynomial(Integer))

This one apparently does not.

fricas

d':=
     Ω /
     X /
     U

$\label{eq3}\frac{��}{U \ X}$

(3)

Type: Fraction(Polynomial(Integer))

Repeat the calculation, assuming that U is symmetric.

fricas

U:=eval(U,sp('u,[2,1])=sp('u,[1,2]))

$\label{eq4}U$

(4)

Type: Polynomial(Integer)

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.

Dimension

This quantity no longer depends on !

fricas

d:=
    Ω /
    U

$\label{eq5}\frac{��}{U}$

(5)

Type: Fraction(Polynomial(Integer))

This one apparently does not.

fricas

d':=
     Ω /
     X /
     U

$\label{eq6}\frac{��}{U \ X}$

(6)

Type: Fraction(Polynomial(Integer))