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TwistedSnakeRelation

Edit detail for TwistedSnakeRelation revision 3 of 4

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Editor: Bill Page Time: 2011/05/09 07:20:15 GMT-7
Note: better graphic

removed:
-
-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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added:
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*


changed:
-and convenient notation
and some convenient notation

Non-degeneracy of the pairing (snake relation)

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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

axiom

)library CARTEN MONAL PROP LOP

   CartesianTensor is now explicitly exposed in frame initial 
   CartesianTensor will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/CARTEN.NRLIB/CARTEN
   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/LOP.NRLIB/LOP

and some 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(OVAR ['1,'2], ℚ)

$\label{eq2}\hbox{\axiomType{LinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 1, 2 ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))$

(2)

Type: Type

axiom

𝐞:ℒ 𝐋      := basisOut()

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

(3)

Type: List(LinearOperator?(OrderedVariableList?([1,2]),Expression(Integer)))

axiom

𝐝:ℒ 𝐋      := basisIn()

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

(4)

Type: List(LinearOperator?(OrderedVariableList?([1,2]),Expression(Integer)))

axiom

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

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

(5)

Type: LinearOperator?(OrderedVariableList?([1,2]),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?(OrderedVariableList?([1,2]),Expression(Integer))

Pairing

A scalar product (pairing) is represented 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?(OrderedVariableList?([1,2]),Expression(Integer))

In general we do not require that it be symmetric.

Co-pairing

Solve the "twisted 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?(OrderedVariableList?([1,2]),Expression(Integer))

axiom

Í :=
  (  I Ω   ) /
  (  I X   ) /
  (   U I  )

$\label{eq9}\begin{array}{@{}l} \displaystyle {{\left({{u^{1, \: 2}}\ {u_{1, \: 2}}}+{{u^{1, \: 1}}\ {u_{1, \: 1}}}\right)}\ {|_{1}^{1}}}+ \ \ \displaystyle {{\left({{u^{1, \: 2}}\ {u_{2, \: 2}}}+{{u^{1, \: 1}}\ {u_{2, \: 1}}}\right)}\ {|_{2}^{1}}}+ \ \ \displaystyle {{\left({{u^{2, \: 2}}\ {u_{1, \: 2}}}+{{u^{2, \: 1}}\ {u_{1, \: 1}}}\right)}\ {|_{1}^{2}}}+ \ \ \displaystyle {{\left({{u^{2, \: 2}}\ {u_{2, \: 2}}}+{{u^{2, \: 1}}\ {u_{2, \: 1}}}\right)}\ {|_{2}^{2}}}$

(9)

Type: LinearOperator?(OrderedVariableList?([1,2]),Expression(Integer))

axiom

Ì:=
  (   Ω I  ) /
  (   X I  ) /
  (  I U   )

$\label{eq10}\begin{array}{@{}l} \displaystyle {{\left({{u^{2, \: 1}}\ {u_{2, \: 1}}}+{{u^{1, \: 1}}\ {u_{1, \: 1}}}\right)}\ {|_{1}^{1}}}+ \ \ \displaystyle {{\left({{u^{2, \: 1}}\ {u_{2, \: 2}}}+{{u^{1, \: 1}}\ {u_{1, \: 2}}}\right)}\ {|_{2}^{1}}}+ \ \ \displaystyle {{\left({{u^{2, \: 2}}\ {u_{2, \: 1}}}+{{u^{1, \: 2}}\ {u_{1, \: 1}}}\right)}\ {|_{1}^{2}}}+ \ \ \displaystyle {{\left({{u^{2, \: 2}}\ {u_{2, \: 2}}}+{{u^{1, \: 2}}\ {u_{1, \: 2}}}\right)}\ {|_{2}^{2}}}$

(10)

Type: LinearOperator?(OrderedVariableList?([1,2]),Expression(Integer))

axiom

equate(f,g)==map((x,y)+->(x=y),ravel f, ravel g);

Type: Void

axiom

eq1:=equate(Í,I)

axiom

Compiling function equate with type (LinearOperator(
      OrderedVariableList([1,2]),Expression(Integer)),LinearOperator(
      OrderedVariableList([1,2]),Expression(Integer))) -> List(Equation
      (Expression(Integer)))

$\label{eq11}\begin{array}{@{}l} \displaystyle \left[{{{{u^{1, \: 2}}\ {u_{1, \: 2}}}+{{u^{1, \: 1}}\ {u_{1, \: 1}}}}= 1}, \:{{{{u^{1, \: 2}}\ {u_{2, \: 2}}}+{{u^{1, \: 1}}\ {u_{2, \: 1}}}}= 0}, \: \right. \ \ \displaystyle \left.{{{{u^{2, \: 2}}\ {u_{1, \: 2}}}+{{u^{2, \: 1}}\ {u_{1, \: 1}}}}= 0}, \:{{{{u^{2, \: 2}}\ {u_{2, \: 2}}}+{{u^{2, \: 1}}\ {u_{2, \: 1}}}}= 1}\right]$

(11)

Type: List(Equation(Expression(Integer)))

axiom

eq2:=equate(Ì,I)

