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	SandBox Quaternion Algebra is Frobenius in Many Ways
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Snake Relation

Edit detail for Snake Relation revision 4 of 11

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Editor: Bill Page Time: 2011/04/22 10:45:12 GMT-7
Note: twisted dimension

changed:
-d:𝐋:=
-
-       Ω      /
-       U    

d:=
    Ω /
    U

added:

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

axiom

)library MONAL PROP LIN

   Monoidal is now explicitly exposed in frame initial 
   Monoidal will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL
   Prop is now explicitly exposed in frame initial 
   Prop will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/PROP.NRLIB/PROP
   LinearOperator is now explicitly exposed in frame initial 
   LinearOperator will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/LIN.NRLIB/LIN

and convenient notation

axiom

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

Type: Void

axiom

macro Ξ(f,i,n)==[f for i in n]

Type: Void

axiom

macro sb == subscript

Type: Void

axiom

macro sp == superscript

Type: Void

Let 𝐋 be the domain of 2-dimensional linear operators

axiom

dim:=2

$\label{eq1}2$

(1)

Type: PositiveInteger?

axiom

macro ℒ == List

Type: Void

axiom

macro ℚ == Expression Integer

Type: Void

axiom

𝐋 := LinearOperator(dim, OVAR [], ℚ)

$\label{eq2}\hbox{\axiomType{LinearOperator}\ } (2, \hbox{\axiomType{OrderedVariableList}\ } ([ ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))$

(2)

Type: Type

axiom

𝐞:ℒ 𝐋      := basisVectors()

$\label{eq3}\left[{|_{1}}, \:{|_{2}}\right]$

(3)

Type: List(LinearOperator?(2,OrderedVariableList?([]),Expression(Integer)))

axiom

𝐝:ℒ 𝐋      := basisForms()

$\label{eq4}\left[{|_{\ }^{1}}, \:{|_{\ }^{2}}\right]$

(4)

Type: List(LinearOperator?(2,OrderedVariableList?([]),Expression(Integer)))

axiom

I:𝐋:=[1]   -- identity for composition

$\label{eq5}{|_{1}^{1}}+{|_{2}^{2}}$

(5)

Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))

axiom

X:𝐋:=[2,1] -- twist

$\label{eq6}{|_{1 \ 1}^{1 \ 1}}+{|_{2 \ 1}^{1 \ 2}}+{|_{1 \ 2}^{2 \ 1}}+{|_{2 \ 2}^{2 \ 2}}$

(6)

Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))

A scalar product (pairing) is denoted by

axiom

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

$\label{eq7}{{u^{1, \: 1}}\ {|_{\ }^{1 \ 1}}}+{{u^{1, \: 2}}\ {|_{\ }^{1 \ 2}}}+{{u^{2, \: 1}}\ {|_{\ }^{2 \ 1}}}+{{u^{2, \: 2}}\ {|_{\ }^{2 \ 2}}}$

(7)

Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))

Co-pairing

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

axiom

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

$\label{eq8}{{u_{1, \: 1}}\ {|_{1 \ 1}}}+{{u_{1, \: 2}}\ {|_{1 \ 2}}}+{{u_{2, \: 1}}\ {|_{2 \ 1}}}+{{u_{2, \: 2}}\ {|_{2 \ 2}}}$

(8)

Type: LinearOperator?(2,OrderedVariableList?([]),Expression(Integer))

axiom

Í:=(I*Ω)/(U*I);

Function:  contract : (Integer,%,Integer,%,Integer) -> % is missing from domain: CartesianTensor(1,2,Expression(Integer))
   Internal Error
   The function contract with signature $(Integer)$(Integer)$(Integer) is 
      missing from domain CartesianTensor12(Expression (Integer))

This is equivalent to a matrix inverse (transposed!)

axiom

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

Function:  contract : (Integer,%,Integer,%,Integer) -> % is missing from domain: CartesianTensor(1,2,Expression(Integer))
   Internal Error
   The function contract with signature $(Integer)$(Integer)$(Integer) is 
      missing from domain CartesianTensor12(Expression (Integer))

Check that the snake relation holds

axiom

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

Function:  contract : (Integer,%,Integer,%,Integer) -> % is missing from domain: CartesianTensor(1,2,Expression(Integer))
   Internal Error
   The function contract with signature $(Integer)$(Integer)$(Integer) is 
      missing from domain CartesianTensor12(Expression (Integer))

Dimension

This quantity depends on !

axiom

d:=
    Ω /
    U

Function:  contract : (Integer,%,Integer,%,Integer) -> % is missing from domain: CartesianTensor(1,2,Expression(Integer))
   Internal Error
   The function contract with signature $(Integer)$(Integer)$(Integer) is 
      missing from domain CartesianTensor12(Expression (Integer))

This one apparently does not.

axiom

d':=
     Ω /
     X /
     U

Function:  contract : (Integer,%,Integer,%,Integer) -> % is missing from domain: CartesianTensor(1,2,Expression(Integer))
   Internal Error
   The function contract with signature $(Integer)$(Integer)$(Integer) is 
      missing from domain CartesianTensor12(Expression (Integer))