$\label{eq12}\begin{array}{@{}l} \displaystyle \left[{{{{u^{2, \: 1}}\ {u_{2, \: 1}}}+{{u^{1, \: 1}}\ {u_{1, \: 1}}}}= 1}, \:{{{{u^{2, \: 1}}\ {u_{2, \: 2}}}+{{u^{1, \: 1}}\ {u_{1, \: 2}}}}= 0}, \: \right. \ \ \displaystyle \left.{{{{u^{2, \: 2}}\ {u_{2, \: 1}}}+{{u^{1, \: 2}}\ {u_{1, \: 1}}}}= 0}, \:{{{{u^{2, \: 2}}\ {u_{2, \: 2}}}+{{u^{1, \: 2}}\ {u_{1, \: 2}}}}= 1}\right]$

(12)

Type: List(Equation(Expression(Integer)))

axiom

snake:=solve(concat(eq1,eq2),concat Ξ(Ξ(sb('u,[i,j]), i,1..dim), j,1..dim));

Type: List(List(Equation(Expression(Integer))))

axiom

if #snake ~= 1 then error "no solution"

Type: Void

axiom

Ω:=eval(Ω,snake(1))

$\label{eq13}\begin{array}{@{}l} \displaystyle {{{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 1}}}-{{{u^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 2}}}- \ \ \displaystyle {{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{2 \ 1}}}+{{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{2 \ 2}}}$

(13)

Type: LinearOperator?(OrderedVariableList?([1,2]),Expression(Integer))

axiom

matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j), i,1..dim), j,1..dim)

$\label{eq14}\left[ \begin{array}{cc} {{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}& -{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}} \ -{{u^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}&{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}$

(14)

Type: Matrix(LinearOperator?(OrderedVariableList?([1,2]),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)

$\label{eq15}\left[ \begin{array}{cc} {u^{1, \: 1}}&{u^{2, \: 1}} \ {u^{1, \: 2}}&{u^{2, \: 2}}$

(15)

Type: Matrix(LinearOperator?(OrderedVariableList?([1,2]),Expression(Integer)))

axiom

mU:=inverse map(retract,Um)

$\label{eq16}\left[ \begin{array}{cc} {{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}& -{{u^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}} \ -{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}&{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}$

(16)

Type: Union(Matrix(Expression(Integer)),...)

axiom

Ωm:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i,1..dim), j,1..dim)

$\label{eq17}\begin{array}{@{}l} \displaystyle {{{u^{2, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 1}}}-{{{u^{2, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{1 \ 2}}}- \ \ \displaystyle {{{u^{1, \: 2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{2 \ 1}}}+{{{u^{1, \: 1}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}}\ {|_{2 \ 2}}}$

(17)

Type: LinearOperator?(OrderedVariableList?([1,2]),Expression(Integer))

axiom

-- compare
test(Ω=Ωm)

$\label{eq18} \mbox{\rm true}$

(18)

Type: Boolean

Check that the twisted snake relation holds

axiom

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

$\label{eq19} \mbox{\rm true}$

(19)

Type: Boolean

axiom

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

$\label{eq20} \mbox{\rm true}$

(20)

Type: Boolean

Dimension

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

$\scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-0.92)(0.82,0.92) \psbezier[linewidth=0.04](0.0,-0.1)(0.0,-0.9)(0.8,-0.9)(0.8,-0.1) \psbezier[linewidth=0.04](0.0,0.1)(0.0,0.9)(0.8,0.9)(0.8,0.1) \psline[linewidth=0.04cm](0.0,0.1)(0.0,-0.1) \psline[linewidth=0.04cm](0.8,0.1)(0.8,-0.1) \end{pspicture} }$

axiom

d:=
    Ω /
    U

$\label{eq21}2$

(21)

Type: LinearOperator?(OrderedVariableList?([1,2]),Expression(Integer))

This "twisted dimension " depends on !

$\scalebox{1} % Change this value to rescale the drawing. { \begin{pspicture}(0,-1.22)(0.82,1.22) \psbezier[linewidth=0.04](0.0,-0.4)(0.0,-1.2)(0.8,-1.2)(0.8,-0.4) \psbezier[linewidth=0.04](0.0,0.4)(0.0,1.2)(0.8,1.2)(0.8,0.4) \psbezier[linewidth=0.04,linestyle=dashed,dash=0.16cm 0.16cm](0.0,0.4)(0.0,-0.2)(0.8,0.2)(0.8,-0.4) \psbezier[linewidth=0.04](0.8,0.4)(0.8,-0.2)(0.0,0.2)(0.0,-0.4) \end{pspicture} }$

axiom

d':=
     Ω /
     X /
     U

$\label{eq22}{{2 \ {u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{2, \: 1}}^2}-{{u^{1, \: 2}}^2}}\over{{{u^{1, \: 1}}\ {u^{2, \: 2}}}-{{u^{1, \: 2}}\ {u^{2, \: 1}}}}$

(22)

Type: LinearOperator?(OrderedVariableList?([1,2]),Expression(Integer))